JEE Physics Electrostatic Potential and Capacitance Notes Set 01

Capacitance

A capacitor is a pair of conductors separated by an insulator that stores charge and energy. Its capacitance is \( C = \dfrac{Q}{V} \), where Q is the magnitude of charge on either plate and V the potential difference. It depends only on the geometry and the medium, not on Q or V. The unit is the farad (F).
1. Isolated sphere of radius R: \( C = 4\pi\varepsilon_0 R \) (times K in a medium of dielectric constant K).
2. Parallel plates of area A and separation d: \( C = \dfrac{\varepsilon_0A}{d} \).
3. Concentric spheres of radii a and b (b greater): \( C = 4\pi\varepsilon_0\dfrac{ab}{b-a} \). Cylindrical capacitor of length l: \( C = \dfrac{2\pi\varepsilon_0 l}{\ln(b/a)} \).

Energy Stored

\[ U = \tfrac12QV = \tfrac12CV^2 = \frac{Q^2}{2C} \] The energy density in the field is \( u = \tfrac12\varepsilon_0E^2 \). The work done by a battery in charging is QV, of which half is stored and half is lost as heat in the circuit, whatever the resistance.

Dielectrics

1. A dielectric of constant K multiplies the capacitance by K when it fills the gap: \( C = K\varepsilon_0A/d \).
2. A slab of thickness t in a gap d: \( C = \dfrac{\varepsilon_0A}{d - t + t/K} = \dfrac{\varepsilon_0A}{d-t(1-1/K)} \). A metal slab corresponds to K infinite, so \( C = \varepsilon_0A/(d-t) \).
3. Two layers parallel to the plates act as capacitors in series; two side-by-side halves act as capacitors in parallel.
4. Battery connected (V constant): Q rises to KQ, E stays V/d, energy rises K times.
5. Battery disconnected (Q constant): V and E fall to 1/K, energy falls to U/K.

Combination of Capacitors

1. Series: same charge on each, \( \dfrac1{C_s} = \dfrac1{C_1}+\dfrac1{C_2}+\dots \); voltage divides inversely as the capacitance. The equivalent capacitance is smaller than the smallest one.
2. Parallel: same voltage, \( C_p = C_1+C_2+\dots \); charge divides in proportion to capacitance.
3. For two capacitors in series \( C_s = \dfrac{C_1C_2}{C_1+C_2} \).
4. Maximum voltage rating: n capacitors in series withstand n times the voltage of one; to build a given capacitance at a high voltage, use rows in series and put rows in parallel.

Sharing of Charge

When two charged conductors (or capacitors) are connected, charge flows until their potentials are equal. Common potential \( V = \dfrac{C_1V_1+C_2V_2}{C_1+C_2} \) (with the signs of the charges). Charge divides in proportion to capacitance, so for two spheres \( q_1/q_2 = R_1/R_2 \), surface charge density \( \sigma\propto 1/R \) and the electric field at the surface also varies as 1/R. Energy is lost: \( \Delta U = \dfrac{C_1C_2(V_1-V_2)^2}{2(C_1+C_2)} \), which is zero only when the potentials were already equal.

Charging and Discharging

In an RC circuit the time constant is \( \tau = RC \). During charging \( q = CV(1-e^{-t/RC}) \); during discharging \( q = q_0e^{-t/RC} \). In the steady state a capacitor carries no current, so it behaves like an open circuit, and the final charge is \( C \) times the voltage across its branch.