Plus Two

Students can Download Chapter 4 Determinants Notes, Plus Two Maths Notes helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus Two Maths Notes Chapter 4 Determinants

Introduction
Determinants have wide applications in engineering, Science, Economics, Social Science, etc. In this chapter we study about the various properties of determinants, minors, cofactors, applications in finding the area of triangle, adjoint, and inverse of a square matrix, and consistency and in consistency of linear equations.

A. Basic Concepts
I. Determinant
Determinant is a real number associated with a square matrix. The determinant of matrix A is denoted by |A|. The value of a determinant is obtained by the sum of products of elements of a row (column) with corresponding cofactors.

• |AB| = |A||B|
• |Aⁿ| = |A|ⁿ

Properties:
(i) The value of a determinant remains the same if its rows and columns are interchanged.
ie; |A^T| = |A|.

(ii) If any two rows (or columns) of a determinant are interchanged, then sign of determinant changes.

(iii) If any two rows (or columns) of a determinant is identical, then value of determinant is zero.

(iv) If each element of a row (or columns) of a determinant is multiplied by a constant k, then its value gets multiplied by k.

(a) If A is a square matrix of order n, then
|KA| = kⁿ|A|

(v) If any two rows (or columns) of a determinant is proportional, then value of determinant is
zero.

(vi) If some or all elements of a row or column of a determinant are expressed as sum of two (or more) terms, then the determinant can be expressed as sum of two (or more) determinants.

(vii) If, to each element of any row or column of a determinant, the equimultiples of corresponding elements of other row (or column) are added, then value of determinant remains the same, ie; the value of determinant remains same if we apply the operation R_i → R_i + kR_j or C_i → C_i + kC_j.

(viii) If the elements of a row or column are multiplied with cofactors of any other row or column, then their sum is zero.
ie; for example; a₁₁C₂₁ + a₁₂C₂₂ + a₁₃C₂₃ = 0.

If the elements of a row or column are multiplied with cofactors of the corresponding row or column, then their sum is |A|.
ie; for example; a₁₁C₁₁ + a₁₂C₁₂ + a₁₃C₁₃ = |A|.

1. Minor of an element:
The minor of an element a_ij is the determinant obtained by deleting the ith row and jth column, usually denoted by M_ij.

2. Cofactor of an element:
A signed Minor is called cofactor, ie; C_ij = (-1 )^{i + j} M_ij. The matrix obtained by replacing all elements by its cofactor is called cofactor matrix.

3. Adjoint Matrix:
The transpose of a cofactor matrix is. called Adjoint Matrix, usually denoted by adj(A)

Properties:

• A × adj(A) = adj(A) × A = I|A|
• If A is a square matrix of order n, then |adj(A)| = |A|^{n – 1}
• adj(AB) = adj(A)adj(B)

II. Inverse of a Matrix
A square matrix A is invertible if |A| ≠ 0 and A inverse is denoted by A^-1 ie; A^-1 = \(\frac{a d j(A)}{|A|}\)
Properties:

• (A^-1)^-1 = A
• (AB)^-1 = B^-1A^-1
• (A^T)^-1 = (A^-1)^T
• If A is a square matrix of order n, then adj(adj(A)) = |A|^n-2 × A.

III. Application of Determinants
1. Area of a triangle whose vertices are (x₁, y₁), (x₂, y₂), (x₃, y₃) is

(i) If Area = 0 then the points are collinear.

2. Solving of system of linear equations using
matrix method:
Consider the system of linear equations
a₁x + b₁y + C₁z = d₁
a₂x + b₂y + c₂z = d₂
a₃x + b₃y + c₃z = d₃
Convert the linear equation into matrix form AX = B, where

Then the solution is given by X = A^-1B

• If |A| ≠ 0 the system is consistent and has unique solution.
• If |A| = 0 and adj(A) × B ≠ 0,the system is inconsistent and has no solution.
• If |A| = Oandadj(A) × B = 0 ,the system may be consistent and has infinitely many solutions.

In order to find these infinitely many solutions, replace one of the variable by k (say z = k) and solve any two of the given equations for x and y in terms of k

Students can Download Chapter 3 Matrices Notes, Plus Two Maths Notes helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus Two Maths Notes Chapter 3 Matrices

Introduction
The term ‘matrix’ was first used in 1850 by the famous English Mathematician James Joseph Sylvester. In 1858 Arther Cayley began the Systematic development of the theory of matrices. Matrix was first used for the study of linear equations and linear transformations. Now it is largely used in disciplines like statistics, physics, chemistry, psychology, etc.

A. Basic Concepts
I. Matrix
A matrix is an ordered rectangular array of numbers or functions. The numbers or functions are called the elements or the entries of the matrix.
1. Order of a Matrix:
A matrix having m rows and n column is called a matrix of order m × n, generally denoted by

Where 1 ≤ i ≤ m, 1 ≤ j ≤ n  i, j ∈ N.

II. Types of Matrix

• Column Matrix: A matrix having only one column is called Column Matrix.
• Row Matrix: A matrix having only one row is called Row Matrix.
• Square Matrix: A matrix having equal number of row and column is called Square Matrix.
• Diagonal Matrix: A Square matrix having all its non-diagonal entries zero is called Diagonal Matrix.
• Scalar Matrix: A Square matrix having all its non-diagonal entries zero and equal diagonal elements is called Scalar Matrix.
• Identity Matrix: A Square matrix having all its non-diagonal entries zero and diagonal elements unity is called an Identity Matrix.
• Zero Matrix: A matrix having all elements zero is called Zero Matrix.

III. Operations on matrices
1. Equality of Matrices:
Two matrices are equal if they are of same order and corresponding elements are equal.

2. Addition:
Addition is possible only if the two matrices are of same order and the operation is done by adding the corresponding elements in each Matrix. The addition of Matrix A and B is denoted by A + B.
Properties:

• Matrix addition is Commutative.
• Matrix addition is Associative.
• Zero Matrix is the additive identity.
• – A is the additive inverse of matrix A.

3. Scalar Multiplication:
The multiplication of a matrix by a scalar number k is done by multiplying each entries of A by k and matrix thus obtained is kA.

4. Difference:
Difference is possible only if the two matrices are of same order and the operation is done by subtracting the corresponding elements in each Matrix. The difference of Matrix A and B is denoted by A – B.

5. Multiplication:
Multiplication is possible only if the number of column of first matrix is equal to the number of rows of the second. The operation is done by multiplying the element in the first row of the first matrix with the corresponding elements in the first column in the second matrix.

This is continued till the rows in the first matrix finish. The multiplication of Matrix A and B is denoted by A × B or AB.
Properties:

• Matrix multiplication is Non-Commutative.
• Matrix multiplication is Associative, le; A(BC) = (AB)C
• Matrix multiplication is Distributive over addition, ie; A(B + C) = AB + AC
• Identity Matrix is the multiplicative identity, le; AI = IA.

IV. Transpose of a Matrix
The transpose of a matrix A is obtained by interchanging the row and column of A and is denoted by A^T.
Properties:

• [A^T]^T = A
• [kA]^T = kA^T
• [A + B]^T = A^T + B^T
• [AB]^T = B^T A^T

1. Symmetric Matrix:
A square matrix is said to be symmetric if [A]^T = A.
Properties:
In a symmetric matrix the corresponding elements on both sides of the main diagonal will be same.

2. Skew Symmetric Matrix:
A square matrix is said to be symmetric if [A]^T = -A.
Properties:

• In a Skew Symmetric matrix the corresponding elements on both sides of the main diagonal differ only in sign.
• For any square matrix A with real entries, A + A^T is Symmetric matrix, and A – A^T is Skew Symmetric matrix.
• Any square matrix can be expressed as the sum of a Symmetric and Skew symmetric matrix.
  ie; A = \(\frac{1}{2}\)(A + A^T) + \(\frac{1}{2}\)(A – A^T)
• If A and B are Symmetric matrices of the same order, AB is Symmetric if and only if AB = BA.
• If A and B are Symmetric matrices of the same order, (AB + BA) is Symmetric and (AB – BA) is Skew Symmetric.

V. Elementary Operation on Matrix
There are 6 operations on matrix, 3 for row and 3 for column.

1. The interchange of any two rows or two columns, symbolically denoted as R_i ↔ R_j or c_i ↔ c_j.
2. The multiplication of the elements of any row or column by a non-zero number, symbolically
   denoted as R_i ↔ kR_j or C_i ↔ kC_j.
3. The addition to the elements of any row or column, the corresponding elements of any other row or column multiplied by any non-zero number, symbolically denoted as R_i ↔ R_i + kR_j or C_i ↔ C_i + kC_j.

VI. Invertible Matrices
A square matrix B is said to be the inverse of a matrix A if AB = I = BA, then B is generally denoted
as A^-1.

1. Inverse of a square matrix, if it exists, is unique.
2. If A and B are invertible matrices of the same order, then (AB)^-1 = B^-1A^-1

Students can Download Chapter 6 Data Base Management System for Accounting Notes, Plus Two Accountancy Notes helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus Two Accountancy Notes Chapter 6 Data Base Management System for Accounting

Database/ Data source – Introduction
A database is a collection of related data. It is organised in such a way that its contents can easily be accessed, managed and updated. In LibreOffice, database is also called data source. Database consists of interrelated data tables that are structured in a manner that ensures-data consistency and integrity. LibreOffice base, MS Access, Oracle, SQL server, etc. are the commonly used softwares for data base management.

Database Management System (DBMS)
DBMS is a collection of programs. It enables the users to create and maintain a data base. It is a general purpose software system that facilitates the process of defining, constructing and manipulating database for various applications.

Advantage of database/data source

1. All of the information is together
2. The information is portable
3. Information can be accessed at any time
4. Many users can access the same database at the same time.
5. Reduced data entry, storage and retrieval cost.

Disadvantages of database/Data source

1. Designing of database is a complex and time consuming process
2. Initial training is required for all the users
3. Installation cost is high

Basic concepts of LibreOffice Base

1. Entities: Anything in the real world is called entities. It may be person, place or things.
   Eg: Employee is an entity, Orange is an entry
2. Attributes: These.define the characteristics of an entity.
   Eg: Name, Age, Caste, Salary etc.
3. Identifiers: The unique attribute of an entity is called identifier. This is also called primary key.
   Eg: Admission number of a student, Aadhar Number of a person etc.
4. Relationships: These are the logical links between two entities or tables.

Components /Elements of LibreOffice Base

1. Fields: Individual pieces of data in a database are called fields.
2. Table: rows and columns to present fields in a database is called table. When creating a table, the characteristics of each field to be defined.
3. Forms: Forms are used to enter or modify data (fields) in to tables. Forms allow the user to display the data in a Table or Query.
4. Query: Query is a question. Queries are used to view, change and analyse data in different ways. It creates a new table from the existing tables based upon the question/ request asked to the data base.
5. Reports: It is used to create and present information based on queries in a easily readable format.

Planning (or designing) a database/ Data source
The first step in creating a database is to list down the various fields which are necessary for creating a database. The listed fields are used to create tables of the database. While entering fields into Tables, a primary key or an identifier is to be set for each table.

The primary key field cannot be left blank. The relationships of entities or tables can be created with the support of primary key. The relationships may be

• One -to-One
• One -to-Many
• Many-to-Many

The database created on the basis of relationships between different data tables is called relational database. The database design can be used to describe the structure of different parts of the overall data base. Avoiding the duplication of attributes/ fields is key criteria of database design.

Creating a new database
To Create a new database, select File → New → Database from the menubar, or click the arrow next to the New icon on the standard tool bar and select Database from the drop-down menu. Both methods open the Database wizard. On the first page of the Database wizard, Select create → a new databases → click Next.
The second page has two questions

1. Yes, register the data base for me
2. Open the database for editing.

Choose any one from the above and click Finish. In LibreOffice Base, the entire database is encompassed in a file with extension .odb. This file format is actually a container of all elements of the database, including Fields, Tables, Forms, Queries and Reports.

Creating database Tables
In DBMS, data is organised in Tables. A Table is a collection of data about a specific topic. Tables organise data into columns and rows. Each table is given a name. This is used to refer the table. The name depicts the content of the table. A database must have at least one table and may have several.

To work with Table, clicks the Tables icon in the Database list or Press Alt+A. The three tasks that we can perform on a table are in the Task list given below.

• Using the wizard to create a Table
• Creating a table by copying an existing table
• Creating tables in Design view.

1. Using the wizard to create a new Table:
Step 1: Open LiberOffice Base
Application → Office → LibreOfficeBase

Step 2: From the Data base wizard Screen, Create a database file or open an existing database file.
Select Database → Select Create a new database option and click on Finish button → then we get save dialogue box.

Step 3: Type appropriate file name. The default extension ‘.odb’ will be automatically added.

Step 4: Select location and click on Save

2. Using the wizard to open an Existing database:
Step 1: Open LibreOffice Base. From Database
wizard → Select open an existing database file option and click on Open button.

Step 2: Choose the file from the destination and click open.

3. Creating a table by Copying an existing table:
Step 1: Open LibreOffice Base
Application → Office → LibreOffice Base

Step 2: Click on the Tables icon in the database pane to see the existing tables
Select Database → Click on the Table icon → Right click on the Table from the Existing Tables.

Step 3: Choose Copy form the pop-up menu

Step4: Move the mouse pointer below the table, → right click → Select Paste. The copy Table dialogue opens.

Step 5: Change table name and click Next

Step 6:

Step 7: Click Create the new table is created

Step 8: Click the Save button at the top of the main database window.

4. Creating Tables in Design View:
Step 1: Click Tables from Data base Pane Database Pane → click Tables.

Step 2: Click Create Tables in design view in Tasks area. The design view of the new table will appear in the work area of the window.

Step 3: In design view, we can see three columns like Field Name, Field Type and Description. Create required field for the Table. Click Field Name cell → center Field names from top to bottom.

Step 4: Right click on the Field Name required to set as unique identifier for the tables, Select Primary Key option from the pop-up menu.

Step 5: In the Field Type, select appropriate field type from the combo box.

Step 6: In Description Field, we canenter appropriate description for each attributes.

Step 7: Save the table by providing table name.

