Using the Electric Potential to Find the Vector Electric Field

Here is the normal routine in introductory physics class.

Start off with the electric field due to point charges. You can find the total electric field at some location by first determining the vector value of the electric field due to individual charges. Then, using the superposition principle, the total electric field is just the vector sum of these fields. It’s not always trivial since the electric field is a vector.

After that, you introduce the idea of the electric potential (with respect to infinity). Again, you find the electric potential due to multiple points and then use superposition to find the total electric potential. But in this case, it’s much easier to add them since the potential is a scalar quantity — you don’t need to worry about direction of vectors and stuff.

So, let’s say you have an electric dipole consisting of two equal, but opposite charges. Could you find the electric potential at some point and then use this potential to find the value of the electric field? Yup. That’s exactly what I’m going to do.

Defining Electric Potential and Electric Field

I’m going to start with the definition of the change in electric potential. It’s really the definition of work (physics work) per unit charge. The work is a path integral of a force and in this case that force is the electric force on a charge. That gives the following definition of the change in potential between points A and B.

Since the change in potential is a path integral of the electric field, the electric field must be a derivative of the electric potential. But just a plain derivative wouldn’t give you a scalar value. That’s where the del operator comes in. It’s essentially a 3-dimensional operator where the components are derivatives.

With that, the electric field at some point would be:

But you can’t just take the derivative of the electric potential if you have just a single value. It would be like trying to find the slope of a single point — you just can’t do it. So, there are two options. First, you could know the electric potential as a function (of x, y, and z) then you can just take the derivatives like a normal human being. Second, if you just have numerical values for the electric potential then you can calculate a numerical derivative. This second method is going to be my plan.