Electric Field

NCERT References

1. 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

1. 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

1. 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

1. 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

1. 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

1. 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.