3.7.2.2FieldsStretch

Electric Potential

Electric potential works just like gravitational potential — energy per unit charge, rather than energy per unit mass — with one crucial twist: because charge can be positive or negative, electric potential can be positive, negative, or exactly zero, in a way gravitational potential never can.

40 min Video by ZPhysics 3.7.2 Electric fields

A Level Physics: Electric Potential and EnergyWatch the full walkthrough before the notes below.

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What you'll be able to do

Define electric potential and use $V = \frac{Q}{4 π ε _{0} r}$
Calculate the potential energy of a charge in an electric field
Calculate the work done moving a charge between two points
Explain the relationship between potential gradient and field strength
Describe equipotential surfaces and their relationship to field lines

Electric potential

$Electric potential$ , $V$ , at a point is the work done per unit (positive) charge bringing a small test charge from infinity (defined as zero potential) to that point. Around a positive point charge, potential is $positive$ everywhere (work must be done against repulsion to bring another positive charge in); around a negative point charge, potential is $negative$ everywhere (a positive test charge is pulled in, releasing energy).

V = \frac{Q}{4 π ε _{0} r}

Electric potential due to a point charge Q, at distance r.

1Use

V = \frac{Q}{4 π ε _{0} r}

V = \frac{- 4.0 \times 1 0 ^{- 9}}{4 π \times 8.85 \times 1 0 ^{- 12} \times 0.30}

Answer

V \approx - 120

V (negative, since the source charge is negative).

Tip — Unlike gravitational potential (always negative), electric potential takes the SAME sign as the source charge creating it — positive charges give positive potential, negative charges give negative potential.

Potential energy and work done

The potential energy of a charge $q$ placed at a point of potential $V$ is $E_{p} = q V$ . Moving a charge between two points requires (or releases) energy equal to the charge multiplied by the potential difference between those points — exactly analogous to the gravitational case, but now the result can come out negative (energy released) or positive (energy required) depending on the signs of the charge and the potential difference involved.

E_{p} = q V W = q (V_{2} - V_{1}) = q Δ V

Electric potential energy, and work done moving a charge between two points.

1Use

W = q (V_{2} - V_{1})

W = 2.0 \times 1 0^{- 6} \times (80 - 150) = 2.0 \times 1 0^{- 6} \times (- 70)

Answer

W = - 1.4 \times 1 0^{- 4}

J — negative, meaning energy is released by the charge as it moves (it moves "downhill" in potential, from high to low).

Tip — A positive charge moving from high to low potential releases energy (like a mass falling under gravity); moving it the other way requires energy to be supplied.

Potential gradient and equipotentials

The field strength at a point equals the negative of the rate of change of potential with distance — the $potential gradient$ . Where potential changes rapidly over a short distance, the field is strong; where potential barely changes, the field is weak. This is exactly consistent with the uniform-field formula $E = V / d$ : a constant potential gradient across parallel plates gives a constant field strength.

An $equipotential$ surface (or line, in a 2D diagram) joins all points of equal potential. Equipotentials are always perpendicular to field lines, and — since there is no potential difference to overcome — moving a charge along an equipotential requires exactly zero work.

Tip — Field lines and equipotentials are always perpendicular — sketching one immediately tells you the shape of the other. Around a point charge, equipotentials are concentric circles (spheres in 3D); between parallel plates, they are straight lines parallel to the plates.

Equation recap

V = \frac{Q}{4 π ε _{0} r}

Electric potential due to a point charge.

E_{p} = q V

Electric potential energy of a charge q.

W = q Δ V

Work done moving a charge through a potential difference.

Common mistakes to avoid

Assuming electric potential, like gravitational potential, is always negative.

Electric potential takes the sign of the source charge — positive for a positive charge, negative for a negative one — and can be positive, negative, or zero.

Forgetting the sign of the moving charge when calculating work done through a potential difference.

W = qΔV — a NEGATIVE charge moving through a positive ΔV releases energy, the opposite of what a positive charge would do in the same field.

Drawing equipotentials parallel to field lines.

Equipotentials are always PERPENDICULAR to field lines, never parallel to them.

Key takeaways

Electric potential V = Q/(4πε₀r), taking the same sign as the source charge.
Electric potential energy Ep = qV; work done moving a charge W = qΔV.
Field strength is the (negative) potential gradient — a steep potential change means a strong field.
Equipotentials join points of equal potential, are always perpendicular to field lines, and require zero work to move a charge along.

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Electric Potential

40 min Video by ZPhysics 3.7.2 Electric fields