Electric potential

Electric potential and potential energy

The electric potential V at a point is the work done per unit positive charge in bringing a test charge from infinity to that point, without acceleration. Unit: volt (1 V = 1 J/C). Potential is a scalar, so potentials from several charges simply add (with signs).

• Point charge: V = kq/r.
• Short dipole: V = kp cos θ / r²; it is zero everywhere on the equatorial line.
• System of charges: V = k(q₁/r₁ + q₂/r₂ + …).

Potential difference: the work done in moving charge q from A to B is W = q(V_B – V_A).

Relation with field: E = –dV/dr. The field points in the direction in which the potential falls fastest.

Equipotential surfaces are surfaces on which the potential is the same everywhere. No work is done in moving a charge along one. Field lines are always perpendicular to equipotential surfaces. For a point charge they are concentric spheres; for a uniform field they are parallel planes.

Potential energy of two charges: U = kq₁q₂/r. It is positive for like charges and negative for unlike charges. A dipole in a uniform field has U = –pE cos θ.

Worked example: potential at 0.5 m from 5 nC: V = 9 × 10⁹ × 5 × 10⁻⁹ / 0.5 = 90 V. Moving 2 μC through 50 V needs W = 2 × 10⁻⁶ × 50 = 1 × 10⁻⁴ J.

Key points

• V = kq/r (scalar, ∝ 1/r); compare E = kq/r² (vector).
• W = qΔV; 1 eV = 1.6 × 10⁻¹⁹ J.
• E is perpendicular to equipotential surfaces.
• U = kq₁q₂/r for a pair of charges.

Exam tips

• When asked for potential due to several charges, add numbers with signs; no vectors needed.
• “Work done along an equipotential surface” is always zero.

Common mistakes

• Using r² in the potential formula.
• Assuming V = 0 implies E = 0 (at the midpoint of a dipole V = 0 but E ≠ 0).
• Forgetting the sign of charges when finding potential energy.