Electrostatic Potential, Capacitance & Energy

Electric potential at a point is the work done per unit positive charge in bringing a test charge from infinity to that point against the electric field. V = W/$q_{0}$ = -integral(E . dr) from infinity to the point. SI unit: volt (V) = J/C. Dimensional formula: [M $L^{2}$ T^(-3) A^(-1)]. Potential is a scalar quantity — this makes it easier to calculate than the vector electric field.

Potential Due to a Point Charge: V = kQ/r. This is positive for positive Q, negative for negative Q, and zero at infinity. Unlike E (which is a vector), potential contributions from multiple charges add algebraically: $V_{total}$ = k*sum($\frac{Q_{i}}{r_{i}}$).

Potential Due to Standard Configurations:

• Uniform ring on axis: V = kQ/$\sqrt{x^{2} + R^{2}}$
• Uniform sphere (outside, r >= R): V = kQ/r
• Uniform sphere (inside, r < R): V = kQ($3R^{2}$ - $r^{2}$)/($2R^{3}$) (solid); V = kQ/R (hollow, constant)
• Infinite line charge: V = -($\frac{\lambda}{2}$$\pi$$\epsilon_{0}$)*ln(r) + C (reference needed; potential is not zero at infinity)
• Dipole at (r, $\theta$): V = kp*cos($\theta$)/$r^{2}$ (for r >> a)

Equipotential Surfaces: Surfaces where V is constant. Properties: (1) E is always perpendicular to equipotential surfaces; (2) No work is done moving a charge along an equipotential surface; (3) Equipotential surfaces never intersect; (4) Conductor surface is always equipotential in equilibrium. For a point charge, equipotentials are concentric spheres. For a uniform field, they are parallel planes.

Relationship Between E and V: E = -dV/dr (in one dimension) or E = -grad(V) = -(dV/dx $x_{hat}$ + dV/dy $y_{hat}$ + dV/dz $z_{hat}$). The electric field points in the direction of decreasing potential. The magnitude of E equals the rate of change of V with distance.

Electrostatic Potential Energy: For a system of charges, U = sum over all pairs of k*$q_{i}$*$\frac{q_{j}}{r_{ij}}$. For two charges: U = kq1q2/r. For three charges: U = kq1q2/r12 + kq1q3/r13 + kq2q3/r23. The energy is positive for like charges (work done against repulsion) and negative for unlike charges (work done against attraction). The number of pair terms for N charges = N(N-1)/2.

Capacitance: A capacitor stores charge and energy. Capacitance C = Q/V, where Q is the charge on one plate and V is the potential difference. SI unit: farad (F) = C/V. Dimensional formula: [M^(-1) L^(-2) $T^{4} \; A^{2}$]. 1 F is enormous; practical capacitors use uF, nF, pF.

Parallel Plate Capacitor: C = $\epsilon_{0}$A/d, where A is plate area and d is separation. With dielectric of constant K: C = K$\epsilon_{0}$*A/d. The field between plates: E = $\frac{\sigma}{\epsilon_{0}}$ = V/d.

Other Capacitor Geometries:

• Spherical capacitor (radii a, b): C = 4*$\pi$*$\epsilon_{0}$*ab/(b-a)
• Cylindrical capacitor (radii a, b, length L): C = 2*$\pi$*$\epsilon_{0}$*L/ln(b/a)
• Isolated sphere of radius R: C = 4*$\pi$*$\epsilon_{0}$*R

Combination of Capacitors:

• Series: 1/$C_{eq}$ = 1/C₁ + 1/C₂ + ... (charge same, voltage adds)
• Parallel: $C_{eq}$ = C₁ + C₂ + ... (voltage same, charge adds)

Energy Stored in a Capacitor: U = ($\frac{1}{2}$)$CV^{2}$ = ($\frac{1}{2}$)QV = $Q^{2}$/(2C).
Energy density (per unit volume) in electric field: u = ($\frac{1}{2}$)$\epsilon_{0}$$E^{2}$. SI unit: J/$m^{3}$.

Dielectrics: When a dielectric (K > 1) is inserted: (1) With battery connected: V stays constant, C increases by K, Q increases by K, E stays constant, U increases by K. (2) With battery disconnected: Q stays constant, C increases by K, V decreases by K, E decreases by K, U decreases by K.