Electrostatic Potential and Capacitance - Quick Revision
Potential and potential energy
- Electric potential V at a point = work done per unit positive charge to bring it from infinity to that point. Unit: volt (V = J/C). Potential is a scalar.
- Point charge: V = kq/r (k = 9 x 10^9). Positive for q > 0, negative for q < 0.
- System of charges: V is the algebraic sum, V = k * sum(q_i / r_i).
- Short dipole: V = k p cos(theta)/r^2; zero in the equatorial plane (theta = 90 deg), falls off as 1/r^2.
- Potential energy of two charges: U = k q1 q2 / r12 (positive for like, negative for unlike charges).
- Charge in external field: U = qV. Dipole in field: U = -p.E = -pE cos(theta).
- 1 eV = 1.6 x 10^-19 J is the energy gained by an electron across 1 V.
Field-potential relation and equipotentials
- E = -dV/dr; field points in the direction of steepest decrease of potential.
- An equipotential surface has constant V; no work is done moving a charge on it, and E is always normal to it.
Conductors and dielectrics
- Inside a conductor E = 0; excess charge resides on the surface; the whole conductor is at one potential. Just outside, E = sigma/eps0 (normal). A cavity is shielded (E = 0 inside).
- Dielectrics polarise in a field, reducing the net field, hence raising capacitance.
Capacitance
- C = Q/V; unit farad (F). Parallel plate (vacuum): C0 = eps0 A/d (eps0 = 8.85 x 10^-12 F/m).
- Dielectric: C = K C0 with K > 1.
- Series: 1/C = sum(1/C_i) (smaller). Parallel: C = sum(C_i) (larger).
- Energy stored: U = (1/2)CV^2 = (1/2)QV = Q^2/(2C). Energy density: u = (1/2) eps0 E^2.