5. Defining Relationships between Tables:
Relationships are used for connecting tables in database to get the advantage of data redundancy. Having completed the designs of all data tables, the next step is to establish relationships between different tables.
Step1: Click on the Tools menu and then Relationships

Step 2: Relation Design window opens and in the work area, a Add Tables dialogue box will appear.

Step 3: Select a Table and clik Add bottom to add it in the relationships.

Step 4: Add two tables in this manner after that click the close button.

Step 5: Create a relationship between two tables, Position the mouse pointer over desired field in table object, hold down the left mouse button, drag the pointer right to targeted field of the next table object and then release the mouse button.
This can also be done Insert → New Relation menu.

Creation of Forms in LibreOffice Base
Forms are used to input data into the database. In the language of database, a Form is a front end for data entry and editing. A Form is a window or screen that contain numerous fields or spaces to enter data. Each field holds a field label so that any user gets an idea of its contents.
The following two methods are used to create forms in LibreOffice Base

• Create Form in Design View
• Use wizard to create Form. The easy way to create Forms is use wizard to create Form.

1. Use Wizard to Create Form:
Step 1: Select Forms options from Database Pane

Step 2: Click on Use Wizard to Create Form Then. Form wizard window will appear.

Step3: Under the Table or queries, select Tables. The fields in the selected tables are listed in Available Fields list. Select the required field on by one and click on

Step 4 – After selecting the required field proceed by clicking Next.

Step 5 – Add sub Form fields. This step is similar to step 3

Step 6 – Get joined Fields.
This step is for tables and queries for which no relationships has been defined. The wizard skip this step. Because we have already defined the relationship.

Step 7: Arrange controls: A control in a Form consist of two parts: label and field. This step in creating the Form determines where a controls label and field are placed relative each other. Four choices are available.

Step 8: Set data entry: It is better to accept the default settings. Click Next.

Step 9: Apply styles: The background colour, field boarder etc. can be selected from this options.

Step 10: Set the name of the form: we can give the name of the Form we are creating. The name must be unique and must have a relation with the data to be stored → click Finish. The Form opens in Edit mode.

Entering data in a form
The text box can be used to add data in the Form. Click on the text Box Icon, and click on the work area. The cursor will be positioned on the Top left of the work area. Then the matter is to be added. The text entered can be formatted. Images can also be inserted in the Form by selecting Image Insert Icon.

Creation of Query in LibreOffice Base
Queries are used to get specific information from the database. Queries are also used to manipulate the database content. Structured Query Language (SQL) is the most widely-used query language. LibreOffice Base also uses SQL command for querying its database. The query operations can be done in two different ways.

• Using the wizard to create a query
• Using the design view

1. Using the wizard to create a query:
Step 1: Selection of Fields
The first step in Query wizard is field selection. All the tables included in the data base can be seen in the Table list select the appropriate Table. All the fields of the selected table can be seen in the Available Fields list.

The user can select the fields needed from the list using the tools arranged right to Available fields list. The selected fields can be seen in Fields in the Query list. The order of the selected fields can change using the tools (∧ and ∨) right to Fields in the query list. Then, press Next button or Finish button.

Step 2: Select the sorting order: In this step, the field name to sort the query result can be selected. (Skip this step, if no sorting is needed).

Step 3: Select the search conditions: This step specifies the search conditions to filter the query. (Skip the step, if no filtering is needed).

Step 4: Details of summary: This page specifies whetherto display all records of the query, or only the results of aggregate functions. This page is only displayed when there are numerical fields in the query that allow the use of aggregate functions.

Step 5: Grouping conditions: Specfies wheter to group the query. The data source must support the SQL statement “order by clause” to enable this page of the wizard.

Step 6: Assign aliases if desired: – Thisjsage helps to assign aliases to field name. Aliases are optional, and can provide more user- friendly names, which are displayed in place of field names.

Step 7: Overview: This wizard page gives an overview of the query made. It helps to enter a name of the query, and specifies whether to display or to modify the query afterthe wizard is finished.

Step 8: Press Finish button after the completion of Query wizard entry. The Query will be saved. The user can run this query at any time.

Step 9: Run the saved Query: Select Queries option from the left panel of the LibreOffice Base window. The saved query can be seen in the right side. Double click on the query name to run the query.

2. To create query in design view:

• Step 1: Use the option Create Query in Design view from Base window to create query in design view.
• Step 2: Use Add Table or Query dialogue to include table(s) to query design.
• Step 3: Include the fields and formula in the top row and give aliases in the second now, if needed.
• Step 4: Press Run Query button to execute the query.

Creation or Reports in LibreOffice Base
Information from a database can be generated through the Reports in LibreOffice Base. The reports can be printed and formatted as perthe requirements of user. The reports can be edited, printed and exported to PDF format.
The reports can be created by the following two ways.

• Create Report in Design view
• Use wizard to ere ate Report

1. Use wizard to create Report in LibreOffice Base:

• Step 1: Click the icon Reports in Database pane
• Step 2: Click on Use wizard to create Report option in Task area.
• Step 3: Select table or query from the drop down option for which reports need to be created.
• Step 4: Select the required fields
• Step 5: Enter title for the report in the field Title of the report and click on Finish button.
• Step 6: The report generated is in the read only mode. It can be edited by clicking on Edit Document option.

In short, When the Report wizard is opened, the Report Builder is also opened. As we make our selections in the wizard these appear in layout in the Report Builder. After finishing the selections, save the report, name it and then close it.

Accessing other Data Sources
LibreOffice Base allows other data sources to be accessed and linked into LibreOffice documents. To acess a data source which is not a .odb file.

• Step 1: File → New → Database opens the Data base wizard window.
• Step 2: Select Connect to exisiting data base. Click the arrow next to Data base type field and select the Database type from drop down list. Click Next.
• Step 3: Click Browse and select the database click Next.
• Step 4: Accept the default settings: Register the data base for me, and open the database for editing. Click Finish. Name and save the database in the location of our choice.

1. Accessing a spreadsheet as a data source:

• Step 1: Choose File → New → Data base.
• Step 2: Select Connect to an existing database. Select Spread sheet as a database type.
• Step 3: Click Browse to locate the spreadsheet we want to access. If spreadsheet is password protected, Check the password required box, click Next.
• Step 4: If the spreadsheet requires a users name, enter it. If a password is also required, check its box. Click Next → save the file.

2. Registering *.odb databases:
Databases created by LibreOffice base are in the *.odb format. Other programs can also produce database in this format. Registering a*.odb data base is simple.

• Step 1: Select Tools → Options → Libre Office Base → Database
• Step 2: Under Registered data Bases, Click New.
• Step 3: Browse to where the database is located.
• Step 4: Make sure the registered name is correct .Odb
• Step 5: Click OK.

Using data sources in LibreOffice
Any data source registered in spreadsheet or text document can use in other LiberOffice components including writer and calc.

1. Viewing data sources:
Open a document in writer or calc. To view the data sources available, Press F4 or select View → Data sources from the pull down menu. This brings up a list of registered databases. To view each data base, click on the arrow to the left of the database’s name.

2. Editing Data sources:
Some data sources can be edited in the Data view window. A record can be edited, added or deleted. Editing the data requires only a click in the cell whose data should be changed. To delete the record, right click on the gray box to the left of row to highlight the entire row, and select Delete row to . remove the selected row.

3. Launching Base to work on data sources:
We can launch LibreOffice Base at any time from the Data source window. Just right click on a database, or its tables or Queries icons and select Edit Database File. In Base, we can edit, add, and delete tables, queries, forms and reports.

4. Using Datasources in writer and Calc.:
Data can be placed into writer and Calc documents from the tables in the data source window. In writer, values from individual fields can be inserted. Or a complete table can be created in the writer document. One common way to use a data source is to perform a mail merge.

Students can Download Chapter 7 The p Block Elements Notes, Plus Two Chemistry Notes helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus Two Chemistry Notes Chapter 7 The p Block Elements

The p-block – Elements of group 13 to 18 of the periodic table. The outer electronic configuration of a p-block element is ns²np^1-6.

Anomalous behaviour of the first element of a group: This is due to

1. Small in size
2. High electronegativity,
3. High ionisation enthalpy and
4. Non-availability of d – orbitals.

Diagonal Relationship:
In some cases, the first element of a group resembles diagonally with the element of the next group and of the next period.

Group 15 – Elements – Nitrogen Family:
Elements are – N, P, As, Sb, Bi
N₂ comprises 78% by volume of the atmosphere. N and P are essential constituents of animals and plants. N – Present in proteins, P – Present in bones.

Characteristics:
1. Atomic radii increases with increase in Atomic Number.

2. Ionisation Enthalpy decreases down the group due to gradual increase in atomic size. Because of the extra stable half-filled p orbitals electronic configuration and smaller size, the ionisation enthalpy is less than that of group 14 elements in the corresponding periods.

3. Electronegativity decreases down the group.

Physical Properties:
All are polyatomic, metallic character increases from N to Bi, density increases from N to Bi, M.P. and B.P. increases down the group, except N all other elements show allotropy.

Chemical Properties:
Oxidisation states and trends in chemical reactivity:
The common oxidation states of 15 group elements are (-3), (+3) and (+5). The stability of +5 oxidisation state decreases and that of +3 state increases down the group due to inert pair effect. Nitrogen exhibits +1, +2, +4 oxidation states also when it react with O₂.

The maximum covalence of N restricted to 4 since only 4 orbitals (one S and three P) are available for bonding.

Anomalous Properties of Nitrogen:
It is due to its small size, high electronegativity, high ionisation enthalpy and non-availability of ‘d’ orbitals. Nitrogen has unique ability to form pπ – pπ multiple bond. It cannot form dπ – pπ bond. P and A scan form dπ – dπ bond.

(i) Reactivity towards hydrogen:
EH₃ hydrides, the central atom is sp³ hybridised, molecules assume trigonal pyramidal geometry with a lone pair on the central atom. Stability-decreases from NH₃ to BiH₃.

This is because, down the group the E-H bond dissociation enthalpy decreases due to increase in size of the central atom. Consequently, reducing character increases from NH₃ to BiH₃. The basicity decreases in the order NH₃ > PH₃ > AsH₃ > SbH₃> BiH₃.

As the electro negativity of the central atom decreases on moving down the group, the bond pair-bond pair repulsion decreases. Hence the bond angle decreases in the order NH₃ > PH₃ > AsH₃.

(ii) Reactivity towards oxygen:
They form E₂O₃ & E₂O₅ type oxides. The oxide in the higher oxidisation state of the element is more acidic than that in lower oxidation state.

(iii) Reactivity towards halogens:
They form EX₃ and EX₅ type halides. Nitrogen does not form pentahalide due to non-availability of d-orbital.

(iv) Reactivity towards Metals:
They react with some metals exhibiting – 3 oxidation state, e.g. Calcium nitrate (Ca₃N₂), Calcium phosphide (Ca₃P₂), Sodium arsenide (Na₃As₂).

Dinitrogen (N₂):
It is produced commercially by the liquefaction and fractional distillation of air.
In laboratory, N₂ is prepared by

NH₄Cl(aq) + NaNO₂(aq) → N₂(g) + 2 H₂O(l)+ NaCl(aq)

Properties:
Colourless, odourless, non-toxic gas; inert at room temperature because of high bond enthalpy of N ≡ N.

Uses:
Manufacture of NH₃, liquid N₂ is used as refrigerant to preserve biological materials, food items and in cryosurgery.

Ammonia:
Laboratory preparation:
2NH₄Cl + Ca(OH)₂ → 2NH₃ + 2H₂O + CaCl₂
(NH₄)₂SO₄ + 2NaOH → 2NH₃ + 2H₂O + Na₂SO₄

Industrial (large scale) preparation by Haber’s process:
N₂(g) + 3H(g) ⇌ NH₃(g); Δ_fH^Ⓗ = -46.1 kJ/mol^-1 Catalyst used earlier- spongy iron with molybdenum promoter. Catalyst used now – iron oxide with small amounts of K₂O and Al₂O₃.

High pressure and low temperature will favour the formation of NH₃ as the forward reaction is exothermic and is accompanied by decrease in number of moles (Le Chatelier’s principle). Hence, a pressure of 200 × 10⁵ Pa (about 200 atm) and a temperature of ~ 700 K are employed to increase the yield of NH₃.

Properties:
Colourless, pungent smelling gas, trigonal pyramidal geometry, highly soluble in water.
NH₃(g) + H₂O(l) \(\rightleftharpoons\) NH⁺₄(aq) + OH^–(aq)
Lewis base – due to the presence of a lone pair of electrons on N. It can form complex compounds with metal ions. This finds application in the detection of
Cu²⁺ and Ag⁺.

Uses:
To produce various nitrogeneous fertilizers, manufacture of inorganic nitrogen compounds (e.g. HNO₃), liquid NH₃ is used as a refrigerant.

Oxides of Nitrogen:

1. Dinitrogen oxide (N₂O) or laughing gas – Oxdation state (+1) – Colourless gas, neutral.
2. Nitrogen monoxide(NO) – Oxdation state (+2) colourless gas, neutral.
3. Dinitrogen Trioxide(N₂O₃) – Oxdation state (+3), blue solid, acidic in nature.
4. Nitrogen dioxide(NO₂) – Oxdation state (+4) brown gas, acidic. It contains odd number of valence electrons. On dimerisation, it is converted to stable N₂O₄ molecule with even number of electrons.
5. Dinitrogen tetroxide(N₂O₄) – Dimer of NO₂ – Oxdation state (+4), colourless solid/liquid, acidic.
6. Dinitrogen pentoxide (N₂O₅) – Oxdation state (+5), colourless solid, acidic.

Nitric Acid:
It is the most important oxoacid of N.
Laboratory preparation:
KNO₃/NaNO₃ + H₂SO₄(conc.) → KHSO₄/NaHSO₄ + HNO₃
Industrial preparation – Ostwald’s process:
(1) NH₃ oxidised to NO by air.

