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UY1: Electric Potential Energy

Connect electric work and potential energy in uniform fields and point-charge systems, with careful sign handling and reference choices.

  • University Physics Year 1
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Learning objectives

  • Construct electric-field and potential models for discrete and continuous charge distributions.
Why this matters + quick links

This page gives the UY1 working model/result for Electric Potential Energy. You reuse it when you build fields/potentials by symmetry or superposition, and when you connect fields to forces, energy, and circuits.

1) At a glance

  • Prerequisites: force between charges (Coulomb’s Law) and the meaning of work/energy (Concept of Work)
  • Outcomes: compute Δ U and W_field with correct sign, and use U(r) = kq₁q₂/r with a stated reference
  • Key results (electrostatics):
W_field = -Δ U, U(r) = (1/4πε₀)q₁q₂/r (U(∞) = 0), Δ U = q Δ V
  • Common trap: mixing up “work done by the field” with “work done by an external agent” or forgetting the sign of q₁q₂
  • Electric force from static charges is conservative.
  • Opposite-sign pairs have negative potential energy (bound state idea).

Motivation / intuition

Energy methods let you answer “how much work?” without tracking force direction at every point along a path. In electrostatics, you can rank configurations (stable vs unstable) just by the sign and size of U.

2) Setup

  • Decide system first (which charges are included).
  • State reference for potential energy (usually U(∞) = 0).
  • Keep signs explicit:
  • If Δ U < 0, field did positive work.
  • If Δ U > 0, external agent must supply work.

3) Core derivation/explanation

General relation:

Δ U = U_b-Uₐ = -∫ₐ^b vector F · d vector l

For two point charges, radial force magnitude is

F(r) = (1/4πε₀)|q₁q₂|/r²

Using signed product in the integral and U(∞) = 0:

U(r) = (1/4πε₀)q₁q₂/r

Interpretation:

  • q₁q₂ > 0 ⇒ U > 0 (repulsive pair, energy stored by bringing together).
  • q₁q₂ < 0 ⇒ U < 0 (attractive pair, energy released when assembling).

In uniform field, potential energy change of charge q is

Δ U = qΔ V

with the same sign logic.

Quick checks (units + limits/sign)
  • Units: [U] = [kq₁q₂/r] = (N m² C⁻²)C²/m = N m = J.
  • Limits/signs: with the standard reference U(∞) = 0, you must have U → 0 as r → ∞; if q₁q₂ < 0, then U(r) < 0 and bringing charges closer (smaller r) releases energy.

4) Worked example(s)

Example A: Two point charges

q₁ = +2.0 μC, q₂ = -3.0 μC, separation r = 0.50 m.

U = ((8.99 × 10⁹)(2.0 × 10⁻⁶)(-3.0 × 10⁻⁶))/0.50 = -1.08 × 10⁻¹ J

Negative value indicates bound attractive configuration.

Example B: From potential difference

A charge q = +2.0 nC moves through Δ V = -120 V.

Δ U = qΔ V = (2.0 × 10⁻⁹)(-120) = -2.4 × 10⁻⁷ J

So field does

W_field = -Δ U = +2.4 × 10⁻⁷ J

5) Practice set (with hints + answers)

  1. Charges + 1 μC and + 1 μC are 0.20 m apart. Find U. Hint: U = kq₁q₂/r. Answer: U ≈ 4.50 × 10⁻² J.

  2. A charge q = -5.0 nC moves through Δ V = +40 V. Find Δ U. Hint: multiply qΔ V with signs. Answer: Δ U = -2.0 × 10⁻⁷ J.

  3. If W_field < 0, is potential energy increasing or decreasing? Hint: use W_field = -Δ U. Answer: increasing (Δ U > 0).

6) Summary + next steps

  • Electric potential energy is a system quantity, not a single-particle property alone.
  • Reference choice and signs determine physical interpretation.
  • The formulas here transition directly to electric potential by dividing by charge.

Next: Electric Potential Energy With Several Point Charges Previous: Electric Field Of Two Oppositely Charged Infinite Sheets Back To Electromagnetism