Reading fields from potential graphs and maps

Field strength and potential describe the same field in two ways, one as force per unit charge and one as energy per unit charge. They are linked: the field is strongest where the potential changes most steeply. That link lets you read a field from a graph of potential or from a map of equipotential lines.

Field strength is the negative potential gradient

Move a charge q a small distance Δ x along the direction of a field E. The field does work qEΔ x on it, and this equals the loss of electric potential energy, -qΔ V. So E = -(Δ V)/(Δ x), and in the limit

The electric field strength at a point is equal to the negative of the potential gradient there. The minus sign says that the field points from high potential to low potential. This also shows why V m⁻¹ is the same unit as N C⁻¹.

Potential across a uniform field

Calculated straight potential profile decreasing from ten volts to zero across five centimetres.

Scroll across the figure to read all labels.

Open full-size graph

What to noticeThe minus sign gives field direction: towards increasing x and decreasing potential.

View figure data

Figure data: Potential across a uniform field

Distance from plate, x (m)	V(x) = 10 − 200x
0.000	10.0
0.001	9.8
0.002	9.6
0.003	9.4
0.004	9.2
0.005	9.0
0.006	8.8
0.007	8.6
0.008	8.4
0.009	8.2
0.010	8.0
0.011	7.8
0.012	7.6
0.013	7.4
0.014	7.2
0.015	7.0
0.016	6.8
0.017	6.6
0.018	6.4
0.019	6.2
0.020	6.0
0.021	5.8
0.022	5.6
0.023	5.4
0.024	5.2
0.025	5.0
0.026	4.8
0.027	4.6
0.028	4.4
0.029	4.2
0.030	4.0
0.031	3.8
0.032	3.6
0.033	3.4
0.034	3.2
0.035	3.0
0.036	2.8
0.037	2.6
0.038	2.4
0.039	2.2
0.040	2.0
0.041	1.8
0.042	1.6
0.043	1.4
0.044	1.2
0.045	1.0
0.046	0.8
0.047	0.6
0.048	0.4
0.049	0.2
0.050	0.0

Figure data: Potential across a uniform field

Distance from plate, x (m)	V(x) = 10 − 200x
0.000	10.0
0.001	9.8
0.002	9.6
0.003	9.4
0.004	9.2
0.005	9.0
0.006	8.8
0.007	8.6
0.008	8.4
0.009	8.2
0.010	8.0
0.011	7.8
0.012	7.6
0.013	7.4
0.014	7.2
0.015	7.0
0.016	6.8
0.017	6.6
0.018	6.4
0.019	6.2
0.020	6.0
0.021	5.8
0.022	5.6
0.023	5.4
0.024	5.2
0.025	5.0
0.026	4.8
0.027	4.6
0.028	4.4
0.029	4.2
0.030	4.0
0.031	3.8
0.032	3.6
0.033	3.4
0.034	3.2
0.035	3.0
0.036	2.8
0.037	2.6
0.038	2.4
0.039	2.2
0.040	2.0
0.041	1.8
0.042	1.6
0.043	1.4
0.044	1.2
0.045	1.0
0.046	0.8
0.047	0.6
0.048	0.4
0.049	0.2
0.050	0.0

Here the potential falls from 10 V to zero over 0.050 m. The gradient is -200 V m⁻¹, so the field is + 200 V m⁻¹: 200 V m⁻¹ in the + x direction. When the graph is curved, the field at a point comes from the gradient of the tangent there.

For a point charge, V = Q/4πε₀r, and -dV/dr = Q/4πε₀r²: the same field you found from Coulomb’s law.

Equipotentials

An equipotential is a line, or surface, joining points at the same potential. Moving a charge along an equipotential changes neither its potential nor its electric potential energy, so the field does no work on it. That is only possible if the force has no component along the equipotential: field lines always cross equipotentials at right angles.

When equipotentials are drawn at equal steps of potential, their spacing shows the field strength. Close together means a steep potential gradient and a strong field. Around a point charge, equipotentials are concentric circles that get further apart with distance. In the uniform field between parallel plates, they are equally spaced and parallel to the plates.

Test the equipotential prediction

Use the stated equal and opposite charges, leave Show equipotentials on, and keep Probe x at 0. Change Probe y or drag the test-charge probe along the y-axis. Compare V at the probe, E at the probe and the field arrow. The plotted charge-mode graphs follow a horizontal slice at the probe’s current height, so they show how potential changes across the axis rather than along your vertical path.

Common mistakes

• Dropping the minus sign: the field points towards lower potential.
• Thinking a large potential means a strong field; the field depends on how fast the potential changes.
• Comparing line spacing on a map whose equipotentials are not at equal steps of potential.
• Drawing field lines that cross equipotentials at an angle other than 90°.
• Using (Δ V)/(Δ r) across a wide gap in a non-uniform field as if it were the field at a point.