Where Is the Electric Potential Equal to Zero?

In introductory physics textbooks often have some version of the following two problems.

• Here are two electric charges. Where is the electric field equal to zero (technically the zero vector)?
• Here are two electric charges. Where is the electric potential (with respect to infinity) equal to zero?

Yes, the electric field is a vector and the electric potential is a scalar — so you would think that the question about potential might be simpler, but not so. If you only have two electric charges, the electric field vector can only be zero on an axis connecting the two charges.

Here is my explanation of the location of the zero electric field.

But enough about the electric field. The fun part is the electric potential. It’s actually sort of complicated and at this point, I don’t even know all the answers. Let’s figure this stuff out.

Here is the situation. Two charges, q_1 and q_2. I’m going to put one of them at the origin and the other on the x-axis. Like this.

I will start with the simple solution — it’s the answer to the question “where on the x-axis is the electric potential equal to zero?” The electric potential at any point is the sum of the potential due to each point charge. Yes, I’m assuming that this is the potential with respect to infinity — I just have to get that out of the way. But here is the potential for a point charge.

In this expression the 1/(4πε_0) is a constant (also represented by just “k”). Of course “q” is the charge and r is the magnitude of the vector distance from the charge to the point where you want to find the potential. So, for two charges, I…