12.6.2 Equipotential Lines

It is quite common to see equipotential lines overlaid on a field map. As the name suggests, an equipotential line is a line joining points with the same electric potential V. Any number of these lines can be drawn for any number of V. The important thing is to draw them at equal spacing of V (e.g. draw the equipotential lines for V=2.5 V, 3.0 V, 3.5 V… i.e. constant ΔV of 0.5 V).  If we do this, equipotential lines can be quite expressive too:

  • The spacing between equipotential lines conveys the magnitude of the field strength. In regions where the field strength is strong, the equipotential lines are closely packed. In regions where the field strength is weak, the lines are far apart. (This is because \displaystyle E=-\frac{{\Delta V}}{{\Delta x}} and DV is fixed by design)
  • The direction of the field is always perpendicular to the equipotential line, towards lower potential.

(This is because there is no potential gradient along an equipotential line, by definition. So the electric field must have zero component parallel to the equipotential line.)

As shown above, the equipotential lines for the field produced by a point charge turn out to be a set of concentric circles. Since the field strength decreases away from the charge, those concentric circles also become more and more spaced out. From the values marked on those equipotential lines, one can tell that potential is decreasing radially outward. From this, one can infer that direction of the field is radially outward.

Above is the field map for a positively charged metal sphere and a metal plate. The sphere is held at 7 V while the plate is held at 0 V. Note that

  • The field lines always cut the equipotential lines perpendicularly.
  • Since the entire sphere is held at 12 V, its surface is actually the 12 V equipotential surface. That’s why the field lines “leave” the surface of the sphere perpendicularly.
  • Since the entire plate is held at 0 V, its surface is also an equipotential surface (0 V). That’s why the field lines “enter” the plate horizontally.
  • The resulting field pattern can be interpreted as that of a positive point charge (spherically distributed charges is equivalent to a point charge at the centre of the sphere) “distorted” by the attraction towards the negatively charged plate (The positively charged sphere induces negative charges on the plate).
  • Both the density of the field lines and the spacing between the equipotential lines show that the field strength generally decreases away from the sphere. It also shows that the field on the left of the sphere is weaker than the field between the sphere and the plate.

Demonstrations

Potential Mapping

Video Explanation

Why Must Field and Equipotential Lines Cut Perpendicularly?

Concept Test

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