A charge of 1 µC is placed at origin and another charge of 3 µC pla...

Question

A charge of 1 µC is placed at origin and another charge of 3 µC placed at (20 m, 0, 0) in external uniform electric field of 40 V/m î, with potential at origin = 0. Find the electrical potential energy of the system.

Accepted Answer

Given:
• q₁ = 1 µC = 1 × 10⁻⁶ C at origin (0, 0, 0)
• q₂ = 3 µC = 3 × 10⁻⁶ C at (20 m, 0, 0)
• External electric field: E = 40 V/m along +X axis
• V at origin = 0

Step 1 — Potential due to External Field:
V_ext(x) = −E × x = −40x
At origin: V_ext(0) = 0 ✓
At x = 20 m: V_ext(20) = −40 × 20 = −800 V

Step 2 — Interaction Energy (Coulombic):
U₁₂ = kq₁q₂/r = (9×10⁹ × 1×10⁻⁶ × 3×10⁻⁶) / 20
= 27×10⁻³ / 20 = 1.35 × 10⁻³ J

Step 3 — Energy of q₁ in external field (at origin, V = 0):
U₁_ext = q₁ × V_ext(0) = 1×10⁻⁶ × 0 = 0

Step 4 — Energy of q₂ in external field:
U₂_ext = q₂ × V_ext(20) = 3×10⁻⁶ × (−800) = −2.4 × 10⁻³ J

Step 5 — Total Potential Energy:
U_total = U₁₂ + U₁_ext + U₂_ext
= 1.35×10⁻³ + 0 + (−2.4×10⁻³)
= −1.05 × 10⁻³ J

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