The Potential to Make Electric Fields Easier to Deal With | Electromagnetism by Parth G

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Some mathematical identities combined with Maxwell's equations allow us to define electric and magnetic potentials... but why are they useful?

Hi everyone! In a recent video, I talked about how the magnetic vector potential was a different way to view magnetic fields, and why Quantum Mechanics (specifically the Aharonov-Bohm effect) made us question whether the original fields were the "fundamental" quantity to be studied. That video can be found here:

In this video, I wanted to provide some more background around potentials, by specifically focusing on the electric scalar potential. We will start by looking at how the scalar potential is defined for electrostatics. It comes about by using a general identity from vector calculus - namely that the curl of the gradient of a scalar is always zero. Then, we combine this with the Maxwell equation from electromagnetism that states that the curl of any electric field is equal to the negative rate of change of the magnetic field in the same region of space. For electrostatics, i.e. for the scenario where the magnetic field does not change over time, we can set the right hand side of this equation to be zero.

It's worth noting that there is more complicated mathematics for the scenario where the magnetic field does change over time, but for now we focus on when the rate of change is zero. In that scenario, the general vector calculus identity looks similar to the Maxwell equation. We can see that the electric field could be defined as the gradient of some scalar field. This scalar field is known as the electric potential.

The electric potential has some interesting properties. Firstly, it tells us something about how charges would interact with the field. At each point in space, we can assign a potential value. And positive charges move from regions of larger potential to regions of smaller potential. The difference between the potential at two points in space is known as the potential difference, and this is exactly what we talk about when studying electric circuits. Positive charges (e.g. conventional current) move from points of higher potential to points of lower potential.

Interestingly though, the potential field does not have a fixed value at any point. All that matters is the difference between potential values at any two points. For this reason, we can choose any point to be the point of "zero potential", and everything else is measured relative to that. This is similar to choosing sea level as our height of 0m, even though we could choose any height to be zero. All other heights are measured relative to this, but any choice of zero is a valid one.

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