Electric Potential & Potential Energy

You’ll be able to

Distinguish electric potential (a scalar, in volts) from the electric field (a vector)
Relate electric potential energy to charge and potential difference
Compute the potential of a point charge and the energy to move a charge through a potential difference

Potential: energy per charge

Lifting a charge against an electric field stores electric potential energy, just as lifting a mass stores gravitational PE. The electric potential V at a point is that potential energy per unit charge: V = U/q, measured in volts (1 V = 1 J·C⁻¹). Crucially, potential is a scalar — it has no direction. That makes it far easier to work with than the field: to combine the potentials of several charges you simply add numbers, keeping track of signs, with no vector arrows.

Potential of a point charge & energy of a charge

V = k · Q / r and ΔU = q · ΔV

Keep the sign of Q in V = kQ/r: a positive charge makes V positive, a negative charge makes V negative. ΔU is the energy change when a charge q moves through a potential difference ΔV.

Potential difference does the work

What actually matters for energy is the potential difference (voltage) between two points, ΔV. Moving a charge q across it changes its potential energy by ΔU = qΔV. A positive charge naturally “falls” from high potential to low, losing PE and gaining kinetic energy — the electrical version of a ball rolling downhill. A negative charge does the opposite. The field points from high to low potential, always downhill on the potential landscape.

Uniform field and potential

E = V / d

Between two parallel plates a distance d apart with a voltage V across them, the field is uniform with magnitude V/d. This is why field units can be written as V·m⁻¹.

Worked example

How much work must be done to move a +2.0 µC charge from a point at 100 V to a point at 400 V?

1.Find the potential difference: ΔV = 400 − 100 = 300 V.
2.Apply ΔU = qΔV = (2.0 × 10⁻⁶ C)(300 V).
3.Multiply: 2.0 × 10⁻⁶ × 300 = 6.0 × 10⁻⁴ J.
4.Because ΔU is positive, an external agent must supply this much work (the charge is pushed to higher potential).

Answer: 6.0 × 10⁻⁴ J of work is required to move the positive charge to the higher potential.

Checkpoint

A +3.0 µC charge is moved through a potential difference of +200 V. What is the change in its electric potential energy?

Watch out

Potential (V, volts) and potential energy (U, joules) are not the same thing. Potential is a property of a location in the field; potential energy belongs to a charge sitting at that location. They are linked by U = qV.

Checkpoint

The electric potential a distance r from a point charge is V. At a distance 2r from the same charge, the potential is:

On the exam

Remember the different distance dependences: the field of a point charge falls off as 1/r², but its potential falls off as 1/r. Energy questions use potential (scalar, add algebraically); force questions use the field (vector).

Answer the 2 checkpoints as you read.