Electrostatics (Electric Field & Potential)

Topics covered: Electric charge, Coulomb’s law, electric field, Gauss’s law, dipole, electric potential and equipotential surfaces (JEE Main focus).

1. Basics of Electrostatics

Electrostatics deals with electric charges at rest. A charge at rest produces an electric field around it.

Properties of Electric Charge

• Conservation: Total charge of an isolated system remains constant.
• Quantisation: Q = ±ne, where e = 1.6 × 10⁻¹⁹ C
• Invariance: Charge is independent of reference frame.

Types of Materials

• Conductors: Large number of free electrons
• Insulators: Very few free electrons
• Semiconductors: Conductivity between conductor and insulator

2. Coulomb’s Law

Force between two point charges:

F = (1 / 4πϵ₀) × (q₁q₂ / r²)

• Attractive if q₁q₂ < 0
• Repulsive if q₁q₂ > 0

In a medium:

F = (1 / 4πϵ₀ϵᵣ) × (q₁q₂ / r²)

ϵᵣ = relative permittivity (dielectric constant)

3. Methods of Charging

• By friction: Transfer of electrons by rubbing
• By conduction: Charge redistribution on contact
• By induction: Charging without contact

4. Charge Distribution

• Linear charge density: λ = charge / length
• Surface charge density: σ = charge / area
• Volume charge density: ρ = charge / volume

5. Electric Field

Electric field intensity:

E = F / q₀

Unit: N/C

Electric Field Due to Various Charge Distributions

• Point charge: E = (1 / 4πϵ₀) × (q / r²)
• Charged ring (on axis): E = (1 / 4πϵ₀) × (qx / (x² + R²)^3/2)
• Infinite line charge: E = λ / (2πϵ₀r)
• Spherical shell:
  • Inside (r < R): E = 0
  • Outside (r ≥ R): E = (1 / 4πϵ₀) × (Q / r²)
• Solid sphere:
  • Inside: E ∝ r
  • Outside: E = (1 / 4πϵ₀) × (Q / r²)

6. Electric Dipole

A dipole consists of two equal and opposite charges separated by a small distance.

Dipole moment: p = q × d (from −q to +q)

Electric Field Due to Dipole

• Axial point: E = (1 / 4πϵ₀) × (2p / r³)
• Equatorial point: E = (1 / 4πϵ₀) × (p / r³)

Dipole in Uniform Electric Field

• Torque: τ = pE sinθ
• Potential energy: U = −pE cosθ

7. Electric Flux & Gauss’s Law

Electric flux:

Φ = ∮ E · dS

Gauss’s Law:

Φ = Q_enclosed / ϵ₀

Applications

• Charge resides on the outer surface of a conductor
• Energy density: u = ½ϵ₀E²
• Field near conducting surface: E = σ / ϵ₀
• Field near non-conducting sheet: E = σ / 2ϵ₀

8. Electric Potential

Electric potential at a point is work done per unit charge.

V = −∫ E · dr

Relation Between E and V

E = −dV/dr

9. Electric Potential Due to Charge Distributions

• Point charge: V = (1 / 4πϵ₀) × (q / r)
• Ring (on axis): V = (1 / 4πϵ₀) × (Q / √(x² + R²))
• Spherical shell:
  • Inside: V = constant
  • Outside: V = (1 / 4πϵ₀) × (Q / r)
• Solid sphere:
  • Inside: V ∝ (3R² − r²)
  • Outside: V = (1 / 4πϵ₀) × (Q / r)

Electric Potential Due to Dipole

• Axial point: V = (1 / 4πϵ₀) × (p / r²)
• Equatorial point: V = 0
• General point: V = (1 / 4πϵ₀) × (p cosθ / r²)

10. Equipotential Surfaces

• All points have same potential
• No work done along equipotential surface
• Electric field is perpendicular to equipotential surface