Learning Objectives
- Define electrostatic potential and potential difference
- Calculate potential due to various charge configurations
- Understand equipotential surfaces and their properties
- Learn about capacitors, capacitance, and their combinations
- Study energy stored in capacitors and effect of dielectrics
Key Concepts
Electric Potential
Electric potential at a point: V = W/q₀ = kQ/r (work done per unit positive charge in bringing it from infinity).
SI unit: volt (V) = J/C. It is a scalar quantity.
Potential difference: V_A - V_B = W_AB/q₀ (work done in moving unit charge from B to A).
Relation to field: E = -dV/dr. The electric field points in the direction of decreasing potential.
Potential Due to Charge Configurations
Point charge: V = kq/r
System of charges: V = k Σ(qᵢ/rᵢ) (scalar addition)
Dipole (axial): V = kp cos θ/r² (for r >> a)
Uniformly charged sphere (outside): V = kQ/r; (on surface): V = kQ/R; (inside): V = kQ(3R² - r²)/(2R³)
Equipotential Surfaces
Surfaces where potential is the same at every point. No work is done in moving a charge along an equipotential surface.
Properties: perpendicular to electric field lines, never intersect, closer together where field is stronger.
For a point charge: concentric spheres. For uniform field: parallel planes.
Electrostatic Potential Energy
Two-charge system: U = kq₁q₂/r
Three-charge system: U = k(q₁q₂/r₁₂ + q₂q₃/r₂₃ + q₁q₃/r₁₃)
Charge in external field: U = qV(r)
Capacitance
C = Q/V (charge stored per unit potential difference). SI unit: farad (F).
Parallel plate capacitor: C = ε₀A/d (A = plate area, d = separation).
With dielectric (κ): C = κε₀A/d = κC₀.
Spherical capacitor: C = 4πε₀(ab)/(b - a) (for inner radius a, outer radius b).
Cylindrical capacitor: C = 2πε₀L/ln(b/a).
Combination of Capacitors
Series: 1/C_eq = 1/C₁ + 1/C₂ + ... (same charge, voltage divides).
Parallel: C_eq = C₁ + C₂ + ... (same voltage, charge divides).
Energy Stored in a Capacitor
U = ½CV² = ½QV = Q²/(2C)
Energy density: u = ½ε₀E² (energy per unit volume in the electric field).
Effect of Dielectric
Inserting a dielectric (κ > 1) between plates:
- With battery connected: V stays same, C increases (κC₀), Q increases, E stays same
- With battery disconnected: Q stays same, C increases (κC₀), V decreases (V₀/κ), E decreases (E₀/κ)
Summary
Electric potential is a scalar quantity representing potential energy per unit charge. Equipotential surfaces are perpendicular to field lines. Capacitance measures the ability to store charge. Parallel plate capacitance depends on area, separation, and dielectric. Capacitors combine in series (reciprocal addition) and parallel (direct addition). Energy stored is ½CV². Dielectrics increase capacitance by factor κ.
Important Terms
- Electric Potential: Work done per unit charge from infinity, V = kQ/r
- Equipotential Surface: Surface of constant potential
- Capacitance: Ability to store charge, C = Q/V
- Dielectric: Insulating material that increases capacitance
- Dielectric Constant (κ): Factor by which capacitance increases with dielectric
- Energy Density: Energy stored per unit volume, u = ½ε₀E²
Quick Revision
- V = kQ/r; E = -dV/dr
- C = ε₀A/d (parallel plate); C = κε₀A/d (with dielectric)
- Series: 1/C = 1/C₁ + 1/C₂; Parallel: C = C₁ + C₂
- U = ½CV² = ½QV = Q²/2C
- Energy density: u = ½ε₀E²
- Dielectric with battery off: Q same, V ↓, C ↑; with battery on: V same, Q ↑, C ↑