(2) NO is converted to NO₂
2NO(g) + O₂(g) ⇄ 2NO₂(g)

(3) NO₂ dissolved in water to give HNO₃
3NO₂(g) + H₂O(l) → 2HNO₃ (aq) + NO(g)

Properties:
Colourless liquid, strong acid in aqueous solution. Concentrated HNO₃ is a strong oxidising agent and attacks most metals except noble metals like Au and Pt. The products of oxidation depend upon the concentration of the acid, temperature and the nature of the material undergoing oxidation, e.g.

• 3Cu + 8HNO₃(dilute) → 3Cu(NO₃)₂ + 2NO + 4H₂O
• Cu + 4HNO₃(conc.) → CuCu(NO₃)₂ + 2NO₂ + 2H₂O
• 4Zn + 10HNO₃(dilute) → 4Zn(NO₃)₂ + 5H₂O + N₂O
• Zn + 4HNO₃(conc.) → Zn(NO₃)₂ + 2H₂O + 2N₂O

Some metals (e.g., Cr, Al) do not dissolve in concentrated nitric acid because of the formation of a passive film of oxide on the surface.

Structure:
In the gaseous state, HNO₃ exists as a planar molecule.

Uses:
Manufacture ammonium nitrate (fertilizer), preparation of explosives, preparation of nitroglycerine, pickling of stainless steel, etching of metals, oxidiser in rocket fuels.

Phosphorus:
Allotropic forms – White P, red P and black P

White Phosphorus:
Transient white waxy solid, poisonous, insoluble in water, soluble in CS₂, glows in dark (chemiluminescence), kept underwater, less stable and therefore more reactive than other solid phases under normal conditions because of angular strain in discrete tetrahedral P₄ molecules (angle 60°), readily catches fire in air and gives dense while fumes of P₄O₁₀.
P₄ + 5O₂ → P₄O₁₀

Red Phosphorus:
Obtained by heating white P at 573 K in an inert atm for several days, possesses iron grey lustre, odourless, non-poisonous, less reactive than white P, does not glow in dark, polymeric consisting of chains of P₄ tetrahedra.

Black Phosphorus:
Obtained when red P is heated under high pressure, two forms α – black phosphorus (formed when red P is heated in a sealed tube at 803 K) and β – black phosphorus (prepared by heating white P at 473 K under high pressure).

Phosphine (PH₃):
Prepared by the reaction of calcium phosphide with water or dilute HCl.
Ca₃P₂ + 6H₂O → 3Ca(OH)₂ + 2PH₃
Ca₃P₂ + 6HCl → 3CaCl₂ + 2PH₃

Laboratory preparation:
By heating white P with con centrated NaOH solution in an inert atmosphere of CO₂.
P₄ + 3NaOH + 3H₂O → PH₃ + 3NaH₂PO₂

Properties:
Colourless gas with a rotten fishy smell, highly poisonous, weakly basic, the structure is similar to NH₃ and gives phosphonium compounds with
acids. PH₃ + HBr → PH₄Br
Uses: in Holme’s signals, in smoke screens.

Phosphorus Halides:
It forms two types of halids PX₃ and PX₅ (X = F, Cl, Br)

Phosphorus Trichloride (PCl₃):
Obtained by passing dry Cl₂ over heated white P.
P₄ + 6Cl₂ → 4PCl₃

Or, by the action of thionyl chloride on white P,
P₄ + 8SOCl₂ → 4PCl₃ + 4SO₂ + 2S₂Cl₂

Properties
Colourless oily liquid, hydrolyses in the presence of moisture giving fumes of HCl.
P₄ + 3H₂O → H₃PO₃ + 3HCl
It has pyrimidal shape and P is sp3 hybridised.

Phosphorus Pentachloride (PCl₅):
Preparation:
White P₄ + 10Cl_2(dry) → 4PCl₅

Properties:
yellowish white powder. In moist air it hydrolysed giving POCl₃ and finally gets converted to phosphoric acid (H₃PO₄)
PCl₅ + H₂O → POCl₃ + 2HCl
POCl₃ + 3 H₂O → H₃PO₄ + 3 HCl
In gaseous and liquid phases, the shape of the molecule is trigonal bipyramidal. There are two types of P-Cl bonds, equatorial bond and axial bond. Axial bonds are longer than equitorial bonds due to more repulsion. In solid state it exits as ionic solid, [PCl₄]⁺[PCl₆]^–.

Oxoacids of Phosphorus:

• Hypophosphorous/Phosphinic acid(H₃PO₂) – Monobasic
• Orthophosphorous/Phosphonic acid(H₃PO₃) – Dibasic
• Pyrophosphorous acid(H₄P₂O₅) – Dibasic
• Hypophosphoric acid(H₄P₂O₆) – Tetrabasic
• Orthophosphoric acid(H₃PO₄) – Tribasic
• Pyrophosphoric acid(H₄P₂O₇) – Tetrabasic
• Metaphosphoric acid(HPO₃)_n – Tribasic

The p-H bonds are not ionisable and have no role in basicity. Only those H atoms in P-OH form are ionisable and cause basicity.

These acids in +3 oxidation state of P tend to disproportionate to higher and lower oxidation states, e.g. Orthophosphorous acid on heating disproportionates to give orthophosphoric acid (P in +5 state) and phosphine

The acids with P-H bond .have strong reducing property, e.g. H₃PO₂. It reduces AgNO₃ to Ag.

Structure of Oxoacids:

Group 16 Elements (Chalcogens):
O, S, Se, Te and Po.

1. Occurrence:
O₂ – Most abundant element on earth crust (46.6%), dry air contains 21% by volume. S – Present as sulphates, sulphides (e.g. CaSO₄, PbS, ZnS). Se &Te-in metal selenides and tellurides, Po-radio active, formed by the decay of thorium and uranium minerals.

2. 6 General electronic configuration-ns²np⁴. In group:
Atomic and ionic radii increases, ionisation enthalpy, electron gain enthalpy and electronegativity decreases – O has the highest electronegativity next to F.

3. Physical Properties:
O is a diatomic gas, non metal. S-solid, non-metal. Se and Te are metalloids. Po-radioactive metal.

4. Chemical Properties:
Oxidation states and trends in chemical activity – exhibits variable oxidation states, stability of -2 oxidation state decreases down the group. O-shows +2 in OF₂, -1 in peroxides and – 2 in other compounds. Other elements show +2, +4, +6 states.

5. Anomolous Behaviour of Oxygen:
It is due to small size high electronegativity, non availability of d-orbital and high polarising power.

(i) Reactivity with Hydrogen:
group 16 elements form H₂E type hydrids (E = O, S, Se, Te, Po). Their acidic character increases from H₂O to H₂Te due to decrease in H-E bond dissociation enthalpy. All hydride except H₂O posses reducing property. Reducing nature increases from H₂S to H₂Te.

Due to small size and high electro naegativity of oxygen, H₂O molecules are highly associated through hydrogen bonding resulting in its liquid state and high boiling point.

While, due to large size and low electronegativity of S association through hydrogen bonding is hot possible in H₂S. Hence it exists as a gas and has low boiling point than H₂O.

(ii) Reactivity with Oxygen:
They form EO₂ & EO₃ type oxides. Ozone, O₃ and SO₂ are gases. Both type of oxides are acidic in nature.

(iii) Reactivity Towards the Halogens:
They form EX₂, EX₄ and EX₆ type halides. The stability of halides decrease in the order F^– > Cl^– > Br > l^–

Dioxygen (O₂):
Preparation:
(i) Heating KClO₃, KMnO₄, KNO₃ etc.

(ii) Thermal decomposition of metal oxides.
2Ag₂O → 4Ag + O₂
2PbO₂ H → 2PbO + O₂

(iii) Decomposition of H202
2H₂O₂ → 2H₂O + O₂.

Large scale preparation:
Electrolysis of water, O₂ liberated at anode.

Properties:
Colourless, odourless gas; paramagnetic, directly reacts with nearly all metals except Au and Pt.

Simple Oxides:
Binary compound of O with another element, e.g. MgO, Al₂O₃. Basic oxide – oxide that combine with water give a base. e.g. MgO. Acidic oxide – oxide that combine with water to give acid, e.g. SO₂, CO₂.
SO₂ + H₂O → H₂SO₃
In general, metallic oxides are basic and non-metallic oxides are acidic.

Ozone (O₃):
Allotropic form of O, too reactive, prepared by passing a slow dry stream of O₂ through a silent electrical discharge.
3O₂ → 2O₃
ΔH = +142 kJ mol^-1

Properties:
Pure O₃ is a pale blue gas, dark blue liquid and violet-black solid, thermodynamically unstable compared to O₂.

Oxidising property:
Due to the ease with which it liberates atoms of nascent oxygen 03 acts as a powerful oxidising agent.
O₃ → O₂ + [O]
e.g. It oxidises lead sulphide to lead sulphate.
PbS(s) + 4O₃(g) → PbSO₄(s) + 4O₂(g)

Estimation of O₃:
When O₃ reacts with excess of Kl solution buffered with a borate buffer (pH 9.2), l₂ is liberated which can be titrated against a standard solution of sodium thiosulphate.
2l^–(aq) + H₂O(l) + O₃(g) → 2OH^–(aq) + l₂(s) + O₂(g)

Uses:
As a germicide, disinfectant and for sterilising water; for bleaching oils, ivory, starch etc. as oxidising agent in the manufacture of KMnO₄.

Sulphur-Allotropic Forms:
Rhombic Sulphur (α – Sulphur):
yellow, insoluble in water, dissolve to some extent in benzene and alcohol, readily soluble in CS₂.

Monoclinic Sulphur (β – Sulphur):
Soluble in CS₂, needle shaped crystals.

Structure:
They exists as S₈ molecules, the S₈ ring is puckered and has a crown shape. The cylco-S₆ ring adopts a chair form.

Sulphur Dioxide (SO₂):
Preparation:
1. Burning of S in air
S(s) + O₂(g) → SO₂(g)

2. Treating sulphite with diluted H₂SO₄.
SO₃^2- + 2H⁺ → H₂O + SO₂

Properties:
Colourless gas with pungent smell, highly soluble water, when passed through water forms sulphurous acid.
SO₂(g) + H₂O(l) → H₂SO₃(aq)

React with NaOH:
2NaOH + SO₂ → Na₂SO₃ + H₂O

Other reactions:
3SO₂ + Cl₂ → SO₂Cl₂

Users:
In pertroleum refining and sugar industry, in bleaching wool and silk, in manufacturing H₂SO₄.

Oxoacids of Sulphur:
Sulphur forms a number of oxoacids such as H₂SO₃, H₂SO₄, H₂S₂O₃, H₂S₂O₇

Manufacture of Sulphuric Acid:
Sulphuricacid is known as king of chemicals. It is manufactured by Contact Process.

Steps Involved:
(i) Burning of S or Sulphide ores in air to form SO₂

(ii) Conversion of SO₂ to SO₃ by the reaction with O₂ in presence of V₂O₅ catalyst.

ΔH = -196.6 KJ mol^-1
Low temperature (720 K) and high pressure (2 bar) are the favourable conditions for maximum yield.

(iii) Absorption of SO₃ in H₂SO₄ to give oleum (H₂S₂O₇)
SO₃ + H₂SO₄ → H₂S₂O₈
Dilution of oleum with water gives H₂SO₄ of desired concentration. H₂S₂O₇ + H₂O → 2 H₂SO₄

Properties:
Colourless, oily liquid, dissolves in water with the evolution of large quantity of heat, dibasic acid, in aqueous solution, it ionises in two steps:
H₂SO₄(aq) + H₂O(l) → H₃O⁺(aq) + HSO₄^– (aq)
HSO₄^–(aq) + H₂O(l) → H₃O⁺(aq) + SO₄^2-(aq)
Concentrated H₂SO₄ is a strong dehydrating agent.

Uses:
Manufacture of fertilisers; petroleum refining; manufacture of pigments, paints, dyestuff; detergent industry; storage batteries; laboratory reagent

Group 17 Elements (Halogens): F, Cl, Br, I and At (radio active), highly reactive non-metallic elements.

1-6 Occurrence:
F-in fluorides (CaF₂, Na₃AIF₆). Sea water contains chlorides, bromides and iodides of Na & K, electronic configuration – ns²np⁵, in a group from top to bottom atomic and ionic radii increases, ionisation enthalpy decreases.

Electron gain enthalpy – halogen have maximum negative electron gain enethalpy. Cl has highest electron gain enthalpy. Electro negativity decreases down the group. F is the most electronegative element in the periodic table.

Physical Properties:
F₂, Cl₂ – gases, Br₂ – liquid and l₂ – solid. F₂ and Cl₂ react with water Br₂ and l₂ sparingly soluble in water.

Oxidation States and Trends in Chemical Reactivity:
All the halogens exhibit-1 oxdn. state. But, Cl, Br and I exhibit +1, +3, +5 and +7 also. They react with metals and non-metals to form halides. The reactivity of the halogens decreases down the group.

Anomalous Behaviour of Fluorine:
It is due to smaller in size, high electronegativity, low F-F bond dissociation enthalpy and non-availability of d-orbitals, due to which it cannot expand its octet. It exhibits only-1 oxidation state.

(i) Reactivity Towards Hydrogen:
All form hydrogen halides (HX) which dissolve in water to form hydrohalic acids. The acidic strength of acids:
HF < HCl < HBr < Hl .The stability of halides decreases down the group due to decrease in (H-X) dissociation enthalpy in the order: H-F > H-Cl > H-Br > H-l.

(ii) Reactivity Towards Oxygen:
They form many oxides but most of them are unstable. Fluorine form OF₂ and O₂F₂. Chlorine form oxides Cl₂O, ClO₂, Cl₂O₆ and Cl₂O₇, which are highly reactive oxidising agents, ClO₂ is used as a bleaching agent for paper pulp, textiles.

(iii) Reactivity Towards Metals:
Metal halides are formed,
e.g. Mg(s) + Br₂(l) → MgBr₂(s)

(iv) Reactivity of Halogens Towards Other Halogens:
Halogens combine amongst themselves to form a number of compounds known as interhalogens. Five types: XX’, XX3, XX’5, XX’7 where X is a halogen of larger size and X’ of smaller size.

