Electric field and electric potential describe the same electrostatic situation from different viewpoints, but they are not interchangeable. This guide explains the difference between them, shows how to read electric field diagrams and equipotential maps, and gives a reliable workflow for sketching fields, calculating work and potential energy, and checking signs in electrostatics problems.
Overview
The electric field tells you the force that a small positive test charge would experience at a point. It is a vector, so it has both magnitude and direction:
E = F/q
Its SI units are newtons per coulomb (N/C), which are equivalent to volts per metre (V/m). The field direction is defined as the direction of the force on a positive test charge. A negative charge placed in the same field feels a force opposite to the field direction.
Electric potential, often written V, tells you the electric potential energy available per unit charge at a point:
V = U/q
Potential is a scalar, so it has a value but no direction. Its unit is the volt (V), equivalent to a joule per coulomb (J/C). Potential differences are often more useful than an absolute potential because only differences in potential affect work and changes in energy.
| Feature | Electric field | Electric potential |
|---|---|---|
| Type of quantity | Vector: magnitude and direction | Scalar: magnitude and sign |
| Common symbol | E | V |
| Definition | Force per unit positive charge | Potential energy per unit charge |
| SI unit | N/C or V/m | Volt, J/C |
| Diagram representation | Arrows or field lines | Equipotential lines or surfaces |
| Direction | Shown directly by arrows | No direction; values indicate relative level |
| Relationship | E points toward decreasing potential | V changes in the direction of the electric field |
The mathematical connection is expressed in one dimension as E = −dV/dr. More generally, the electric field is the negative gradient of potential, E = −∇V. The minus sign matters: electric field points from higher potential toward lower potential.
What to track
1. Charge sign and geometry
Before drawing anything, mark every charge and record its sign and approximate position. For an isolated positive point charge, field lines point radially outward. For an isolated negative point charge, they point radially inward. The potential behaves differently: a positive charge produces positive potential, while a negative charge produces negative potential, assuming the usual reference of zero potential at infinity.
For a point charge, the magnitudes are
E = k|Q|/r² and V = kQ/r.
The field is a vector and must be added using vector components. Potential is a scalar and can be added algebraically:
Vtotal = kQ1/r1 + kQ2/r2 + ...
This difference is one of the most useful shortcuts in electrostatics. At a point where fields from two charges cancel, the potential may still be nonzero. Conversely, a point can have zero potential while the electric field is nonzero.
2. Field-line direction and spacing
When you read an electric field diagram, follow the arrowheads. Field lines leave positive charges and enter negative charges, or extend toward or away from infinity when no opposite charge is present. At any point, the electric field is tangent to the field line there.
Field-line spacing gives a qualitative indication of field strength. Closely spaced lines represent a stronger field; widely spaced lines represent a weaker field. This is a visual convention rather than a universal numerical scale, so do not extract an exact field magnitude unless the diagram provides a scale.
Field lines do not cross. If they crossed, the field would have two directions at the same point. Lines also do not run along the surface of an equipotential line in an electrostatic diagram; they cross equipotential lines at right angles.
3. Equipotential values and spacing
An equipotential line connects points with the same electric potential. Moving a charge along that line requires no work by the electric field because the potential difference is zero. Equipotential lines can be closed curves around a point charge or distorted curves around several charges. In three dimensions they become equipotential surfaces.
Equipotential lines are not automatically equally spaced in physical distance. Where they are packed closely together, the potential changes rapidly with distance and the electric field is strong. Where they are farther apart, the field is weaker, provided the potential intervals between adjacent lines are equal or otherwise known.
To estimate the field between nearby equipotentials, use the approximate relation
|E| ≈ |ΔV|/Δs,
where ΔV is the potential difference and Δs is the perpendicular separation between the lines. Always measure the shortest distance between equipotentials, because the field is perpendicular to them.
4. Work and potential energy
For a charge q moving from an initial point i to a final point f, the change in electric potential energy is
ΔU = qΔV = q(Vf − Vi).
The work done by the electric field is
Wfield = −ΔU = q(Vi − Vf).
A positive charge moving naturally in the direction of the electric field moves toward lower potential, so the field does positive work and its potential energy decreases. A negative charge feels a force opposite to the field and can naturally move toward higher electric potential while still losing potential energy. This is why checking the sign of q is essential.
