Electric charges and fields class 12 notes

Basic Properties of Electric Charge

Electric charge is a fundamental property of matter. It exists in two forms, positive and negative, and is quantized, meaning it can only exist in discrete amounts. The smallest unit of charge is the charge of an electron, denoted by `\( e \)`, which has a magnitude of approximately `\( 1.602 \times 10^{-19} \)` coulombs (C). Charges of the same sign repel each other, while charges of opposite signs attract. Charge is always conserved. The magnitude of the charge doesn't change with its speed.

Coulomb's Law

Coulomb's law describes the force between two point charges. It states that the magnitude of the electrostatic force `\( F \)` between two point charges `\( q_1 \)` and `\( q_2 \)` separated by a distance `\( r \)` in a vacuum is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. Mathematically, it is expressed as:

`\[F = \frac{k \cdot |q_1 \cdot q_2|}{r^2}\]`

Where:

- `\( F \)` is the magnitude of the electrostatic force,

- `\( q_1 \)` and `\( q_2 \)` are the magnitudes of the charges,

- `\( r \)` is the distance between the charges, and

- `\( k \)` is Coulomb's constant, approximately `\( 8.9875 \times 10^9 \)` Nm²/C².

Forces Between Multiple Charges

When dealing with multiple charges, the total force on a given charge is the vector sum of the forces exerted by all other charges. This principle is based on the superposition principle, which states that the net force on a charge due to a system of charges is the vector sum of the forces exerted by each individual charge.

Mathematically, for `\( n \)` number of charges, the total force `\( F_{total} \)` on a charge `\( q_i \)` due to all other charges is given by:

`\[F_{total} = \sum_{j=1, j \neq i}^{n} \frac{k \cdot |q_i \cdot q_j|}{r_{ij}^2}\]`

Where:

- `\( r_{ij} \)` is the distance between charges `\( q_i \)` and `\( q_j \)`.

Electric Field

The electric field `\( E \)` at a point in space is a vector quantity that describes the force experienced by a positive test charge placed at that point. It is defined as the force per unit positive charge. Mathematically, it is expressed as:

`\[\mathbf{E} = \frac{\mathbf{F}}{q_0}\]`

Where:

- `\( \mathbf{F} \)` is the force experienced by the test charge `\( q_0 \)`, and

- `\( \mathbf{E} \)` is the electric field vector.

Electric Field Lines

Electric field lines are a visual representation of the electric field. They are directed away from positively charged objects and toward negatively charged objects. The density of electric field lines indicates the strength of the electric field: the closer the lines, the stronger the field.

The properties of electric field lines are as follows:

1. They start on positive charges and end on negative charges.

2. They never intersect.

3. They are more concentrated where the electric field is stronger.

Electric Flux

Electric flux is a measure of the electric field passing through a surface. It is defined as the product of the electric field `\( E \)` and the area `\( A \)` of the surface, multiplied by the cosine of the angle between the electric field and the surface normal. Mathematically, it is expressed as:

`\[\Phi = \mathbf{E} \cdot \mathbf{A} = E \cdot A \cdot \cos(\theta)\]`

Where:

- `\( \Phi \)` is the electric flux,

- `\( \mathbf{E} \)` is the electric field vector,

- `\( \mathbf{A} \)` is the area vector of the surface,

- `\( E \)` is the magnitude of the electric field,

- `\( A \)` is the magnitude of the area, and

- `\( \theta \)` is the angle between `\( \mathbf{E} \)` and `\( \mathbf{A} \)`.

Electric Dipole

An electric dipole consists of two equal and opposite point charges separated by a distance `\( d \)`. The electric dipole moment `\( p \)` is defined as the product of one of the charges and the separation between them. Mathematically, it is expressed as:

`\[\mathbf{p} = q \cdot \mathbf{d}\]`

Where:

- `\( \mathbf{p} \)` is the electric dipole moment,

- `\( q \)` is the magnitude of one of the charges, and

- `\( \mathbf{d} \)` is the displacement vector separating the charges.