Chlorine:
Preparation:
(i) By heating manganese dioxide with concentrated HCl.
MnO₂ + 4HCl → MnCl₂ + 2H₂O + Cl₂

(ii) By the action of HCl on KMnO₄.
2KMnO₄ + 16HCl → 2KCl + 2MnCl₂ + 8H₂O + 5Cl₂

Manufacture:
(i) Deacon’s Process – By oxidation of HCl gas by atm oxygen in the presence of CuCl₂ at 723K.

(ii) Electrolytic process
 (liberated at anode)

Properties:
Greenish yellow gas with pungent and suffocating odour, reacts with metal, and non metals

With excess NH₃, Cl₂ gives N₂ and NH₄Cl whereas with excess Cl₂, NH₃ gives NCl₃ (explosive) and HCl.

With cold and dilute alkalies chlorine produces a mixture of chloride and hypochlorite but with hot and concentrated alkalies it gives chloride and chlorate.

With dry slaked lime, it gives bleaching powder.
2Ca(OH)₂ + 2Cl₂ → Ca(OCl)₂ + CaCl₂ + 2H₂OCl₂ is a powerful bleaching agent.
Cl₂ + H₂O → 2HCl + [O]
Coloured substance + [0] → colourless substance

Uses:
For bleaching wood pulp, cotton and textiles; for the preparation of insectiside, pesticides and other organic solvents, e.g. CHCl₃, DDT, BHC etc.

Hydrogen Chloride (HCl):
Preparation:

HCl gas is dried by passing through a cone. H₂SO₄.
Properties :
Colourless and pungent smelling gas, soluble in water and ionises as follows:
HCl + H₂O → H₃O⁺ + Cl^–
It reacts with NH₃ to give white fumes of NH₄Cl.
NH₃ + HCl → NH₄Cl
It decomposes salt of weaker acids.
Na₂CO₃ + 2HCl → 2NaCl + H₂O + CO₂
NaHCO₃ + HCl → NaCl + H₂O + CO₂

Uses: manufacture of Cl₂, NH₄Cl and glucose; for extracting glue.

Oxoacids of Halogen:
Due to high electronegativity and smaller in size fluorine forms only one oxoacid, HOF known as fluoric acid or hypofluorous acid.

Some oxoacids of Chlorine:

• Hypochlorous acid: HOCl (Cl in +1 state)
• Chlorous acid: HClO₂ (Cl in +3 state)
• Chloricacid: HClO₃ (Cl in +5 state)
• Perchloricacid: HClO₄(Cl in +7 state).

Interhalogen Compounds:
Two different halogens react to form inter halogen compounds, e.g. ClF, ClF₃, BrF₅, IF₇

Preparation:
By the direct combination of halogens.

Properties:
Covalent molecules, diamagnetic, volatile solids or liquids at 25°C except ClF which is a gas. They are more reactive than halogens because X-X bond is weaker than X-X bond. Due to electronegativity difference the X – X bond is polarised, hence it is reactive.

Their stability increases as the size difference of the halogens increases due to increase in the polarity of the bond. e.g. IF₃ is more stable than ClF₃.

Group 18 Elements (Noble Gases):
He, Ne, Ar, Kr, Xe and Rn (radio active). Except He all other noble gas have 8 electrons in the valence shell. Due stable electronic configuration all these are gases and chemically unreactive, (exeption – Kr, Xe, Rn)

Occurrence:
All except Rn occur in the atmosphere. The main source of He-natural gas. Rn- obtained as a decay of product of Radium.

Electronic Configuration-ns²np⁶ (except He-1s² ), ionisation enthalpy-high due to stable electronic configuration-it decreases down the group, atomic radii-increases down the group, electron gain enthalpy-almost zero since no tendency to accept an electron.

Physical Properties:
Monoatomic, colourless, odourless and tasteless gases, sparingly soluble in water, very low melting and boiling points because the only type of interatomic interaction in these elements is weak dispersion forces.

Chemical Properties:
Least reactive due to stable electronic configuration, high ionisation enthalpy and more positive electron gain enthalpy.

N. Bartlett prepared Xe⁺PtF₆ by mixing PtF₆ and Xe.

(a) Xenon – Fluorine Compounds:
Xe forms three binary fluorides XeF₂, XeF₄ and XeF₆ by the direct reaction of Xe with F₂.

XeF₄^– also prepared by reaction with O₂F₂ and XeF₄. XeF₄ + O₂F₂ → XeF₆ + O₂

Structure:
(a) XeF₂ – sp³d hybridisation -linear
XeF₄ – sp³d² hybridisation – square planar
XeF₆ – sp³d³ hybridisation – distorted octahedral

(b) Xenon-Oxygen Compounds:
XeO₃: Prepared by hydrolysis of XeF₄ and XeF₆.
6XeF₄ + 12H₂O → 4Xe + 2XeO₃ + 24HF + 3O₂
XeF₆ + 3H₂O → XeO₃ + 6HF.
XeOF_4^–prepared by partial hydrolysis of XeF₆.
XeF₆ + H₂O → XeOF₄ + 2HCl

Structure:
XeO₃ – sp³ hybridisation – Pyramidal
XeOF₃ – sp³d² hybridisation – Square pyramidal

Uses of Noble Gases:
1. He – for filling airships, aeroplane tyres, in gas-cooled nuclear reactors, for providing an inert atmosphere in the welding of metals and alloys.

2. Ne – for filling discharge tubes and fluorescent bulb for advertisement purpose, in botanical gardens and greenhouses.

3. Ar – to provide inert atmosphere in high-temperature metallurgical processes, for filling electric bulbs, for handling air-sensitive substances in laboratory.

4. Xe and Kr – in light bulbs designed for special purposes.

Students can Download Chapter 5 Accounting Software Package – GNUKhata Notes, Plus Two Accountancy Notes helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus Two Accountancy Notes Chapter 5 Accounting Software Package – GNUKhata

GNUKhata – Introduction
GNUKhata is an accounting software based on Double Entry Book Keeping system. It is a Free and Open Source Software (FOSS).

GNUKhata is developed by Digital Freedom Foundation in association with International Centre for Free and Open Source Software (ICFOSS). It is a free and flexible software for accounting and inventory management.

Features of GNUKhata

1. It is free and open source accounting software.
2. It is based on double entry book keeping
3. All financial reports can be prepared
4. Display of dual ledger facility
5. Attachment of source document to vouchers is possible
6. Linking of sales and purchase transactions to invoice.
7. Export or import of data from spread sheet is possible.
8. It ensures password security and data audit facility.

Installing GNUKhata
GNUKhata is Free and Open Source Software (FOSS). It can be legally downloaded and copied without having to pay anything to anyone. It is very simple to install GNUKhata in Linux. The installer file can be downloaded from the website www.anukhata.in. When we open GNUKhata for the first time, we can see a welcome screen.

Create Organisation
The first step in GNUKhata is to create an the organisation. To create a new organisation, click on “Create organisation” or press shift + control + R. While creating an organisation, the following details are to be given.

• Organisation Name: enter the name of the organisation and press Enter Key.
• Case: Choose the appearance of the organisation name, the options are As -is, Upper case, Lower Case or Title Case.
• Organisation Type: Select the organisation type either Profit Making or Not for Profit.
• Financial year: Enter the opening date of financial year then press enter key, closing date will show automatically which can be edited.
• Inventory: Tick the box of Inventory for maintaining inventory accounts.

1. Create Admin and log in:
There are Four levels of users in GNUKhata. They are Admin, Manager, Operatorand internal Auditor. Each has different authorities. Only one user can log in as “Admin”. There may be any number of users in the role of manager, operator and internal auditor.

“Create Admin” is mandatory. After creating an organisation, the next step is “Create Admin”. Fill all the fields in the Create Admin window and Enter/click on create and login. Now we can see a Menu bar at the top. Click Menu items to activate the Keyboard shortcuts.

2. Organisation Particulars:
We can enter the organisation details like Address, Country, State, city, etc through Edit Organisation Particulars in Master menu. Enter or click on Save to save the details and click on Reset to clear the fields, if necessary.

3. Change Organisation:
Select Change Organisation from Sign Out menu to exit the active organisation. To change the user, select Logout from Sign Out menu.

4. Selecting Organisation:
To select the existing organisation, click on Select Existing Organisation tab and select Organisation name from the drop down menu.

5. Deleting Organisation:
After login as “Admin” user, select Delete Organisation from Administration menu. Confirm the decision to delete the organisation, the organisation will be deleted.

Groups and Sub-Groups
Grouping of account is a method of organising the large number of ledger accounts into sequential arrangement for recording and summarisation of accounting data. GNUKhata has predefined Groups and Sub-Groups.
They are:

• Balance Sheet Groups
• Profit & Loss / Income & Expenditure Account Group

1. Balance sheet Groups in GNUKhata:

2. Profit and loss / Income & Expenditure Account Groups in GNUKhata:

3. Description of the Groups and Sub Groups:

(i) (a) Capital: Amount contributed by proprietor, partners and share holders are recorded in this group.
(b) Corpus: Amount Contributed by the members of a non profit organisation (capital fund) are recorded in this group.

(ii) Current Assets: The assets which are consumed in operations are known as current assets. Accounts of such assets generated in the course of doing business are recorded in this group. The sub groups are

• Bank-(Deposits)
• Cash – (in hand, at factory, petty cash)
• Inventory-(closing stock)
• Loans and Advances-(temporary advance to Staff)
• Sundry Debtors – (Credit sales)

(iii) Current Liability: The liabilities which are to be paid with in a short period (less than one year) are called current liabilities. The sub groups of current liabilities are.

• Provisions: (PF, ESI, TDS dues, etc.)
• Sundry CreditorsforExpenses: (Outstanding expenses)
• Sundry Creditors for Purchase: (Amount payable to suppliers)

(iv) Fixed Assets: Accounts of all fixed assets (life span more than one year) are recorded in this Group. The sub groups are

• Building
• Furniture
• Land
• Plant and Machinery

(v) Investments: Contains accounts of investment made by the organit#ation. The sub groups are

• Investment in Bank Deposits
• Investment in shares and Debentures

(vi) Loan (Assets): Includes accounts of long term loans given

(vii) Loans (Liability): Amount borrowed from financial institutions. The sub Groups are

• Secured: (Loan against Security)
• Unsecured: (No Security)

(viii) Miscellaneous Expenses (Assets): This includes preliminary and preformation Expenses to the extent those are not written off.

(ix) Direct Income: Income from sale of goods, included in this group. If it is a service organisation, income from fees will come under this group.

(x) Indirect Income: All incomes which is not a direct income come under this group.
Eg: rent received, discount received, dividend received etc.

(xi) Direct Expenses: Expenses of purchase or manufacturing of goods are included in this group.
Eg. Wages, carriage inward, consumables etc. GNUKhata opens Opening Stock Account under this group.

(xii) Indirect Expenses: All office, administration sellng and distribution expenses are coming under this group.
Eg: Salary, Interest, depreciation, etc.

(xiii) Reserves: Contains retained earnings reserves and surplus.

System Generated Ledger Accounts in GNUkhata
GNUkhata has 29 predetermined Groups and subGroups. Out of these 13 are Groups and 16 are sub Groups. 25 Predefined Groups and sub groups are related with Balance sheet. Out of these 9 are Groups and 16 are Sub Groups. 4 Groups are related with Profit and Loss Account/ Income and Expenditure Account.

In GNUkhata, there are 4 system generated ledger accounts. We can neither change the name nor delete these accounts. Do not create accounts with the similar names. These are.

Creating Ledger Accounts in GNUKhata
A ledger account contains record of all transactions relating to Assets, Capital, Liabilities, Expenditure and Revenues. A ledger account is to be created under any of the above five groups. Based on the group under which a ledger account is created, the “ balance of the ledger account will be appeared either in Trading, Profit and loss account or in Balance sheet.

1. Steps for Creation of Ledger Accounts:

1. Step 1: Click on Master Menu or Press Shift + Control + M or Press F2
2. Step 2: Select Create Account option. Then a dialogue box appears. Enter all the details.
   • Group Name: Select the name of the group from the drop down list.
   • Sub group name: Select the name of sub gorup from the drop down list depending on the group we selected. A new subgroup can also be created.
   • Account name: Enter the name of account which we want to create
3. Step 3: Click on Save button

2. Display Ledger Accounts:
To display Ledger accounts, select List of Accounts from Report menu. Now we can see a table containing a list of all ledger accounts along with its group name and sub group name.

3. Editing a ledger account:
To edit a ledger account, select Edit Account from Master menu. Here we can change Account name and opening balance. But we can not change the name of Group and sub Group.

4. Deleting a ledger account:
To delete a ledger account, select Edit Account from Master menu. Select the ledger account – we want to delete, click on Delete Button and confirm the deletion. System generated account and the ledger account already used in voucher cannot be deleted.

Types of Vouchers
GNUKhata has the following pre defined voucher types. We can not create a new voucher type.

Voucher Entry
Recording a transaction through voucher is called voucher entry. While recording a voucher the debit part of the transaction recorded first and there after credit part. We can add any number of debits and credits in a voucher.

1. Finding and Editing a Voucher Entry:
To edit a voucher entry, it must be found out first. For this, the given steps are to be followed.

• Step 1: Select Find/ Edit voucher from voucher menu.
• Step 2: Select any one Criteria and press Enter key. All transactions fulfilling the criteria will be displayed.
• Step 3: Select the transaction that we want to edit and press Enter key
• Step 4: Click Edit button to open the transaction in edit mode make necessary changes.
• Step 5: Click on Save button to save changes.

2. Deleting a voucher entry:
To delete a voucher, First find it and click on Delete, after confirmation, the record will be deleted. Deleted vouchers cannot be restored.

3. Add account while in voucher entry:
While recording transactions in voucher entry mode, we can add a ledger account by clicking on Add Account, select the Group, sub Group, enter the account name and opening balance, if any. Click on Save Button and return to Voucher entry mode.

Reports
One of the tabs in the menu bar is Reports. From the Report menu, we will enable to view reports such as Ledger, Trial Balance, Balance sheet, Profit and Loss Account, List of accounts and List of deleted vouchers. For all these reports, we have to specify the periods.

