Electrostatic Potential & Capacitance— Notes, Formulas & Revision

Electric potential V at a point is the work done per unit charge in bringing a +ve test charge from infinity to that point.

For a point charge: V = kq/r. Scalar quantity, signed.

V is positive near +q, negative near −q. SI unit: volt = J/C.

V = kq/r and E = kq/r². Both fall, but V slower than E.

E = −dV/dr — field is negative gradient of potential.

Plot V vs r: hyperbola decreasing from infinity at r=0 to 0 at r=∞.

An equipotential surface has the same V at every point.

Field lines are perpendicular to equipotential surfaces.

Spacing of equipotentials is related to field strength: closer surfaces → larger E.

Field is the negative gradient of potential: E⃗ = −∇V.

In 1D: E_x = −dV/dx. Steeper V slope → stronger field.

If V = const, E = 0.

Potential due to a short dipole at distance r and angle θ from axis: V = kp cosθ / r².

On the axis (θ=0): V = kp/r² (positive on +q side).

On the equator (θ=90°): V = 0.

Work done by external agent moving charge q from A to B: W = q(V_B − V_A).

Path-independent (electrostatic field is conservative).

If V_B > V_A and q > 0: positive work needed (field opposes motion).

Capacitance C = Q/V, measure of charge stored per volt.

Parallel plates: C = ε₀A/d (vacuum). Add dielectric κ: C = κε₀A/d.

Energy stored U = ½CV² = Q²/(2C).

Inserting a dielectric of relative permittivity κ multiplies capacitance: C' = κC₀.

If battery stays connected (V fixed): Q increases by factor κ.

If battery disconnected (Q fixed): V drops by factor κ; energy U drops by κ.

Charging a capacitor stores energy in the electric field.

U = ½CV² = ½QV = Q²/(2C). All three forms are equivalent.

Energy density u = ½ε₀E² (in vacuum) — energy lives in the field, not the plates.

Charging through R: q(t) = CV(1 − e^(−t/τ)), τ = RC.

At t = τ: 63% of final charge. At 5τ: ≈ 99%.

Current decays as I = (V/R)e^(−t/τ).

Discharging: q(t) = Q₀e^(−t/τ).

At τ: 37% of original charge remains.

Energy dissipated through R = ½CV² (all stored energy).

In series: same charge Q on each capacitor (charge conservation).

Voltages add: V_total = ΣVᵢ.

1/C_eq = Σ1/Cᵢ. Equivalent capacitance is smaller than the smallest.

In parallel: same V across each capacitor.

Charges add: Q_total = ΣQᵢ.

C_eq = ΣCᵢ. Total is sum of all.

Multi-vane gang capacitor: capacitance varies as overlap area changes.

C(θ) ∝ |cosθ| for half-circle vanes — used in tuned LC radio receivers.

Allows manual tuning of resonant frequency f₀ = 1/(2π√(LC)).

When a dielectric slab is partially inserted, it gets pulled in (force F = ε₀bV²(κ−1)/(2d)).

This force is independent of insertion length x.

Capacitance increases linearly with x as dielectric fills in.

Plates of opposite charge attract: F = Q²/(2ε₀A) = ½ε₀E²A.

Always attractive, regardless of which plate is +.

Note the factor ½ — only ONE plate's field acts on the other (its own field can't push it).

Each dielectric has a maximum field E_max it can withstand before ionising and conducting.

Breakdown produces a spark (arc) — useful in spark plugs, harmful in capacitors.

Air: ~3 MV/m, mica: ~100 MV/m, glass: ~14 MV/m.

Connected battery: V fixed. Insert dielectric → Q increases (battery does work).

Battery removed: Q fixed. Insert dielectric → V drops by κ; U drops by κ.

These are two distinct physical scenarios with different energy bookkeeping.