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.
2 · Watch
Take it off screen
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.