Where Is There Zero Electric Field Between Two Charges

Coulomb's Law, Electric Field and Superposition

A bead of charge +Q and another of charge +4Q are fixed a distance D apart. Is there a point between them where the electric field is zero? Is there any other point where the electric field is zero?

Imagine we place a charge of one coulomb on the line joining the two charges – then the force acting on this charge (in Newtons) will be equal to the electric field (in Newtons/Coulomb). Clearly, the two forces (from the two charges) the test charge experiences will oppose each other in the area, so it is likely that there is a point (a distance x away from the +Q charge) where they have the same magnitude and exactly cancel.

F=kQ*1/x² = k*4Q*1/(D-x)²

Canceling common factors from both sides gives:

1/x² = 4/(D-x)²

x² = (D-x)²/4

x = (D-x)/2

x = D/3

At this point the force, and hence the field, is zero. Anywhere off this line, the forces from the two charges will not oppose each other (in direction) and thus it is impossible for them to exactly cancel – hence there are no other points where the field is zero.