Finding the Work Done by the Electric Field as You Move a Charge Near a Dipole

I’ll be honest. This connection between the electric potential (change in electric potential) and the electric field can get sort of crazy. But let’s just start with a problem and then solve it in more ways than you wanted.

Here is the problem.

Let’s start with the energy to bring an electron to point B. The energy needed would be equal to the change in electric potential energy which is equal to:

That means I just need to calculate the change in electric potential from infinity to point B. Yes, you could also calculate the work needed to move the charge-I’ll do that also.

Since I am dealing with two point charges, I can use the following expression for the potential due to a point charge (with respect to infinity):

Where k is the Coulomb constant (k = 9 x 10⁹ Nm²/C²) and r is the distance from the point charge to the final location. Since there are two point charges, the total potential will just be the sum of the two potentials. Let me call the positive charge “1” and the negative charge “2”. That means the total potential will be:

From the original problem, q1 = 2 x 10^-9 C and q2 = -2 x 10^-9 C. The distance r1 will be 6 mm and the distance r2 will be 4 mm (need to convert these to…