Practice question

**Relation E = -∇V** shows field points down potential gradient. For system of opposite charges close together, equipotential near midpoint between them has V≈0, but shape distorted, not spherical, reflecting superposition of potentials V = k q₁/r₁ + k q₂/r₂. U = (1/4 π ε₀) (q₁ q₂/r) = 9 × 10⁹ × (6 × 10⁻⁶ × (-3 × 10⁻⁶)/0.12) = 9 × 10⁹ × (-18 × 10⁻¹²/0.12) = -1.35 J . Using V = kQ/r, U = k q₁q₂/r, E = -dV/dr, C = ε₀A/d, 1/C_eq series, C_eq = ΣC parallel, U = ½ C V² and common potential V = Q_total/C_total, result -1.35 J follows, reflecting potential-capacitance relations.