1. Display Ledger Account:

• Step 1: Click on Report tab from the menu bar
• Step2: Select Ledger option from the list, Now view Ledger screen appears
• Step 3: Select the name of ledger account we want to display
• Step 4: Enter from date and to date
• Step 5: Click on view button to view the ledger

2. Display Trial Balance:

• Step1: Click on Report tab from the menu bar
• Step 2: Select Trial balance from the menu
• Step 3: Enter From date and To date
• Step 4: Select the Trial Balance Type
• Step 5: Click on view button to view the Trial Balance

3. Display Profit and Loss Account/Income and Expenditure Account:

• Step 1: Click on Report tab from the menu bar
• Step 2: Select Profit and loss or Income and Expenditure from the list
• Step 3: Enter From date and To date.
• Step 4: Click on view button

4. Display Balance sheet:

• Step 1: Click on Report Tab from the menu bar
• Step 2: Select Balance sheet from the list
• Step 3: Select Balance sheet type
• Step 4: Click on View button

Bank Reconciliation statement (BRS)
Bank Reconciliation statement is prepared by an account holder to reconcile cash book balance and pass book balance on a specific date. It is prepared for bank accounts opened under the Sub Group Bank of the Group Current Assets. To open Bank Reconciliation Statement, select Bank Reconciliation statement from the Master menu.

1. Causes of Difference between Cash Book Balance and Pass Book Balance:

• Cheque issued; but not presented for payment
• Cheque deposited; but not collected
• Direct payment by a customer to the bank
• Interest on deposit credited by the bank
• Dividend, rent, etc collected by the bank
• Payment made on behalf of the customer
• Bank charges as per pass book
• Bills receivables discounted, but dishonored
• Interest on overdraft debited in the pass book

2. Terms associated with BRS:

• Transaction Date: The date of the transaction
• Clearance date: The date on which a particular transaction appears in a pass book
• Reconciliation period: The period for which Bank Reconciliation is done is called Reconciliation period.
• Statement of uncleared items: When the clearance date and Transaction date of a transaction falls with in the Reconciliation period, that transaction is said to be cleared. Otherwise it is uncleared item.

Short cut keys in GNUKhata

Use	Keys
1. Activate Toolbar Tab	F1
2. Create Ledger Account	F2
3. Find/Edit Ledger Account	F3
4. Receipt Voucher	F4
5. Receipt Voucher	F5
6. Sales voucher	F6
7. Purchase Voucher	F7
8. Contra Voucher	F8
9. Journal Voucher	F9
10. Find/ Edit voucher	F10
11. View Ledger Account	F11
12. Display Trial Balance	F12
13. Sales Returns voucher	Ctrl + 1
14. Purchases Returns voucher	Ctrl + 2
15. Credit Note voucher	Ctrl + 3
16. Debit Note voucher	Ctrl + 4
17. Cost centre statement	Ctrl + 5
18. Cash Flow statement	Ctrl + 6
19. List of Accounts	Ctrl + 7
20. Create/ Edit cost centre	Alt + P
21. Bank Reconciliation statement	Alt + R
22. Manual	Alt + M
23. Master Tab	Ctrl + M
24. Inventory Tab	Ctrl + I
25. Transaction Tab	Ctrl + T
26. Report Tab	Ctrl + R
27. Administration Tab	Ctrl + D
28. Help Tab	Ctrl + H
29. Sign out Tab	Ctrl + S
30. Sign out Tab	Ctrl + L

Students can Download Chapter 2 Electric Potential and Capacitance Notes, Plus Two Physics Notes helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus Two Physics Notes Chapter 2 Electric Potential and Capacitance

Introduction
The electric field strength is a vector quantity, while electric potential is a scalar quantity. Both these quantities are inter related.

Electrostatic Potential

1. Electric potential: The electric potential at a point is the work done by an external agent in moving a unit positive charge from infinity to that point against the electric field (without acceleration)
Explanation: If W is the work done in moving a charge ‘q’ from infinity to a point, then potential at
that point is, V = \(\frac{w}{q}\)

Potential difference: Electric potential difference between two points is the work done in moving a unit positive change from one point to other.
Potential difference between the points A and B is
V_AB = V_A – V_B
V_A and V_B are the potentials at the points A and B respectively.

Potential energy difference: Potential energy difference is the work done to bring a q charge from one point to another point with out acceleration.
Relation between potential difference and potential energy difference:

where U_A and U_B are the potential energies at the points A and B respectively.

Electric field is conservative: Electric field is conservative. A conservative field is defined as the field in which work done is zero in a complete round trip.

(or)

A conservative field is one in which work done is independent of path.

Potential Due To A Point Charge

Let P be a point at a distance Y from a charge +q. Let A be a point at a distance ‘x’ from q ,and E is directed along PA. Consider a positive charge at A. Then the electric field intensity at A’ is given by

If this unit charge is moved (opposite to E) through a distance dx, the work done dw = – Edx
[-ve sign indicates that dx is opposite to E ]
So the potential at ‘P’ is given by

Potential due to an electric dipole

Consider dipole of length ‘2a’. Let P be a point at distance r₁ from +q and r₂ from -q. Let ‘r’ be the distance of P from the centre ‘O’ of the dipole. Let θ be angle between dipole and line OP.

Therefore total potential,

From ∆ABC , we get (r₂ – r₁) = 2a cosθ
we can also take r₂ = r₁ = r (since ‘2a’ is very small) Substituting these values in equation (1), we get

Case 1: If the point lies along the axial line of the dipole, then θ = 0°

Case 2: If the point lies along the equatorial line of the dipole, then θ = 90°

V = 0

Potential Due To A System Of Charges

Consider a system of charges q₁, q₂,……,q_n with position vectors r_1P, r_2P……..,r_nP relative to some origin. The potential V₁ at P due to the charge q₁ is

where r_1P is the distance between q₁ and P. Similarly, the potential V₂ at P due to q₂,

where r_2P is the distances of P from charges q₂. By the superposition principle, the potential V at P due to the total charge configuration is the algebraic sum of the potentials due to the individual charges
ie. V = V₁ + V₂ +……+ V_n

Equipotential Surface
The surface over which the electric potential is same is called an equipotential surface.
Properties:

1. Direction of electric field is perpendicular to the equipotential surface.
2. No work is done to move a charge from one point to another along the equipotential surface.

Example:

1. Surface of a charged conductor.
2. All points equidistant from a point charge.

Equipotential surfaces for a uniform electric filed:

1. Relation Between Electric Field And Potential:

Consider two points A and B, separated by very small distance dx. Let the potential at A and B be V+ dV and V respectively. The electric field is directed from A to B.
If a unit +ve charge is moved through a distance ‘dx’ against this field, work done,
dw = -Edx _____ (1)
For unit charge dw = dv
∴ dv = – Edx
or

Electric field intensity at a point is the negative rate of change of potential with distance.

Potential Energy Of System Of Charges

1. Potential Energy of System of Two Charges:
The potential energy of a system of two charges is defined as the work done in assembling this system of charges at the given position from infinite separation.

Consider two charges q₁ and q₂ separated by distance r. Imagine q₁ to be at A and q₂ at infinity. Electric potential at B due to charge q₁ is given by

which is the work done in bringing unit positive charge from infinity to B. Therefore the work done in bringing charge q₂ from infinity to B is
W = potential difference × charge
W = (V₁ – V_∞)q₂
potential at inf infinity. V_∞ = 0
W = V₁ × q₂
\(W=\frac{1}{4 \pi \varepsilon_{0}} \frac{q_{1} q_{2}}{r_{12}}\)
This work done is stored as potential energy. Hence potential energy between the charges q₁ and q₂

2. Potential Energy of System of Three Charges:
Consider three charges q₁, q₂, and q₃ separated by distances r₁₂, r₂₃ and r₁₃.

The electric potential energy of this system is the sum of potential of each pair. Hence we can write
U = U₁₂ + U₂₃ + U₁₃

Potential Energy In An External Field

1. Potential energy of a single charge:
Consider a point O in an electric field. Let V be the electric potential at O. Hence work done in bringing a charge q from infinity to the point O is,
W = Vq.
This work done is stored in the form of electrostatic potential energy (U) of the charge q.
∴ The potential energy of the charge q in an electric field is U = Vq
Where V is the potential at that point.

2. Potential energy of a system of two charges in an electric field:
Consider an electric field. Let 1 and 2 be two points in the field and V₁ and V₂ be the potential at these points. Two charges q₁ and q₂ are located at 1 and 2.
Potential energy of the charge q₁ in the external filed is, U₁ = V₁ q₁
Potential energy of the charge q₂ in the external field is, U₂ = V₂q₂
Potential energy between the system of two charges q₁ and q₂

where r₁₂ is the distance between the charges According to the principle of super position, the potential energy of the system of two charges in an electric field is
U = U₁ + U₂ + U₁₂

3. Potential energy of a dipole in an external field:
Consider a dipole of dipole moment ‘P’ suspended in a uniform electric field of intensity ‘E’. Let θ be the angle between P and E.
Then we know torque τ = PE sinθ
Let the dipole be turned through an angle dθ
then work done dw = τdθ
= PE sinθ dθ
Total work done in rotating the dipole from θ₁ to θ₂

W = PE (cosθ₁ – cosθ₂)
This work done is stored as potential energy.

Electrostatics Of Conductors
The electrostatic properties of conductors are given below:
1. Inside a conductor, electrostatic field is zero:
In the static situation, there is no current found inside the conductor. Hence we conclude that the electric field is zero inside the conductor The vanishing of electric field inside the metal cavity is called electrostatic shielding.

2. At the surface of a charged conductor, electrostatic field must be normal to the surface at every point.

3. The interior of a conductor can have no excess charge in the static situation.

4. Electrostatic potential is constant throughout the volume of the conductor and has the same value (as inside) on its surface.

5. Electric field at the surface of a charged conductor
\(\bar{E}=\frac{\sigma}{\varepsilon_{0}} \hat{n}\)
where σ is the surface charge density and is a unit vector normal to the surface in the outward direction.

6. Electric field inside a metal cavity is zero. Vanishing of electric inside a metal cavity is called electrostatic shielding. Sensitive electrical instruments can be protected from external electricfield by placing it in a metal cavity.

Dielectrics And Polarization
Dielectrics (insulator): Dielectrics are non-conducting substances. They have no charge carriers. The molecules of dielectrics may be classified into two classes.

1. Polar molecule
2. Nonpolar molecule

Electric field due to a dipole at a point on the perpendicular bisector of the dipole (at a point on the equatorial line).

Consider a dipole of dipole moment P = 2aq. Let ‘S’ be a point on its equatorial line at a distance ‘r’ from its centre. The magnitudes of electric field at ‘S’ due to +q and -q are equal and acts as shown in figure. To find the resultant electric field resolve.

Their normal components cancel each otherwhere as their horizontal components add up to give the resultant field at ‘S’.
E = E_Acos θ + E_Bcos θ = 2 E_B cos θ

The direction of the field due to the dipole at a point on the equatorial line is opposite to the direction of dipole moment.

1. Dielectrics in external electric field
(a) Nonpolar dielectrics in external field:

Considers nonpolar dielectrics in an external electricfield. In electricfield, the positive and negative changes of a nonpolar molecule are displaced in opposite directions. Thus dipole moment is induced in a nonpolar molecule. The induced dipole moments of different molecules add up giving a net dipole moment.

(b) Polar dielectrics in external electric field:

The permanent dipoles in a polar dielectrics are arranged randomly. So total dipole moment is zero. But when we apply external electric field, the individual dipole tends to align in the direction of electricfield. The induced dipole moments of different molecules add up giving a net dipole moment.

Electric susceptibility: Non-polar dielectrics and polar dielectrics can produce net dipole moment in the external electric field. The dipole moment per unit volume is called polarization and is denoted as P. For linear isotropic dielectrics.
\(\bar{P}=\chi_{0} \bar{E}\)
where χ_e is a constant and is known as the electric susceptibility of the dielectric medium.
How does the polarized dielectric modify the external field inside it?

Consider a dielectric slab placed inside a uniform external electric field E₀. This field produces a uniform polarization as shown in the figure. Any region inside the dielectric, the net charge is zero.

This is due to the cancellation of positive charge of one dipole with negative charge of adjacent dipole. But the positive ends of the dipole do not cancel at right surface and the negative ends at the left surface.

This surface charges (-σ_p and +σ_p) produce a field \(\left(\vec{E}_{i n}\right)\) opposite to the external field. Hence total electric field is reduced inside the dielectric field which is shown in the figure below.
ie; E₀ + E_in ≠ 0

How does a metal modify the external electric field applied on it?

When a conductor placed in a electric field, the free charges are moved in opposite direction as shown in figure. This rearrangement of charges in a metal produce an internal field (E_in) inside the metal. This internal field cancels the external field. Thus the net electric field inside the metal becomes zero.
ie; E₀ + E_in = 0

Capacitors And Capacitance
Capacitor: Capacitor is a system of two conductors separated by an insulator for storing electric charges.
Capacitance of a capacitor:

Consider two conductor having charges +Q and -Q and potentials V₁ and V₂. The amount of charge Q on a plate is directly proportional to the potential difference (v₁ – v₂) between the plates,
ie. Q α V₁ – V₂
(or) Q α V (where V = V₁ – V₂)

The constant C is called the capacitance of the capacitor. If V = 1, we get Q = C. Hence capacitance of a capacitor may be defined as the amount of charge required to raise the potential difference between two plates by one volt.
Dielectric strength:
What happens to the charge stored in capacitor when the p.d. between two plates increases?
When the p.d. between two plates increases, electric field in between two plates increases. This high electric field can ionize the surrounding air (or medium) and accelerate the charges to the oppositely charged plates and neutralize the charge on the plate. This is called electric break down.

The maximum electric field that a dielectric medium can withstand without break down (of its insulating property) is called its dielectric strength. The dielectric strength of air is 3 × 10⁶ v/m.

The Parallel Plate Capacitor
Electric field due to a capacitor:

Consider a parallel plate capacitor consists of two large conducting plates 1 and 2 separated by a small distance d. Let +σ and -σ be the surface charge densities of first and second plates respectively. (Here, we take, electric field towards right is taken as positive and left as negative.)
Region I: This region lies above plate 1.
E = E₊ + E_– ie.

