Foundational Physics

Short Answer: Equipotential lines are perpendicular to electric field lines because electric potential does not change as you move along an equipotential. If the electric field had a component along that line, it would do work on a charge moving along it, causing the potential to change. In an electrostatic field, the electric field therefore points across equipotentials, not along them, wherever the field is nonzero.

What does equipotential mean?

Imagine marking several places in a region where the electric potential is exactly the same. A curve joining those places is an equipotential line. If you move between points along this line, the electric potential difference is zero.

For a charge that moves between those points, that also means there is no change in its electric potential energy due to the electrostatic field.

What does the electric field tell you?

The electric field gives the direction of the electric force on a positive test charge. If a positive charge moves a small distance along the electric field, the field can do positive work and the electric potential decreases.

But if the charge moves a small distance perpendicular to the field, the electric force has no component along the displacement, so the field does no work during that movement.

Why must the two line types meet at right angles?

Suppose an electric field line met an equipotential line at some angle other than 90 degrees. Then part of the electric field would point along the equipotential line.

That field component would do work on a test charge as it moved along the line, and the electric potential would change. But the potential is supposed to be the same everywhere along that line.

To avoid this contradiction, the electric field must be perpendicular to the equipotential.

What would this look like between capacitor plates?

In the approximately uniform field between two large, oppositely charged parallel plates, electric field lines point straight from the positive plate toward the negative plate. Equipotential lines in a two-dimensional view run parallel to the plates.

The two sets cross at right angles. Moving across the gap changes electric potential; moving along a given equipotential does not. Near the plate edges, the field may curve, and the equipotentials curve with it while remaining locally perpendicular.

The takeaway

Electric field lines point in the direction where potential changes most rapidly; equipotential lines follow directions where potential does not change. That is why they are perpendicular wherever the electrostatic field is nonzero.

When This Breaks

At a point where the electric field is zero, there is no uniquely defined field direction to compare with an equipotential. In three dimensions, we speak of equipotential surfaces, and the electric field is normal to those surfaces.

The explanation also assumes an electrostatic situation, where electric potential can be defined consistently throughout the region. In the presence of changing magnetic fields, induced electric fields need not be described everywhere by a single scalar electric potential.

Keep Exploring: Electric Fields

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