Question

(a) Draw electric field lines and equipotential surfaces for a system of two equal positive charges separated by some distance.

(b) Using the drawing in (a) above, explain that electric field need not be a constant over each point of an equipotential surface.

Hint:

Draw lines going out from both charges, bending apart. Show an equipotential that passes through the midpoint, where E = 0, and note E is not zero elsewhere on it.

Answer 1

Step 1: Electric field lines of two equal positive charges (part a):
Mark two points +q and +q a distance apart on a horizontal line. Field lines start on each charge and go outward. Near each charge they spread out radially. Lines from one charge do not cross lines from the other.
They bend away from each other, so no line joins the two charges. Far from the pair, the lines look like those of a single charge of 2q going radially outward.
At the midpoint between the charges, the two fields are equal and opposite. So the field is zero there. This is the neutral point.

Step 2: Equipotential surfaces (part a):
Equipotential surfaces are always perpendicular to the field lines. Near each charge they are small closed curves (spheres in 3-D) around that charge.
At a certain potential, the two small closed surfaces touch and form one peanut-shaped (figure-eight) surface that passes through the neutral point. Further out, they are big closed surfaces that look almost like spheres around the pair.

Step 3: Field need not be constant over a surface (part b):
The magnitude of the field is \(E = -\dfrac{dV}{dl}\). It depends on how close two nearby equipotential surfaces are. If they are close, E is large. If they are far apart, E is small.
On the figure-eight surface, take a point on the line joining the charges, at the neutral point, where E = 0. Take another point on the same surface, away from the line, where the surfaces are packed more closely. There E is not zero.
So the field takes different values at different points of the same equipotential surface.

Final Answer:
The field lines repel each other and the equipotential surfaces are perpendicular to them. The field is zero at the midpoint but non-zero elsewhere on the same equipotential, so E is not constant over an equipotential surface.