Recovering the Field from the Potential Gradient
The field is minus the gradient of the potential, so a map of potential contains the field everywhere. Equipotential surfaces are perpendicular to field lines, and the field is strongest where they crowd together.
What a learner can do afterwards
- Differentiates a potential function to recover each field component
- Sketches field lines on a given equipotential map and states the relation between them
- Explains why no work is done moving a charge along an equipotential
1 · Read
The electric field is minus the gradient of the potential: each component is the slope along one axis with a flipped sign. Slice V along x and its slope gives E_x with a minus in front, and the same along y and z. A flat potential along some direction means zero field there. Where equipotential surfaces crowd together the field runs strong, and where they spread out it runs weak. This one rule replaces memorising separate field formulas case by case.
Recover a point charge field by differentiating its potential V = kQ over r. Along r the slope is dV/dr = minus kQ over r squared, so E_r = kQ over r squared, pointing outward. Between parallel plates the field is nearly uniform, so voltage drops in a straight line: drop = field times distance. Double the spacing at fixed field and the energy gain of a released charge doubles with it.
An equipotential surface joins every point at the same voltage, and field lines always cross these surfaces at right angles, running from high toward low. Moving a charge along one costs no work, since start and finish share the same energy. A conductor in balance sits at one potential throughout with zero field inside, and field lines meet its surface perpendicularly.
Read any contour map in three moves: pick two neighbouring lines, divide the voltage step by the gap for the strength, and face downhill for the direction. A 10 V step across 2 m means about 5 V/m toward the lower line. Ask which side sits higher before trusting any sign.
Flip the voltage slopes for the field, cross equipotentials squarely, and divide steps by gaps.
More to read
7.4 Determining Field from Potential
The electric field points in the direction where the potential drops fastest. The field is minus the gradient of the potential, so each component is a partial derivative with a sign flip. This turns a scalar map of voltage values into a full vector field. Where equipotential surfaces crowd together the field is strong, and where they spread out it is weak. To recover the field in practice, slice the potential along each axis and measure its slope. The slope along x gives Ex, along y gives Ey, and along z gives Ez. A flat potential means zero field in that direction. This single rule replaces memorising separate field formulas for many charge arrangements.
7.5 Equipotential Surfaces and Conductors
An equipotential surface joins every point at the same voltage. No work is needed to move a charge along it, because start and finish share the same potential energy. Field lines always cross these surfaces at right angles, pointing from high toward low potential. Sketching one family lets you draw the other. Conductors add a sharp rule. Inside a conductor in electrostatic balance the field is zero, so the whole volume sits at one potential. Its surface is itself an equipotential, and field lines meet it perpendicularly. Sharp points squeeze the surfaces together, which is why charge and field pile up there.
19.4 Equipotential Lines
Equipotential lines are the flat map version of equipotential surfaces. Each line labels a voltage, and moving along the line costs no energy. Electric field lines cut across them at right angles, flowing downhill from higher to lower voltage. The spacing tells the strength, since tight lines mean a strong field. Reading such a map is a core skill. Pick two neighbouring lines, note the voltage step and the distance between them, and divide to estimate the field strength. Check the sign by asking which side sits higher. With that habit any contour plot becomes a field diagram.
19.2 Electric Potential in a Uniform Electric Field
Between two parallel plates the field is nearly uniform, so the voltage drops in a straight line from one plate to the other. The drop equals field strength times distance, V = Ed. This is the simplest stage on which to see potential, field, and energy trade places. Release a charge between the plates and it runs downhill in voltage, gaining kinetic energy equal to the charge times the drop. Double the spacing at fixed field and you double the energy gain. Particle accelerators and classic charge experiments both lean on this linear relation.
2 · Watch
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Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.