Electrostatic Potential

Electric potential at point P is the work done per unit positive charge in bringing a test charge from infinity to P against the electric field. SI unit: volt (V = J C⁻¹).

Potential at distance r from a point charge q. Positive for positive charge, negative for negative charge. Zero at infinity.

Net potential at a point due to a system of charges is the algebraic (scalar) sum of individual potentials. No direction involved — simpler than field superposition.

Potential at distance r from the centre of a dipole of moment p, at angle θ from the dipole axis. Approximation valid for r ≫ a.

Potential on the axis of a dipole (θ = 0° or 180°). Positive on the +q side, negative on the −q side. Follows from the general expression with cos 0° = 1.

Potential is zero at all points on the equatorial plane (θ = 90°). The contributions from +q and −q are equal in magnitude and opposite in sign.

Electric field component along any direction is the negative rate of change of potential in that direction. The negative sign: field points from high V to low V.

Full vector relation between field and potential. The gradient operator ∇ gives the direction of maximum rate of increase of V; the negative sign reverses it to give E.

Work done by the electric field in moving charge q from A to B. Depends only on endpoints — not on the path. This is the statement that the electrostatic force is conservative.

Electrostatic potential energy stored in a system of two point charges separated by distance r. This is the work done in assembling the configuration from infinity.

Total electrostatic PE of a system of n charges — sum over all unique pairs. For 3 charges: U = U₁₂ + U₁₃ + U₂₃. Counts each pair once (i < j condition).

Potential energy of a charge q placed at a point where the external potential is V. This energy is work done by external agent in bringing q from infinity.

Capacitance is the charge stored per unit potential difference. SI unit: farad (F = C V⁻¹). A property of the conductor geometry, not of Q or V individually.

Capacitance of a parallel plate capacitor with plate area A and separation d in vacuum. Increases with area, decreases with separation.

Capacitance of two concentric spherical shells of radii a (inner) and b (outer). For isolated sphere (b → ∞): C = 4πε₀a.

Capacitance of two coaxial cylinders of length L, inner radius a, outer radius b. Derived from the field of an infinite line charge via Gauss's law.

In series, same charge Q on each capacitor, voltages add. Equivalent capacitance is always less than the smallest individual capacitance.

In parallel, same voltage across each capacitor, charges add. Equivalent capacitance is the sum of all individual capacitances.

Energy stored in a charged capacitor. Three equivalent forms depending on which pair of (Q, C, V) is known. The factor ½ arises because V builds up gradually as charge is added.

Energy stored per unit volume in an electric field. Derived from the parallel plate capacitor energy. Applies to any electric field in vacuum — not just capacitors.

Inserting a dielectric of dielectric constant K between the plates multiplies capacitance by K. K > 1 always. K = 1 for vacuum.

Capacitance when a dielectric slab of thickness t and constant K fills part of the gap d. Treated as two capacitors in series: one vacuum (d − t), one dielectric (t).