Electric Potential

We have met the electric field, which measures the force per unit charge at a point in space. In this lesson we meet its partner, the electric potential, which measures the potential energy per unit charge. The two are related in exactly the same way as g and V_g in the gravitational case. OCR places particular emphasis on the analogy between electric and gravitational fields, and this lesson is the cleanest place to draw it.

This lesson continues OCR H556 Module 6.2.

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Work Done Moving a Charge

Move a small positive test charge q from point A to point B in an electric field. The field exerts a force on it, and so work is done either by the field (if the charge moves in the direction of the force) or against the field (if it moves in the opposite direction).

We define the potential difference between A and B as the work done per unit charge in moving the test charge from A to B:

V_B − V_A = W_{A→B} / q

The electric potential at a point is the work done per unit positive charge in bringing a small test charge from infinity to that point:

V = W_∞ / Q

Unit: volt (V) = joule per coulomb. This is the same volt you have used for e.m.f., potential difference and battery voltage — the concept runs right through GCSE electricity into A-Level electrostatics. A 1.5 V battery gives 1.5 J to every coulomb that passes through it.

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Potential Around a Point Charge

For a single point charge Q, the electric potential at distance r is

V = Q / (4πε₀r)

Note the 1/r dependence, not 1/r². Potential falls off more slowly with distance than field strength does. A very common exam slip is to write V = Q/(4πε₀r²) — that is the field formula, not the potential.

The potential is positive around a positive charge and negative around a negative charge. At infinity, V → 0 — infinity is the natural zero of electric potential.

Distance	V (if Q = +1 nC)
0.10 m	+89.9 V
0.50 m	+18.0 V
1.00 m	+8.99 V
10.0 m	+0.899 V
∞	0 V

Worked Example — Potential Around a Point Charge

Calculate the electric potential 25 cm from a point charge of +4.0 nC.

V = (4.0 × 10⁻⁹) × (8.99 × 10⁹) / 0.25
  = 35.96 / 0.25
  = 143.8 V
  ≈ 144 V

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Potential Energy of a Charge Pair

The potential energy of a charge Q₂ sitting in the field of another charge Q₁ at separation r is the work done bringing Q₂ from infinity to its current position. Because V₁ = Q₁/(4πε₀r) is the potential at r due to Q₁,

E_p = Q₁Q₂ / (4πε₀r)

Signs matter. Two like charges (both positive or both negative) have positive potential energy — work was done against the repulsion to assemble them, and energy would be released if they flew apart. Two unlike charges have negative potential energy — like a bound state of gravitating masses, you would have to do positive work to pull them apart to infinity.

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Worked Example — Energy to Separate Two Charges

Two charges of +2.0 nC and −3.0 nC are 5.0 cm apart. How much work must be done to separate them to infinity?

E_p = Q₁Q₂ / (4πε₀r)
    = (+2.0 × 10⁻⁹)(−3.0 × 10⁻⁹) × (8.99 × 10⁹) / 0.05
    = (−6.0 × 10⁻¹⁸)(8.99 × 10⁹) / 0.05
    = −5.394 × 10⁻⁸ / 0.05
    = −1.08 × 10⁻⁶ J

The potential energy of the bound pair is about −1.08 μJ. To separate them to infinity, we must do an equal but opposite amount of work: +1.08 μJ.

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The Gravitational Analogy — OCR's Favourite Table

OCR H556 Module 6.2 explicitly asks candidates to compare electric fields with gravitational fields. Here is the comparison in full: