### Notes from Toppers

## Electric Field

**NCERT References**

*NCERT Class 12, Physics, Part-I, Chapter 1: Electric Charges and Fields, pages 1-7, 13-17.*

**Definition:**The electric field at a point is defined as the force experienced by a positive test charge placed at that point divided by the magnitude of the test charge.**Mathematical Representation:**$$ \mathbf{E} = \frac{\mathbf{F}}{q} $$- $\mathbf{E}$ is the electric field vector.
- $\mathbf{F}$ is the force vector experienced by the test charge.
- $q$ is the magnitude of the test charge.

**Electric Field Due to Point Charge:**The electric field due to a point charge`Q`

at a distance`r`

from the charge is given by: $$ \mathbf{E} = \frac{1}{4\pi\epsilon_0} \frac{Q}{r^2} \hat{r} $$- $\epsilon_0$ is the permittivity of free space.
- $\hat{r}$ is the unit vector pointing from the point charge to the observation point.

**Electric Field Due to Multiple Charges:**The electric field due to multiple charges is the vector sum of the electric fields due to each individual charge.**Electric Field Due to Continuous Charge Distributions:**The electric field due to a continuous charge distribution is obtained by integrating the electric field due to each small charge element.**Electric Field Lines:**Electric field lines are imaginary lines drawn in such a way that the tangent at any point on a line gives the direction of the electric field at that point.

## Electric Potential

**NCERT References**

*NCERT Class 12, Physics, Part-I, Chapter 2: Electrostatic Potential and Capacitance, pages 8-11.*

**Definition:**Electric potential at a point is defined as the amount of work done in bringing a positive test charge from infinity to that point.**Mathematical Representation:**$$ \phi = \frac{W}{q} $$- $\phi$ is the electric potential.
- $W$ is the work done in bringing the test charge from infinity to that point.
- $q$ is the magnitude of the test charge.

**Relation between Electric Potential and Electric Field:**$$ \mathbf{E} = -\nabla \phi $$- $\nabla$ is the gradient operator.

**Electric Potential Due to Point Charge:**The electric potential due to a point charge`Q`

at a distance`r`

from the charge is given by: $$ \phi = \frac{1}{4\pi\epsilon_0} \frac{Q}{r} $$**Electric Potential Due to Multiple Charges:**The electric potential due to multiple charges is the algebraic sum of the electric potentials due to each individual charge.**Electric Potential Due to Continuous Charge Distributions:**The electric potential due to a continuous charge distribution is obtained by integrating the electric potential due to each small charge element.**Potential Difference:**The potential difference between two points is the difference in their electric potentials.

## Gauss’s Law

**NCERT References**

*NCERT Class 12, Physics, Part-I, Chapter 4: Moving Charges and Magnetism, pages 25-28.*

**Statement:**Gauss’s law states that the net electric flux through any closed surface is equal to the total charge enclosed by that surface.**Mathematical Form:**$$\oint \mathbf{E}\cdot \hat{n} dA = \frac{Q_{in}}{\epsilon_0} $$- $\epsilon_0$ is the permittivity of free space.
- $Q_{in}$ is the total charge enclosed by the surface.
- $\hat{n}$ is the unit normal vector to the surface.
- The integral is taken over the entire closed surface.

**Applications of Gauss’s Law:**- Calculating electric field due to symmetric charge distributions.
- Proving the inverse square law for electric field.

**Gauss’s Law in Differential Form:**$$\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0} $$- $\rho$ is the charge density.

## Electric Flux

**NCERT References**

*NCERT Class 12, Physics, Part-I, Chapter 2: Electrostatic Potential and Capacitance, pages 11-14.*

**Definition:**Electric flux through a surface is defined as the net amount of electric field passing through that surface.**Relation between Electric Flux and Electric Field:**$$\Phi_E = \mathbf{E}\cdot \hat{n} dA $$- $\Phi_E$ is the electric flux.
- $\mathbf{E}$ is the electric field vector.
- $\hat{n}$ is the unit normal vector to the surface.
- $dA$ is the differential area of the surface.

**Gauss’s Law as an Integral Form of Electric Flux:**$$\oint \mathbf{E}\cdot \hat{n} dA = \frac{Q_{in}}{\epsilon_0} $$- $Q_{in}$ is the total charge enclosed by the surface.
- $\epsilon_0$ is the permittivity of free space.

## Capacitors

**NCERT References**

*NCERT Class 12, Physics, Part-II, Chapter 3: Current Electricity, pages 145-154.*

**Definition:**A capacitor is a device that stores electrical energy in an electric field.**Types of Capacitors:**- Parallel-plate capacitor
- Cylindrical capacitor
- Spherical capacitor

**Capacitance of a Parallel-Plate Capacitor:**$$C = \frac{\epsilon_0 A}{d} $$- $C$ is the capacitance of the capacitor.
- $\epsilon_0$ is the permittivity of free space.
- $A$ is the area of each plate.
- $d$ is the distance between the plates.

**Dependence of Capacitance on Various Factors:**- Area of the plates
- Distance between the plates
- Permittivity of the medium between the plates

**Energy Stored in a Capacitor:**$$U=\frac{1}{2} CV^2 $$- $U$ is the energy stored in the capacitor.
- $C$ is the capacitance of the capacitor.
- $V$ is the potential difference across the capacitor.

## Dielectrics

**NCERT References**

*NCERT Class 12, Physics, Part-II, Chapter 3: Current Electricity, pages 154-157.*

**Definition:**A dielectric is an insulating material that can be polarised by an electric field.**Properties of Dielectrics:**- Low electrical conductivity
- High resistivity
- High dielectric constant

**Polarisation of Dielectrics:**Dielectrics get polarised when placed in an electric field.**Effect of Dielectrics on Capacitance:**The presence of a dielectric between the plates of a capacitor increases its capacitance.