Cadence and checkpoints
Use the following workflow whenever you analyze a new charge configuration or field map.
- Set the reference. Decide whether the problem defines a zero-potential point. For isolated point charges, zero at infinity is common, but follow the stated convention.
- Sketch the geometry. Mark charges, distances, symmetry axes, and the point or region being studied. Do not begin with equations before identifying the geometry.
- Draw individual field contributions. At a chosen point, draw the field from each charge. Point away from positive charges and toward negative charges.
- Add vectors carefully. Resolve fields into components when they are not collinear. Use symmetry to identify components that cancel, but do not assume the potential cancels in the same way.
- Calculate potential separately. Add each contribution with its algebraic sign. Potential does not require vector components.
- Sketch equipotentials. Draw curves that are perpendicular to the field lines. Label enough values to show whether potential increases or decreases in a given direction.
- Check the spacing. Equal potential changes over shorter distances indicate larger field magnitude. Make sure the field is strongest where equipotentials are closest.
- Check units and signs. Use N/C or V/m for electric field, V for potential, and J for potential energy. Confirm that your direction agrees with the behavior of a positive test charge.
For a two-charge sketch, begin near each charge, where the field changes most rapidly. Lines from like charges bend away from the central region; lines between unlike charges tend to connect from the positive charge to the negative charge. The exact curvature depends on the charge magnitudes and separation, so treat a hand sketch as a qualitative map unless a scale or calculation is provided.
When reading a potential map, first identify the potential at the starting and ending points. If a charge moves along one equipotential, ΔV is zero and the electric field does no work. If it crosses several equipotentials, add the potential difference between the endpoints rather than summing path length. Electrostatic work is path independent.
How to interpret changes
The most common mistake is to treat electric potential like a directional arrow. Potential has no direction, so statements such as “the potential points left” are not meaningful. Instead, say that potential increases or decreases in a direction. The electric field points toward decreasing potential.
A second mistake is confusing field-line density with potential itself. Dense field lines mean a large field magnitude, not necessarily a large positive potential. Near a negative charge, the field can be strong while the potential is strongly negative. Near a point where contributions from multiple charges cancel, the field may be zero even though the potential is not.
Use these sign checks:
- If a positive charge moves to lower potential, its potential energy decreases.
- If a negative charge moves to higher potential, its potential energy also decreases because q is negative.
- If two points have the same potential, the work done by the electrostatic field between them is zero.
- If equipotential lines are closer together in one region, the field magnitude is larger there when the displayed potential intervals are comparable.
- If your field arrow runs along an equipotential line, redraw it: the field should cross an equipotential at 90 degrees.
For exam problems, separate the diagram task from the calculation task. First identify directions and symmetry; then write the relevant equation; finally substitute values with units. A structured approach like the one in How to Solve Physics Problems can help prevent a correct formula from being applied to the wrong quantity. If the question involves measured field or voltage data, also consider whether uncertainty changes the conclusion; the methods in Physics Lab Uncertainty and Error Analysis provide a useful framework.
When to revisit
Return to this guide whenever a new diagram uses unfamiliar symbols, a problem switches between field and potential, or a sign check produces an unexpected result. A short review is especially useful before an electrostatics unit, an AP Physics or IB Physics revision session, or a university electromagnetism assessment.
For a recurring study routine, revisit your notes weekly while learning the topic and again after completing a new set of charge-configuration problems. At each checkpoint, redraw one isolated-charge field, one two-charge field, and one equipotential map from memory. Then compare your sketch against the rules: field lines follow the force on a positive test charge, equipotentials are perpendicular to fields, and closer equipotentials indicate a larger field when the voltage steps are comparable.
Update a personal formula sheet when your course introduces a new convention, such as a different reference potential, potential from a continuous charge distribution, or the relation between voltage and electric field in a uniform region. Keep the core distinctions visible: field is a vector that describes force per charge; potential is a scalar that describes energy per charge.
Finally, apply the workflow to one fresh diagram without looking at the solution. Label charge signs, mark field directions, identify equipotentials, calculate ΔV and ΔU, and check units. If every step agrees, you are not only recognizing a formula—you are reading the physical map.