Region II: This region lies below the plate 2.
E = E_– + E₊ ie.

E = 0
Electric field in between two plates: In the inner region between the plates 1 and 2, the electric field due to two charged plates add up.

(a) Expression for capacitance of a capacitor: Potential difference between two plates
V= Ed

Effect Of Dielectric On Capacitance

Consider a capacitor of area A and charge densities +σ and -σ. Let d be the distance between the plates. If a dielectric slab is placed inside this capacitor, it undergoes polarization. Let +σ_p and -σ_p be polarized charge densities due to polarization.
Due to polarization electric field in between the plate becomes
\(E=\frac{\sigma}{K \varepsilon_{0}}\) _____(1)
The potential difference between the plates,
V = Ed _____(2)
Sub (1) in (2)
\(\mathrm{V}=\frac{\sigma}{\mathrm{K} \mathrm{s}_{0}} \mathrm{d}\)
Then the capacitance of capacitor

The product ε₀ K is the permittivity of the medium.

Combination Of Capacitors
1. Capacitors in series: Let three capacitors C₁, C₂ and C₃ be connected in series to p.d of V. Let V₁, V₂ and V₃ be the voltage across C₁, C_2, and C₃.

The applied voltage can be written as
V = V₁ + V₂ + V₃ ______(1)
Charge ‘q’ is same as in all the capacitor. So,

Substituting these values in (1),

If these capacitors are replaced by a equivalent capacitance ‘C’, then
V = \(\frac{q}{C}\)
Hence eq(2) can be written as

Effective capacitance is decreased by series combination.

2. Capacitors in parallel:

Let three capacitors C₁, C₂ and C₃ be connected in parallel to p.d of V. Let q₁, q₂, and q₃ be the charges on C₁, C₂ and C₃.
If ‘q’ is the total charge, then’q’can be written as
q = q₁ + q₂ + q₃
But q₁ = C₁V, q₂ = C₂V and q₃ = C₃V
Hence eq (2) can be written as
CV = C₁V + C₂V + C₃V

Effective capacitance increases in parallel connection.

Energy Stored In A Capacitor
Energy of a capacitor is the work done in charging it. Consider a capacitor of capacitance ‘C’. Let ‘q’ be the charge at any instant and ‘V’ be the potential. If we supply a charge ‘dq’ to the capacitor, then work done can be written as,
dw = Vdq
dw = \(\frac{q}{C}\)dq (since V = \(\frac{q}{C}\))
∴ Total work done to charge the capacitor (from ‘0’ to ‘Q’) is

But Q = CV
W = \(\frac{1}{2}\) CV²
This work done is stored in the capacitor as electric potential energy.
∴ Energy stored in the capacitor is,

Van De Graff Generator
Van de Graff generator is used to produce very high voltage.
Principle: If two charged concentric hollow spheres are brought in to contact, charge will always flow from inner sphere to the outer sphere.
Construction and working:

The vande Graff generator consists of a large spherical metal shell, placed on an insulating stand. Let p₁ and p₂ be two pulleys. Pulley p₁ is at the center of the spherical shell S. A belt is wound around two pulleys p₁ and p₂. This belt is rotated by a motor. Positive charges are sprayed by belt. Brush B₂ transfer these charges to the spherical shell. This process is continued. Hence a very high voltage is produced on the sphere.
Why does the charge flow from inner sphere to outer sphere?

Let ‘r’ and ‘R’ be the radius of inner sphere and outer sphere carrying charges q and Q respectively.
The potential on the outer sphere,
V(R) = Potential due to outer charge + potential due to inner charge

Potential on the inner sphere. V(r) = Potential due to outer charge + Potential due to inner charge

∴ Potential difference between the two spheres

The above equation shows that, the inner sphere will be always at higher potential. Hence, charge always flow from inner sphere to outer sphere.

Students can Download Chapter 4 Graphs and Charts for Business Data Notes, Plus Two Accountancy Notes helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus Two Accountancy Notes Chapter 4 Graphs and Charts for Business Data

Introduction
One of the very useful features of Libre Office Calc Spreadsheet is the capability to create charts and graphs of the data. They are visual representations of numerical data, which has atleast two-dimensional relationship. The graph has at least two axes-x and y. X axis is usually horizontal while Yaxis is vertical.

Types of Graphs and Charts
Libre Office Calc provides various types of chart to help us to display data in different ways as per the need of the viewers. The major types available are given below:

1. Column chart:
It is the most commonly used chart type. It shows a bar chart or bar graph with vertical bars. The X axis shows the categories and Y axis shows the value for each category. Column chart are used to compare values across categories.

2. Bar chart:
This type of chart shows a bar graph or column chart with horizontal bar. The Y axis shows categories and the X axis shows the value for each category. It is suitable for comparing multiple values.

3. Line chart:
A line chart shows values in the Y axis and categories in X axis. The Y values of each data series is connected by a line. Line chart shows data changes for a certain period of time.

4. Pie chart:
A pie chart displays the contribution of each value to a total. It represents multiple subgroup of a single variable. It contains only one data series. A pie chart shows values as circular sectors of the total circle. Pie chart may be

• Normal Pie chart
• Exploded Pie chart
• Doughnut chart or Donut chart
• Exploded Doughnut chart

5. Area chart:
The chart shows values as points on the Y axis. The X axis shows categories. The Y values of each data series are connected by a line. The area between each two lines is filled with a colour.

6. Scatter chart:
Scatter chart is also known as XY chart. In this type of chart, both axes display values. This chart is used to show the relationship among two variables.

7. Radar chart:
It is also known as Net chart or Star chart. A radar chart has a separate axis for each category and the axes extend outward from the center of the chart. The value of each data point is plotted on the corresponding axis.

Basic steps for Graphs /Charts/Diagrams using Libre Office Calc
The steps involved in the chart preparation is given below:

• Step 1: Data entry (in spread sheet, in the form of a table)
• Step 2: Data selection (select data for plotting chart)
• Step 3: Plotting the chart (Standard tool bar → Insert → chart wizard)
• Step 4: Chart type (choose a chart type from the list)

Elements of a Chart

Chart elements	Description
1. Axes Titles	Mention the names or title for X, Y and Z axes
2. X, Y, & Z axes	In 2D chart, the horizontal X axis contains categories and vertical Y axis contains dependent values. In 3D chart, Z axis will also be there represents the depth which
3. Chart Area	This is the total space that is enclosed by a chart. It is the background of the chart.
4. Chart wall	In 2D chart, the wall or area is bounded by X and Y axis. In 3D chart the wall is bounded by three axes X, Y and Z
5. Chart floor	The chart floor is the lower area in 3D chart.
6. Main Title/sub Title	It is the explanatory heading of the chart. It identifies the purpose of a chart.
7. Data Marker	Individual values plotted in a chart are called data marker or data point
8. Data Series	Data markers of the same colour or pattern is called data series.
9. Legend	It is an identifier of a piece of information shown in the chart/ graph. The legends are assigned to the data series in a chart.
10. Data Label	The value of the data series plotted in a chart is known as data label.
11. Grid Lines	These are the vertical and Horizontal lines that appear in a chart. It increase the readability of a chart.

Advantages in using Graph/Chart

1. It summarises a large data set in visual form.
2. Charts or graphs can clarify trends better than do tables.
3. It helps to estimate key values at a glance.
4. It shows each data category in a frequency distribution.
5. It permits a visual check of the accuracy and reasonableness of calculations.
6. The charts and graphs allow the investigator to draw a valid conclusion.

Students can Download Chapter 3 Use of Spread Sheet in Business Application Notes, Plus Two Accountancy Notes helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus Two Accountancy Notes Chapter 3 Use of Spread Sheet in Business Application

Business Applications
The following accounting applications are done with the help of spreadsheet

• Payroll Accounting
• Asset Management
• Loan Repayment

Pay Roll Accounting
Payroll is a statement prepared to show the detailed salary calculation of employees. It contains Basic
Pay, Dearness Allowance, Travelling Allowance, Provident Fund Contribution, ESI Premium, etc. The computation of salary payment is based on the number of days an employee has worked, rate per grade, rate of allowances and deductions to be made therefrom.

1. Preparation of Salary Bill:
The preparation of salary bill should provide for the following:

• Maintaining payroll related data such as Employee No., Name, attendance, Basic Pay, DA, and other allowances, deductions to be made, etc.
• Periodic Payroll Computations: It includes the calculation of various earnings and deductions.
• Preparation of salary statement and employee’s salary slip.
• Generation of advice to bank: It contains the net salary to be transferred to individual bank account of employees and other salary related statutory payments such as provident fund, tax, etc.

2. Pav Roll Components:
The following elements are important for salary computation and its payment.

Earnings:

• Basic Pay (BP): It is the pay in the pay scale
• Grade Pay (GP): It is the pay to be added to the basic pay according to the designation.
• Dearness Pay (DP): Portion of dearness allowance merged with Basic Pay
• Dearness allowance (DA): Compensation for erosion in the purchasing power of wage earner due to Price rise.
• House Rent Allowance (HRA): An amount paid as rent of residential accommodation.
• Transport Allowance (TA): It is an amount to faclitate commuting to the palace of work.

Deductions:

• Professional TAX: Statutory deduction levied by State Government.
• Provident Fund (PF): It is a statutory deduction under provident Fund Act. It is deducted from salary as a part of social security.
• Tax Deducted at Source (TDs)- Monthly deduction towards income Tax liability of an employee.
• Recovery of Loan instalment: An amount deducted on account of any loan taken up by employee.

3. Net Salary Calculation:
Step 1: Calculate Gross salary by using the given formula.

Step 2: Calculate Total Deduction by using the following formula.
Total Deduction = Professional Tax + Provident Fund + Tax deducted at source + Loan Recovery + Any other deductions.

Step 3: Calculate net salary by the given formula
Net Salary = Gross salary – Total Deduction.

Asset Accounting
Assets are resources of the organisation. Assets which are used in the business for more than one year, called Fixed assets. The value of Fixed assets may be reduced due to depreciation. The gradual and permanent diminution in the value of assets due to wear and tear is called depreciation.

The depreciation on fixed assets is provided to recognise the cost of the asset consumed during an accounting period since the life of such assets extends beyond single accounting year.
Total amount of Depreciation – Acquisition cost – Salvage value.

1. Methods of calculation of depreciation:

• Straight Line Method (SLM)
• Written Down Value Method (WDV)

Straight Line Method
Under this method a fixed amount is deducted from the value of an asset year after year on account of depreciation and debited to profit and loss account. This method is also called Fixed Instalment method, or Original Cost method. Under this method value of asset will be reduced to zero.
Depreciation = \(\frac{\text { cost of the asset }-\text { Scrap Value }}{\text { Life of the asset }}\)

1. Cost of the asset/Acquisition cost = Purchase Price + Other expenses directly related with the asset (ie., carriage inward, freight, installation, renewal or reparis, Pre operating expenses).

2. Scap value / salvage value: It is the value of an asset which is realisable at the end of its useful life. Salvage value is the estimated residual value of depreciable asset or property at the end of its economical or useful life.

3. The depreciation under straight line method is computed by using the built in LibreOffice calc function SLN.

Written Down Value Method (WDV)
This method is also known as Diminishing balance method or Reducing balance method. Under this method, a fixed percentage is written off every year on the book value of the asset at the beginning of the year. Here the amount of depreciation goes on decreasing and there fore, the book value of asset will not become zero after its working life.
Amount of depreciation = Written Down Value of asset × Rate of depreciation

1. This method is also called Declining Balance (DB) method and uses the LibreOffice Calc function DB to compute the depreciation.

Loan Repayment Schedule
Loan is a sum of borrowed money for a specified period at a pre-specified rate of interest. The loan is repaid through a number of periodic repayment instalments over the loan repayment period.

LibreOffice Calc function PMT is used to calculate loan repayment schedule. The parameters of the function PMT are as follows.

Students can Download Chapter 1 Electric Charges and Fields Notes, Plus Two Physics Notes helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus Two Physics Notes Chapter 1 Electric Charges and Fields

Introduction
You may have seen a spark (or a crackle sound), when we take off our synthetic clothes. Have you ever tried to find any explanation for this phenomenon? Do you know the reason for lightning?

The above phenomenon can be explained on the basis of static electricity. Static means anything that does not change with time. Electrostatics deals with the properties of charges at rest.

Electric Charge
It is found experimentally that the charges are of two types:

1. Positive charge
2. Negative change

The unit of charge is Coulomb (c).
Note: Positively charged body means deficiency of electrons in the body and a negatively charged body means excess of electrons.

Gold-Leaf electroscope: A simple apparatus to detect charge on a body is called a gold-leaf electroscope.

Apparatus: It consists of a vertical metal rod placed in a box. Two thin gold leaves are attached to its bottom end as shown in figure.

Working: A charged object touches the metal knob at the top of the rod. Charge flows on to the leaves and they diverge. The degree of divergence is an indicator of the amount of charge.

Conductors And Insulators
Conductors: Conductors are those substances which allow passage of electricity through them.
Insulators: Insulators are those substances which do not allow passage of electricity through them.

1. Earthing (Or) Grounding:
When a charged body bring in contact with earth, all the excess charge pass to the earth through the connecting conductor. This process of sharing the charges with the earth is called grounding or earthing. Earthing provides protection to electrical circuits and appliances.

Charging By Induction
A body can be charged in different ways.

1. Charging by friction
2. Charging by conduction
3. Charging by induction

1. Charging by friction:
When two bodies are rubbed each other, electronsin one body (in which electrons are held less tightly) transferred to second body (in which electrons are held more tightly).

Explanation: When a glass rod is rubbed with silk, some of the electrons from the glass are transferred to silk. Hence glass rod gets +ve charge and silk gets -ve charge.

2. Charging by conduction:
Charging a body with actual contact of another body is called charging by conduction.
Explanation:

If a neutral conducting body (A) is brought in contact with positively charged conducting body (B), the neutral body gets positively charged.

3. Charging by induction:
The phenomenon by which a neutral body gets charged by the presence of neighboring charged body is called electrostatic induction.
Explanation:
Step I: Place two metal spheres on an insulating stand and bring in contact as shown in figure (a).

