Chapter 2: Electrostatic Potential and Capacitance

ELECTROSTATIC POTENTIAL

Basic Concept

  • Definition: Work done per unit charge to bring a test charge from infinity to a point in electric field
  • Mathematical Expression: V = W/q₀ (where q₀ is test charge)
  • Unit: Volt (V) = Joule/Coulomb (J/C)
  • Scalar Quantity: Potential is a scalar (unlike electric field which is a vector)

Potential Due to a Point Charge

\[ V = \frac{kq}{r} \]
  • k = 1/(4πε₀) = 9 × 10⁹ N·m²/C²
  • Decreases as 1/r (slower than electric field which decreases as 1/r²)
  • Positive for positive charge, negative for negative charge

Potential Due to a System of Charges

\[ V = V_1 + V_2 + V_3 + ... + V_n = k\left(\frac{q_1}{r_1} + \frac{q_2}{r_2} + ... + \frac{q_n}{r_n}\right) \]

Follows superposition principle

Potential Due to an Electric Dipole

\[ V = \frac{k p \cos\theta}{r^2} \]
  • At a point making angle θ with dipole axis
  • p = dipole moment
  • At large distances, potential decreases as 1/r²

Potential Difference

\[ \Delta V = V_2 - V_1 = -\int_1^2 \vec{E} \cdot d\vec{l} \]

Work done = q₀ΔV

EQUIPOTENTIAL SURFACES

Properties

  • Surface where potential is constant
  • Electric field lines are perpendicular to equipotential surfaces
  • No work done in moving a charge along an equipotential surface
  • For point charge: Equipotential surfaces are concentric spheres
  • For uniform field: Equipotential surfaces are parallel planes

Relation Between Field and Potential

\[ \vec{E} = -\nabla V \]

Electric field is negative gradient of potential

In one dimension: E = -dV/dx

POTENTIAL ENERGY

Potential Energy of a System of Charges

\[ U = \frac{1}{2}\sum_i q_i V_i \]

Where Vi is potential at location of qi due to all other charges

For two charges: U = kq₁q₂/r₁₂

Potential Energy in External Field

  • For single charge: U = qV
  • For dipole in uniform field: U = -p·E = -pE·cosθ

ELECTROSTATICS OF CONDUCTORS

Properties of Conductors in Electrostatic Equilibrium

  • Electric field inside is zero
  • Charge resides only on the surface
  • Electric field just outside is perpendicular to surface
  • Surface is an equipotential
  • Electric field just outside = σ/ε₀ (where σ is surface charge density)

DIELECTRICS AND POLARIZATION

Dielectrics

  • Definition: Materials with bound charges, poor conductors of electricity
  • Examples: Glass, rubber, plastic, water
  • Polarization: Alignment of dipoles in electric field
  • Dielectric Constant (K): Measure of material's ability to store electrical energy
    • K = ε/ε₀ (where ε is permittivity of dielectric)
    • Air ≈ 1, Water ≈ 80, Paper ≈ 3.5

Effect of Dielectric

  • Reduces electric field: E = E₀/K
  • Increases capacitance: C = KC₀

CAPACITORS

Definition and Capacitance

  • Capacitor: Device that stores electric charge
  • Capacitance: Charge stored per unit potential difference
  • C = Q/V
  • Unit: Farad (F) = Coulomb/Volt (C/V)

Parallel Plate Capacitor

\[ C = \frac{\varepsilon_0 A}{d} \]
  • A = plate area, d = separation between plates
  • With dielectric: C = Kε₀A/d

Capacitors in Series

\[ \frac{1}{C} = \frac{1}{C_1} + \frac{1}{C_2} + ... + \frac{1}{C_n} \]
  • Equivalent capacitance is less than the smallest individual capacitance
  • Charge is same on each capacitor

Capacitors in Parallel

\[ C = C_1 + C_2 + ... + C_n \]
  • Potential difference is same across each capacitor

ENERGY STORED IN A CAPACITOR

Energy Formulas

\[ U = \frac{1}{2}QV = \frac{1}{2}CV^2 = \frac{Q^2}{2C} \]

Energy Density

  • Energy per unit volume in electric field
  • u = (1/2)ε₀E²
  • With dielectric: u = (1/2)εE² = (1/2)Kε₀E²

KEY FORMULAS

  • Potential: V = kq/r
  • Potential due to dipole: V = kp·cosθ/r²
  • Capacitance: C = Q/V
  • Parallel plate capacitor: C = ε₀A/d
  • With dielectric: C = Kε₀A/d
  • Energy stored: U = (1/2)CV²
  • Capacitors in series: 1/C = 1/C₁ + 1/C₂ + ... + 1/Cn
  • Capacitors in parallel: C = C₁ + C₂ + ... + Cn