Worked example: A dipole with p = 4.0 × 10⁻²⁹ C·m sits in E = 2.0 × 10⁶ N/C. (a) Find U at θ = 0°, 90°, 180°. (b) How much work must an external agent do to rotate it from 0° to 180°? Step 1: pE = (4.0 × 10⁻²⁹)(2.0 × 10⁶) = 8.0 × 10⁻²³ J. U = −pE·cosθ. Step 2: θ = 0°: U = −8.0 × 10⁻²³ J (minimum, stable). θ = 90°: U = 0. θ = 180°: U = +8.0 × 10⁻²³ J (maximum, unstable). Step 3: External work = ΔU = U_final − U_initial = 8.0 × 10⁻²³ − (−8.0 × 10⁻²³) = 1.6 × 10⁻²² J. The agent must supply this; the field opposes the rotation the whole way. Step 4 (check via torque): W = ∫τ·dθ = ∫₀^π pE·sinθ·dθ = 2pE = 1.6 × 10⁻²² J. Matches — energy and torque tell the same story.