Step II: Bring a positively charged rod near to these spheres. The free electrons in the spheres are attracted towards the rod. Hence, one side of the sphere becomes negative and the other side becomes positive as shown in the figure (b).

Step III: Separate the spheres by a small distance by keeping the rod nearto sphere A. The two spheres are found to be oppositely charged as shown in figure (c).

Step IV: Remove the rod, the charge on spheres rearrange themselves as shown in figure (d).

In this process, equal and opposite charges are developed on each sphere.

Basic Properties Of Electric Charge
1. Unlike charges attract and like charges repel.

2. Charge is conserved : Changes can neither be created nor be destroyed.
Explanation: When a glass rod is rubbed with silk, some of the electrons from the glass are transferred to silk. Hence glass rod gets +ve change and silk gets -ve changes.

3. Electric Charge is Quantized: Change on any body is the integral multiple of electronic charge. This is called quantization of charge.
i.e. q = ± ne, n = 1, 2, 3, ……….

4. Additivity of Charges: If a system contains n charges q₁, q₂, q₃,……..q_n, then the total change of the system is q₁ + q₂ + q₃ +……+q_n.

Coulomb’S Law
Statement: The force between two stationary electric charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.
Explanation:

Consider two point charges q₁ and q₁. which are separated by a distance ’r’. The force between the changes.
\(\mathrm{F}=\frac{1}{4 \pi \varepsilon_{0}} \frac{\mathrm{q}_{1} \mathrm{q}_{2}}{\mathrm{r}^{2}}\)
vector form: The force F₁₂ (on the first charge by second} is given by (vector form)

Forces Between Multiple Charges
Super position principle: If the system contains a number of interacting changes, then the force on a given charge is equal to the vector sum of the forces exerted on it by all remaining charges.
Explanation:

Consider a system of three charges q₁ q₂ and q₃ as shown in figure.
The force on q₁ due to q₂
\(\overrightarrow{F_{12}}=\frac{1}{4 \pi \varepsilon_{0} r_{12}^{2}} r_{12}^{\wedge}\)
Similarly the force q₁ due to q₃
\(\overrightarrow{F_{13}}=\frac{1}{4 \pi \varepsilon_{0}} \frac{q_{1} q_{2}}{r_{13}^{2}} r_{13}^{\wedge}\)
The total force F₁ on q₁ (due to q₂ and q₃) can be written as

System of ‘n’ charges: If system contains ‘n’ charges, total force acting on q₁ due to all other charges.

Electric Field
The concept electric field is introduced to explain the interaction between two charges.
Electric field intensity: Strength or intensity of the electric field at any point is defined as the force acting on a unit positive charge placed at that point.
Mathematical expression of electric field intensity:

Consider a charge q (test charge) at a distance ‘r’ from a source charge Q.
The force acting on q due to Q.
\(\mathrm{F}=\frac{1}{4 \pi \varepsilon_{0}} \frac{\mathrm{Qq}}{\mathrm{r}^{2}}\)
If q = 1, the force acting on this unit charge due to Q
\(\mathrm{F}=\frac{1}{4 \pi \varepsilon_{0}} \frac{\mathrm{Q}}{\mathrm{r}^{2}}\)
This force is called electric field intensity at a distance ‘r’ due to the charge Q.
ie; \(E=\frac{1}{4 \pi \varepsilon_{0}} \frac{Q}{r^{2}}\)

1. Electric field due to a system of charges:
Consider a system of charges q₁, q₂………..q_n. Let P be point at distances r_1p, r_2p,………..r_np from charges. q₁, q₂,……..q_n respectively. According to super position principle, total electric field at ‘p’ due to all other charges,

2. Physical Significance Of Electric Field:
Question 1.
What are the importance of the concept of electric field?
Answer:

• Electric field explains the electrical environment of a system of charges.
• Electric field help us to explain the interaction between two charges at rest or in motion.

Electric Field Lines
Properties of Electric Lines of Force

(Field lines due to some simple charge configurations)

1. An electric line of force originates from positive charge and ends on negative charge.
2. The tangent drawn at a point on an electric line of force will give the direction of electric field at that point.
3. Two lines of force never intersect each other. (If they cut each other, at the point of intersection there will be two tangents. This indicates that there will be two directions of electric field at the same point which is impossible).
4. The number of electric lines of force passing normally through an area is directly proportional to the strength of the electric field.
5. The relative density of the field lines indicates the relative strength of electric field.
6. Electric field lines due to static charge never form closed loops.
7. In a uniform electric field, lines of force are parallel.

Electric Flux

Consider a closed surface. Let ∆\(\vec{S}\) be a small area element on the surface. The electric field lines (E) passes through this area element at an angle θ. Electric flux ∆Φ through an area element ∆\(\vec{S}\) is defined by

The direction of area vector d\(\vec{S}\) is perpendicular to the surface.

Electric Dipole
Electric dipole: A pair of equal and opposite charges separated by small distance is called electric dipole. Dipole moment (p): Electric dipole moment (p) is defined as product of magnitude of charge and dipole length.
Dipole moment p = q × 2a
q – charge, 2a – dipole length
Dipole moment is a vector, directed from negative to positive charge.

1. Electric field at a point on the axial line of an electric dipole: Consider an electric dipole of moment P = 2aq. Let ‘S’ be a point at a distance ‘r’ from the centre of the dipole.

Electric field at ‘S’ due to point charge at ‘A’

Directed as shown in figure. Electric field at ‘S’ due to point charge at ‘B’

Directed as shown in figure. Therefore resultant electric field at ‘S’
And its magnitude E = E_B + -E_A

P = q × 2a
We can neglect a2 because a<<r.
∴ Electric field at S,

This can be written in vector form as

The direction is along E_B.
The field due to an electric dipole is directed from negative charge to positive charge along the axial line.

2. Electric field due to a dipole at a point on the perpendicular bisector of the dipole (at a point on the equatorial line).

Consider a dipole of dipole moment P = 2aq. Let ‘S’ be a point on its equatorial line at a distance ‘r’ from its centre. The magnitudes of electric field at ‘S’ due to +q and -q are equal and acts as shown in figure. To find the resultant electric field resolve

Their normal components cancel each otherwhere as their horizontal components add up to give the resultant field at ‘S’.
E = E_Acos θ + E_Bcos θ = 2 E_B cos θ

The direction of the field due to the dipole at a point on the equatorial line is opposite to the direction of dipole moment.

3. Physical significance of dipole: The molecules of dielectrics may be classified into two classes:
(i) Polar molecules: In polar molecule, the centres of negative charges and positive charges do not coincide. Therefore they have a permanent
Example: H₂0, HCl, etc.

(ii) Nonpolar molecule: In nonpolar molecule, the centres of negative charges and positive charges coincide. Therefore they have no permanent electric dipole moment.
Example: C0₂, CH_4, etc.
Note: In the presence of external electric field, a nonpolar molecule becomes a polar molecule.

Dipole In A Uniform External Field

Consider an electric dipole of dipole moment P = 2aq kept in a uniform external electric field, inclined at an angle θ to the field direction.

Equal and opposite forces +qE and -qE act on the two charges. Hence the net force on the dipole is zero. But these two equal and opposite forces whose lines of action are different. Hence there will be a torque.
torque = any one force × perpendicular distance (between the line of action of two forces)
τ = qE × 2 a sin θ
Since P = 2aq
τ = P E sin θ
Vectorialy \(\vec{\tau}=\vec{P} \times \vec{E}\)
This torque tries to align the dipole along the direction of the external field.
Special Case:

• When θ = 0 ; τ =0
• When θ = 90; τ = PE, the maximum.

Note: In uniform electric field dipole has only rotational motion

Dipole in nonuniform electric field:
Question 2.
What happens to dipole if the applied electric field is nonuniform?
Answer:
In nonuniform electric field, the net force and torque acting on the dipole will not be zero. Hence the dipole undergoes for both translational and rotational motion.

Continuous Charge Distribution
Charges on a body may be distributed in different ways according to the nature of body. Depending upon this distribution of charge, we deal with different types of charge densities,

1. Line charge density, λ
2. Surface charge density, σ or
3. Volume charge density, ρ

1. Linear charge density (λ): Charge per unit length is called linear charge density. If ∆Q is the charge contained in a line element ∆l,
Linear charge density λ = \(\frac{\Delta Q}{\Delta l}\)

2. Surface charge density (σ): Charge per unit area is called surface charge density. If ∆Q is the charge contained in a area element ∆s, surface charge density can be written as
σ = \(\frac{\Delta Q}{\Delta S}\)

3. Volume charge density (ρ): Charge per unit volume is called volume change density. If ∆Q is the charge contained in a volume ∆v, volume charge density.
ρ = \(\frac{\Delta Q}{\Delta v}\)

Gauss’s Law
Gauss’s theorem states that the total electric flux over a closed surface is 1/ε₀ times the total charge enclosed by the surface.
Gauss’s theorem may be expressed

Proof:

Consider a charge +q .which is kept inside a sphere of radius ‘r’.

Important points regarding Gauss’s law:

• Gauss’s law is true for any closed surface.
• Total charge enclosed by the surface must be added (algebraically). The charge may be located anywhere inside the surface.
• The surface that we choose for the application of Gauss’s law is called the Gaussian surface.
• Gauss’s law is used to find electric field due to system of charges having some symmetry.
  Gauss’s law is based on the inverse square of distance. Any violation of Gauss’s law will indicate departure from the inverse square law.

Applications Of Gauss’s Law
Gauss’s law can be used to find electric field due to system of some symmetric charge configurations. Some examples are given below.

1. Field Due To An Infinitely Long Straight Uniformly Charged Wire:

Consider a thin infinitely long straight rod conductor having change density λ. (λ = \(\frac{q}{l}\))
To find the electric field at P, we imagine a Gaussian surface passing through P.
Then according to Gauss’s law we can write,

Integrating over the Gaussian surface, we get (we need not integrate the upper and lower surface because, electric lines do not pass through these surfaces.)

2. Field Due To A Uniformly Charged Infinite Plane Sheet

Consider an infinite thin plane sheet of change of density σ. To find electric field at a point P (at a distance ‘r’ from sheet), imagine a Gaussian surface in the form of cylinder having area of cross section ‘ds’.
According to Gauss’s law we can write,

(Since q = σds)
But electric field passes only through end surfaces ,so we get ∫ds = 2ds

E is directed away from the charged sheet, if a is positive and directed towards the sheet if a is negative.

3. Field Due To A Uniformly Charged Thin Spherical Shell: Consider a uniformly changed hollow spherical conductor of radius R. Let ‘q’ be the total charge on the surface.

To find the electric field at P (at a distance r from the centre), we imagine a Gaussian spherical surface having radius ‘r’.
Then, according to Gauss’s theorem we can write,

The electric field is constant, at a distance ‘r’. So we can write,

Case I: Electric field inside the shell is zero.
Case II: At the surface of shell r = R

Students can Download Chapter 10 Wave Optic Notes, Plus Two Physics Notes helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus Two Physics Notes Chapter 10 Wave Optic

Introduction
In 1678, the Dutch physicist Christian Huygens put forward the wave theory of light. We will discuss in this chapter.

Wavefront:
The wavefront is defined as the locus of all points which have the same phase of vibration. The rays of light are normal to the wavefront. Wavefront can be divided into 3.

1. Spherical wavefront
2. Cylindrical wavefront
3. Plane wavefront.

1. Spherical Wavefront:

The wavefront originating from a point source is spherical wavefront.

2. Cylindrical Wavefront:

If the source is linear, the wavefront is cylindrical.

3. Plane wavefront:
If the source is at infinity, we get plane wavefront.

Huygen’s Principle
According to Huygen’s principle

1. Every point in a wavefront acts as a source of secondary wavelets.
2. The secondary wavelets travel with the same velocity as the original value.
3. The envelope of all these secondary wavelets gives a new wavefront.

Refraction And Reflection Of Plane Waves Using Hygens Principle
1. Refraction of a plane wave. (To prove Snell’s law):
AB is the incident wavefront and c₁ is the velocity of the wavefront in the first medium. CD is the refracted wavefront and c₂ is the velocity of the wavefront in the second medium. AC is a plane separating the two media.

The time taken for the ray to travel from P to R is

O is an arbitrary point. Hence AO is a variable. But the time to travel a wavefront from AB to CD is constant. In order to satisfy this condition, the term containing AO in eq.(2) should be zero.

where ¹n₂ is the refractive index of the second medium w.r.t. the first. This is the law of refraction.

2. Reflection of plane wave by a plane surface:

AB is the incident wavefront and CD is the reflected wavefront, ‘i’ is the angle of incidence and ‘r’ is the angle of reflection. Let c₁ be the velocity of light in the medium. Let PO be the incident ray and OQ be the reflected ray.
The time taken for the ray to travel from P to Q is

O is an arbitrary point. Hence AO is a variable. But the time to travel for a wave front from AB to CD is a constant. So eq.(2) should be independent of AO. i.e., the term containing AO in eq.(2) should be zero. AO
∴ \(\frac{A O}{C_{1}}\)(sin i – sin r) = 0
sin i – sin r= 0
sin i = sin r
i = r
This is the law of reflection.
Behavior of wave frond as they undergo refraction or reflection.

a. Wave frond through the prism:

Consider a plane wave passing through a thin prism. The speed of light waves is less in glass. Hence the lower portion of the incoming wave frond will get delayed. So outgoing wavefrond will be tilted as shown in the figure.

b. Wave frond through a thin convex lens:

Consider a plane wave passing through a thin convex lens. The central part of the incident plane wave travels through the thickest portion of lens.

Hence central part get delayed. As a result the emerging wavefrond has a depression at the centre. Therefore the wave front becomes spherical and converges to a point F.

c. Plane wave incident on a concave mirror:

A plane wave is incident on a concave mirror and on reflection we have spherical wave converging to the focul point F.

3. The Doppler Effect:
There is an apparent change in the frequency of light when the source or observer moves with respect to one another. This phenomenon is known as Doppler effect in light.