Dipole in a Uniform External Field

When an electric dipole is placed in a uniform external electric field, it experiences a torque that aligns it with the field. The torque `\( \tau \)` acting on the dipole is given by:

`\[\mathbf{\tau} = \mathbf{p} \times \mathbf{E}\]`

Where:

- `\( \mathbf{p} \)` is the electric dipole moment, and

- `\( \mathbf{E} \)` is the external electric field.

The dipole tends to align itself such that its dipole moment vector points in the same direction as the external field.

Continuous Charge Distribution

In many practical situations, the charge is distributed continuously over a region rather than concentrated at discrete points. Continuous charge distributions are described using charge density `\( \rho \)`, which can be either volumetric, surface, or linear.

For a volumetric charge distribution, the total charge `\( Q \)` within a volume `\( V \)` is given by the integral of the charge density over the volume:

`\[Q = \int_V \rho \, dV\]`

Similarly, for surface charge distribution, the total charge `\( Q \)` on a surface `\( S \)` is given by the integral of the surface charge density over the surface:

`\[Q = \int_S \sigma \, dA\]`

And for linear charge distribution, the total charge `\( Q \)` along a line `\( L \)` is given by the integral of the linear charge density over the line:

`\[Q = \int_L \lambda \, dl\]`

Where:

- `\( \rho \)` is the charge density (charge per unit volume),

- `\( \sigma \)` is the surface charge density (charge per unit area),

- `\( \lambda \)` is the linear charge density (charge per unit length), and

- `\( dV \)`, `\( dA \)`, `\( dl \)` are differential elements of volume, area, and length, respectively.

Gauss's Law

Gauss's law relates the electric flux through a closed surface to the total charge enclosed by that surface. It states that the electric flux `\( \Phi \)` through a closed surface `\( S \)` is equal to the total charge `\( Q_{\text{enc}} \)` enclosed by the surface divided by the permittivity of the medium `\( \varepsilon_0 \)`. Mathematically, it is expressed as:


`\[\Phi = \frac{Q_{\text{enc}}}{\varepsilon_0}\]`


Where:
- `\( \Phi \)` is the electric flux,
- `\( Q_{\text{enc}} \)` is the total charge enclosed by the surface, and
- `\( \varepsilon_0 \)` is the permittivity of the medium.


This law is particularly useful for calculating electric fields in situations with high symmetry, as it provides a convenient method for determining electric fields without directly integrating over-charge distributions.

Applications of Gauss's Law

Gauss's law finds applications in various areas of physics and engineering. Some of the key applications include:

1. Calculating Electric Fields: 

Gauss's law provides a method for calculating electric fields due to highly symmetric charge distributions, such as uniformly charged spheres, cylinders, and planes. By choosing appropriate Gaussian surfaces, one can simplify the calculation of electric fields.

2. Charged Conductors: 

Gauss's law is used to analyze the behavior of electric fields inside conductors. Inside a conductor at electrostatic equilibrium, the electric field is zero, and Gauss's law can be used to determine the charge distribution on the surface of the conductor.

3. Electric Flux Density: 

In the context of materials with dielectric properties, Gauss's law is used to define the electric flux density `\( D \)`. It relates the electric flux to the free charge density `\( \rho_f \)` and the bound charge density `\( \rho_b \)` inside a material:


`\[\Phi = \oint_S \mathbf{D} \cdot d\mathbf{A} = Q_{\text{free}} = \int_V \rho_f \, dV\]`

4. Electrostatic Shielding: 

Gauss's law is applied to analyze the effectiveness of conducting shields in protecting sensitive electronic devices from external electric fields. Conducting shields can redistribute charges to cancel out the external electric field within the shielded region.

5. Charged Dielectrics: 

Gauss's law is used to analyze the behavior of electric fields inside dielectric materials. It helps in understanding polarization effects and the redistribution of charges in response to an applied electric field.


Gauss's law is a powerful tool in electrostatics, providing insights into the behavior of electric fields and charge distributions in various systems and materials.