Electric potential

Electric potential at a point is the electric potential energy per unit charge there: the work done per coulomb in bringing a small positive charge from infinity to that point. It is measured in volts, where 1 V = 1 J/C. Near a point charge Q, the potential is V = Q/(4πε₀r), positive near a positive charge and negative near a negative one.

What is electric potential?

Think of electric potential as a map of how much energy each coulomb of charge would have at each place in an electric field. Potential energy belongs to a particular charge; potential belongs to the place. Multiply the potential by a charge and you get that charge's potential energy:

For absolute values, the zero of potential is taken at infinity, where a charge would feel no force from the field. AQA's A-level specification asks you to understand absolute potential, including its zero value at infinity, and potential difference between two points.

Potential is a scalar: it has a size and a sign but no direction. That makes it much easier to work with than electric field strength, which is a vector.

Electric potential of a point charge

Around a point charge Q (or outside a charged sphere, measuring r from its centre), the potential at distance r is:

Here ε₀ is the permittivity of free space, and 1/(4πε₀) = 8.99 × 10⁹ N m² C⁻², often written k. So V = kQ/r.

• V is proportional to 1/r, not 1/r². Double the distance and the potential halves. (Field strength, E = kQ/r², falls to a quarter.)
• The sign of V follows the sign of Q. Near a positive charge V is positive, and a positive test charge would need work done on it to be brought closer. Near a negative charge V is negative.

For a charge of +4.0 nC, kQ = 8.99 × 10⁹ × 4.0 × 10⁻⁹ = 35.96 V m. In the graph, x is the distance r from the charge in metres and y is the potential V in volts:

Electric potential difference and work done

The potential difference between two points is the difference in their potentials. Moving a charge Q through a potential difference ΔV transfers energy:

This is the same idea as the potential difference in circuits, where 12 V means 12 joules for each coulomb. Moving a positive charge to a higher potential needs work done on it; letting it move to a lower potential releases energy, often as kinetic energy.

The electronvolt

One electronvolt (eV) is the energy a charge of e gains when accelerated through a potential difference of 1 V: 1 eV = 1.60 × 10⁻¹⁹ J. It is the natural unit for the energy of electrons and ions accelerated through a voltage.

Equipotential surfaces

An equipotential is a line or surface joining points at the same potential. Around a point charge they are spheres centred on the charge (circles in a diagram), closer together where the field is stronger. Between two parallel plates they are flat planes parallel to the plates, evenly spaced.

• No work is done moving a charge along an equipotential, because ΔV = 0.
• Equipotentials are always at right angles to field lines.
• Closer equipotentials mean a stronger field.

You can see field lines for any arrangement of charges in the field lines simulator. Equipotentials cross them at right angles everywhere.

Potential gradient and field strength

Electric field strength is the potential gradient: how fast the potential changes with distance. AQA's specification gives the relationship as:

• The gradient of a V against r graph gives the field strength at that point (the field points from high to low potential).
• The area under an E against r graph gives the potential difference between two distances.
• In a uniform field between parallel plates, the gradient is constant, so E = V/d, where V is the potential difference across the plates and d is their separation.

Worked examples

Example 1: potential near a point charge

Find the potential 0.20 m from a charge of +4.0 nC.

V = kQ/r = (8.99 × 10⁹ × 4.0 × 10⁻⁹) ÷ 0.20 = 35.96 ÷ 0.20 = 180 V (2 s.f.).

Example 2: work done moving a charge

A +2.0 nC charge is moved from 0.30 m to 0.10 m away from a +6.0 nC charge. How much work is done?

1. Potential at 0.30 m: (8.99 × 10⁹ × 6.0 × 10⁻⁹) ÷ 0.30 = 179.8 V.
2. Potential at 0.10 m: 53.94 ÷ 0.10 = 539.4 V.
3. ΔV = 539.4 − 179.8 = 359.6 V.
4. ΔW = QΔV = 2.0 × 10⁻⁹ × 359.6 = 7.2 × 10⁻⁷ J. Work must be done on the charge, because two positive charges repel.

Example 3: accelerating an electron

An electron starts at rest and is accelerated through a potential difference of 2000 V. Find its kinetic energy and speed (mass 9.11 × 10⁻³¹ kg).

1. Kinetic energy = eV = 1.60 × 10⁻¹⁹ × 2000 = 3.2 × 10⁻¹⁶ J, which is 2000 eV.
2. From ½mv² = 3.2 × 10⁻¹⁶: v = √(2 × 3.2 × 10⁻¹⁶ ÷ 9.11 × 10⁻³¹) = 2.7 × 10⁷ m/s.

Example 4: potential from two charges

A +5.0 nC charge and a −3.0 nC charge are near a point P. P is 0.25 m from the positive charge and 0.15 m from the negative one. Find the potential at P.

Potentials add as ordinary numbers, with their signs: (8.99 × 10⁹ × 5.0 × 10⁻⁹) ÷ 0.25 + (8.99 × 10⁹ × −3.0 × 10⁻⁹) ÷ 0.15 = 179.8 − 179.8 = 0 V. The potential at P is zero, but the field there isn't: the two fields are vectors pointing in different directions, so they don't cancel.

Electric potential vs gravitational potential

Gravitational potential is always negative because gravity only attracts. Electric potential can be either sign, because charges can repel as well as attract. The gravitational potential guide covers the gravity side.

Common mistakes

• Using 1/r² for potential. V = kQ/r. It is the field strength that uses r².
• Dropping the sign of a negative charge. Put the sign into V = kQ/r; it matters when you add potentials.
• Adding potentials as vectors. Potential is a scalar: just add the values with their signs.
• Measuring r from the surface of a sphere. For a charged sphere, r is measured from its centre.
• Unit slips. nC is 10⁻⁹ C and μC is 10⁻⁶ C; convert cm to m before substituting.

To practise electric fields alongside the rest of the second-year course, try today's A-level Physics: Year 2 short timed paper.

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