Electrostatic Potential and Capacitance
What is examined
Electrostatic Potential and Capacitance tests potential as a scalar, the relation between E and V, equipotentials, capacitance of parallel-plate and spherical conductors, dielectrics, combinations of capacitors and stored energy.
Core rules
- V = W/q₀; potential due to a point charge V = kq/r, with sign. Potential is scalar, so add algebraically.
- E = −dV/dr. Field points from high to low potential; equipotential surfaces are perpendicular to field lines. Work done in moving a charge on an equipotential is zero.
- Potential energy of a dipole U = −pE cos θ; of two charges U = kq₁q₂/r.
- Parallel-plate capacitor C = ε₀A/d; with a dielectric slab of constant K filling the gap, C = Kε₀A/d.
- Series: 1/C = Σ1/Cᵢ, same charge on each. Parallel: C = ΣCᵢ, same voltage.
- Energy U = ½CV² = ½QV = Q²/2C; energy density u = ½ε₀E².
- Isolated charged capacitor: Q constant when a slab is inserted, so V falls and C rises. Battery connected: V constant, so Q rises.
Worked example
A 2 μF and a 3 μF capacitor are in series across 10 V. Find charge and each voltage. C_eq = (2×3)/(2+3) = 1.2 μF. Q = C_eq V = 1.2×10⁻⁶ × 10 = 12 μC. V₂ = Q/C₂ = 12/2 = 6 V; V₃ = 12/3 = 4 V. Check: 6 + 4 = 10 V. Reasoning: series capacitors carry equal charge; the smaller capacitor takes the larger voltage.
Traps
Treating potential as a vector; mixing series and parallel rules; forgetting to convert μF to F; using V constant when the battery is disconnected; and ignoring the sign of q in V = kq/r.