When the source moves away from the observer the wavelength as measured by the source will be larger. The increase in wavelength due to Doppler effect is called as red shift.

When waves are received from a source moving towards the observer, there is an apparent decrease in wavelength, this is referred to as blue shift.

Mathematical expression for Doppler shift:
The Doppler shift can be expressed as

V_radial is the component of source velocity along the line joining the observer to the source.

Coherent And Incoherent Addition Of Waves
Super position principle:
According to superposition principle, the resultant displacement produced by a number of waves at a particular point in the medium is the vector sum of the displacements produced by each of the waves.

Coherent sources:
Two sources are said to be coherent, if the phase difference between the displacements produced by each of the waves does not change with time.

Incoherent sources:
Two sources are said to be coherent, if the phase difference between the displacements produced by each of the waves changes with time.

Constructive interference:
Consider two light waves meet together at a point. If we get maximum displacement at the point of meeting, we call it as constructive interference.

Destructive interference:
Consider two lightwaves meet together at a point. If we get minimum displacement at the point of meeting, we call it as destructive interference.

Mathematical condition for Constructive interference and Destructive interference:

Consider two sources S₁ and S₂. Let P be point in the region of s₁ and s₂. The displacement produced by the source s₁ at P.
y₁ = a cos ωt
Similarly, the displacement produced by the source s₂ at P
y₂ = a cos (ωt + Φ)
Where Φ is the phase difference between the displacements produced by s₁ and s₂
The resultant displacement at P,
Y = y₁ + y₂
= a cos ωt + a cos (ωt + Φ)
= a (cos ωt + cos (ωt + Φ))

Therefore total intensity at P,

Constructive interference:
If we take phase difference Φ = 0, ±2π, ±4π……., we get maximum intensity (4I₀) at P. This is the mathematical condition for constructive interference. The condition for constructive interference can be written in the form of path difference between two waves.

Where n = 0, 1, 2, 3……..

Destructive interference:
If we take phase difference Φ = ±π, ±3π, ±5π………., we get zero intensity at P. This is the mathematical condition for destructive interference. The condition for destructive interference can be written in the form of path difference between two waves.

Where n = 0, 1, 2, 3……..

Interference Of Light Waves And Youngs Double Slit Experiment
Young’s double-slit experiment:

The experiment consists of a slit ‘S’. A monochromatic light illuminates this slit. S₁ and S₂ are two slits in front of the slit ‘S’. A screen is placed at a suitable distance from S₁ and S₂. Light from S₁ and S₂ falls on the screen. On the screen interference bands can be seen.

Explanation:
If crests (ortroughs) from S₁ and S₂ meet at certain points on the screen, the interference of these points will be constructive and we get bright bands on the screen.

At certain points on the screen, crest and trough meet together. Destructive interference takes place at those points. So we get dark bands.

Expression for band width:

S₁ and S₂ are two coherent sources having wave length λ. Let ‘d’ be the distance between two coherent sources. A screen is placed at a distance D from sources. ‘O’ is a point on the screen equidistant from S₁ and S₂.
Hence the path difference, S₁O – S₂O = 0
So at ‘O’ maximum brightness is obtained.
Let ‘P’ be the position of n^th bright band at a distance x_n from O. Draw S₁A and S₂B as shown in figure. From the right angle ∆S₁AP
we get, S₁P² = S₁A² + AP²
S₁P² = D² + (X_n – d/2)² = D² + X_n² – X_nd + \(\frac{d^{4}}{4}\)
Similarly from ∆S₂BP we get,
S₂P² = S₂B² + BP²
S₂P² = D² + (X_n + d/2)²

S₂P² – S₁P² = 2x_nd
(S₂P + S₁P)(S₂P – S₁P) = 2x_nd
But S₁P ≈ S₂P ≈ D
∴ 2D(S₂P – S₁P) = 2x_nd
i.e., path difference S₂P – S₁P = \(\frac{x_{n} d}{D}\) ____(1)
But we know constructive interference takes place at P, So we can take
(S₂P – S₁P) = nλ
Hence eq(1) can be written as

Let x_n+1 be the distance of (n+1)^th bright band from centre o, then we can write

This is the width of the bright band. It is the same for the dark band also.

Diffraction
The bending of light round the comers of the obstacles is called diffraction of light.

1. The single slit diffraction:

Consider a single slit AC having length ‘a’. A screen is placed at suitable distance from slit. B is midpoint of slit, A straight line through B (perpendicular to the plane of slit), meets the screen at O. AD is perpendicular CP.

Calculation of path difference:
Consider a point P on the screen having a angle θ with normal AE. The path difference between the rays (coming from the bottom and top of the slit) reaching at P,
CP – AP = CD
(CP – AP) = a sin θ
path difference, (CP – AP) = a θ______(1)
[for small θ. sin θ ≈ θ]

(I) Position of maximum intensity:
Consider the point ‘O’, the path difference between the rays (coming from AB and BC) reaching at O is zero. Hence constructive interference takes place at ‘O’. Thus maximum intensity is obtained. This point is called central maximum or the principal maximum.

(II) Position of secondary minima:
Let P be a point on the screen such that the path difference between the rays AP and CP be λ.
ie, CP – AP = λ______(2)
Substituting eq (1) in eq (2) we get
θ = λ
(or) θ = \(\frac{\lambda}{a}\)______(3)
Let the slit AC be imagined to be split into two equal halves AB and BC. For every point in AB, there is a corresponding point in BC such hat the distance between the points are equal to a/2 Consider two points K and L such that, KL = a/2. There fore, the path difference between the rays (coming form K and L) at P is,.
LP – KP = \(\frac{a}{2}\)θ_______(4)
Substituting (3) in (4) we get

This means that the rays (coming from K and L) reaching at P are out of phase and cancel each other. Hence the intensity at P becomes zero.
In otherwards, at angle θ = \(\frac{\lambda}{\mathrm{a}}\)
The intensity becomes zero.
Similarly on the lower half of the screen, the intensity is zero for which θ = – \(\frac{\lambda}{\mathrm{a}}\)
The general equation for zero intensity can be written as
θ = \(\pm \frac{n \lambda}{a}\)
Where n = 1, 2, 3,…
For first minima n = 1, and second minima n = 2.

(III) Position of Secondary maxima:
Let P be a point on the screen, such that
CP – AP = \(\frac{3}{2}\)λ
From eq (1),we know (CP – AP) = aθ
Therefore aθ = \(\frac{3}{2}\)λ
The wave front AC can be divided into three equal parts.

The rays from first and second parts will cancel each other and the rays from third part will reach at P. Hence the point P becomes bright.

Similarly the next maximum occurs at θ = \(\frac{5}{2}\)\(\frac{λ}{a}\)
The general equation for maximum can be written
\(\theta=\pm \frac{(2 n+1) \lambda}{2 a}\)

1. (a) Intensity Distribution on the screen of diffraction pattern:

(b) Comparison between interference and diffraction bands:
Interference:

• Interference is due to superposition of waves coming from two wavefronts.
• Interference bands are of equal width.
• Minimum intensity regions are perfectly dark.
• All the bright bands are of equal intensity.

Diffraction:

• Diffraction is due to the superposition of waves coming from different parts of the same wave front.
• Diffraction bands are of unequal width.
• Minimum intensity regions are not perfectly dark.
• All bright bands are not of the same intensity.

2. Seeing The Single Slit Diffraction Pattern:

Take two razor blades and an electric bulb. Hold the two blades as shown in the figure. Observe the glowing bulb through the slit. A diffraction pattern can be seen.

3. Resolving Power Of Optical Instruments:
Resolving power of optical instrument:
The ability of an optical instrument to form distinctly separate images of the two closely placed objects is called is resolving power.

Explanation:

The image of a point object formed by a ideal lens is a point only. But because of diffraction effect, instead of point image, we get a diffraction pattern. Diffraction pattern consists of a bright central circular region surrounded by concentric dark and light rings.

Let us discuss three cases; when we observe two point object through a lens.

1. Unresolved:
If central maxima of two diffraction pattern are overlapped, the image is unresolved. This image can’t be viewed clearly.

2. Just resolved:
If central maxima of two diffraction pattern are just separated, the image is just resolved. In this case image is just distinqushed.

3. Resolved:
If central maxima of two diffraction pattern are separated, the image is resolved. This image can be viewed clearly.

Limit of resolving power of optical instrument:
The minimum distance of separation between two points so that they are just resolved by the optical instrument is known as its limit of resolution. Resolving power is also defined as reciprocal of limit of resolution.

1. Telescope and resolving power:

Telescope consist of two convex lenses called eyepiece and objective .The light falling on objective lens undergoes for diffraction. Hence a diffraction pattern of bright and dark rings is produced around central bright region as shown in figure.
The radius of central bright region,

This radius can be written in terms of angular width,
∆θ ≈ \(\frac{0.61 \lambda}{\mathrm{a}}\)
Where a is the radius and f – focal length of objective lens. λ is the wave length of light used.

This angular width of central bright region is related to resolving power of telescope. When angular width of spot increases, resolving power decreases.

The limit of resolution of telescope, ∆θ ≈ \(\frac{0.61 \lambda}{\mathrm{a}}\)
This equation shows that telescope will have better resolving power if ‘a’ is large and λ is small.

2. Microscope and resolving power:

In microscope the object (microscopic size) is placed slightly beyond f (focal length of objective lens). When the separation between two points in a microscopic specimen is comparable to the wavelength λ of light, the diffraction effect become important.

Where nsinβ is called numerical aperture, n is the refractive index of liquid used in microscope, β is the half angle of the cone of light from the microscopic object with objective lens.
The limit of resolution of microscope d_min = \(\frac{1.22 f \lambda}{2 n \sin \beta}\)
This equation also can be written as d_min = \(\frac{1.22 \lambda}{2 \tan \beta}\)

Note: Telescope is used to resolve objects at far distance but microscope is used to produce magnification of near objects.

4. The Validity Of Ray Optics:
Fresnel distance is the distance beyond which the diffraction properties becomes significant, (ie. the ray optics is converted into wave optics).
Fresnel distance, z_F = \(\frac{\mathrm{a}^{2}}{\lambda}\)
Where ‘a’ is the size of the aperture
For distances much smaller than z_F, the spreading due to diffraction is smaller compared to the size of the beam. It becomes comparable when the distance is approximately z_F. For distances much greater than z_F, the spreading due to diffraction dominates over that due to ray optics.

Polarisation

Consider a long string that is held horizontally, the other end of which is assumed to be fixed. If we move the end of the string up and down in a periodic manner, a wave will propagate in the +xdirection (see above figure). Such a wave can be described by the following equation
y(x,t) = a sin (kx – ωt)
where ‘a’ represent the amplitude and k = 2π/λ represents the wavelength associated with the wave.

Since the displacement (which is along the y-direction) is at right angles to the direction of propagation of the wave, this wave is known as a transverse wave.

Also, since the displacement is in the/direction, it is often called to as a y-polarised wave. Since each point on the string moves on a straight line, the wave is also called to as a linearly polarised wave.

The string always remains confined to the x-y plane and therefore it is also called to as a plane polarised wave.

In a similar manner we can consider the vibration of the string in the x-z plane generating a z-polarised wave whose displacement will be given by
z(x,t) = a sin (kx – ωt)

Unpolorised wave:
If the plane of vibration of the string is changed randomly in very short intervals of time, then it is known as an unpolarized wave.

(a) Polarization property of light:
When light passes through certain crystals like tourmaline, the vibrations of electric field vector are restricted. This property exhibited by light is known as polarization.

Note:

1. Polarization is the property of light which reveals that light is a transverse wave.
2. A sound wave can’t be polarized because sound wave is a longitudinal wave.

Polarizer and analyzer:
When an unpolarized light passes through a tourmaline crystal T₁, the light coming out of T₁ is plane polarized.

In order to check the polarization, another tourmaline crystal T₂ is kept parallel to T₁.

When we look through T₂ we get maximum intensity. Then T₂ is rotated through 90°. If no light is coming, we can say that light from T₁ is plane polarized.

Polarizer: The crystal which produces polarized light is known as polarizer.

Analyzer: The crystal which is used to check weather the light is polarized or not is called the analyzer or detector.

Law of Malus: This law states that when a beam of plane polarized light is incident on an analyzer, the intensity (I) of the emergent light is directly proportional to the square of the cosine of the angle (θ) between the polarizing directions of the polarizer and the analyzer.

I = I_m cos²θ
where I_m is the maximum intensity.

1. Polarisation By Scattering:

The nunpolarized light incident on a dust particle in atmosphere, it is absorbed by electrons in the dust particle. The electrons in the dust particle reradiate light in all directions. This phenomenon is called scattering.

Explanation:
Let a beam of unpolarized light be incident on a dust particle along x-axis. The electrons in the dust particle absorb light and behave as a oscillating dipole. This dipole emit light in all directions.

When an observer observe this particle along y-axis, the observer can receive light from the electron vibrating in z-axis. This light is linearly polarised in z-direction (its plane of polarisation is yz).

This polarised light is represented by dots in the picture. This explains the polarisation of scattered light from the sky.

2. Polarization By Reflection:
At a particular angle of incidence on a medium, the reflected lights is fully polarized. This angle is known as polarizing angle or Brewster’s angle. At polarizing angle, the reflected and refracted rays are mutually perpendicular.

Brewster’s law:
Brewster’s law states that the tangent of the polarizing angle is equal to the refractive index of the material of the reflector.

Let ‘Q ’ be the polarizing angle and ‘n’ be the refractive index of the medium then,
tan θ = n
At polarizing angle, r + θ =90°.

Proof:
Consider an unpolarized light coming from air and is incident on a medium having refractive index n. Let θ be the angle of incidence, Φ be the angle of reflection and ‘r’ be the angle of refraction.
Using snells law, we can write
n = \(=\frac{\sin \theta}{\sin r}\) ______(1)
At the polarizing angle reflected and refracted light are mutually perpendicular
ie. Φ – 90 + r = 180°
∴ r = 90 – Φ______(2)
Substituting eq (2) in eq(1), we get

But we know
Angle of incidence (θ) = angle of reflection(Φ)
∴ n = \(\frac{\sin \theta}{\cos \theta}\)
n = tanθ