# About electric field and electric potential

We know that electric potential is the negative of work done by electric field in moving a unit charge from infinity to that place. This statement shows that electric field causes a potential difference between any two points in space. But in case of electric current, charges move due to presence of an electric field. And that electric field is generated due to potential difference between the terminals of battery.

So what confuses me is that electric field is the cause for generation of potential difference or is it the potential difference that causes electric field?

Please correct me if I am going wrong in some concept.

• That's a bit of a chicken-or-egg question, I would say. Oct 28, 2022 at 6:32
• You should have a look at this related thread: physics.stackexchange.com/questions/387494/… Oct 29, 2022 at 2:41

What's fundamental is the Coulomb force. If we place two charged particles near each other, they experience an attractive or repulsive force, depending on whether they are opposite or like charges. We can measure that force directly, for example if the charges are on the plates of an electroscope, by observing the deflection of the plates.

The electric field is an abstraction of that force, allowing us to predict what force a hypothetical charge might experience if placed near some other charge. A mathematical construct that we can imagine into existence at any time.

The electric potential is then a further abstraction, the integral of the field along a path between two points.

In electrodynamics (as opposed to electrostatics which I've been talking about up to now), we can produce a non-conservative electric field. This happens typically when the field is produced by a changing magnetic field rather than by a charged particle. In this case we can still in principle measure the force produced on a charged particle (You could calculate the forces that are required on each electron in an antenna to account for their motion when they are receiving a radio signal, for example). But the scalar potential isn't even a defined quantity since the integral of the electric field will be different for different paths between your two points.

I would say that potential is easier to measure in practical examples.

Suppose we want to measure the electric field across a resistor of a circuit. I requires a test charge moving freely from one end to the other. But how could the charge move freely inside the resistor?

On the other hand, it is possible to join the ends in parallel with a much higher (and known) resistance. The resultant tiny current generates some magnetic field, and can be measured by a device sensible to magnetic force. Using the relation $$V = RI$$ the voltage can be easily measured. So, the device relates magnetic force to magnetic field, to current, and to voltage. And all this describes a voltmeter.

In an electrostatic situation, charge gives rise to both electric field and electric potential. Charge is the source.

Electric potential is a way of "turning" electric field into a scalar quantity. We are incentivized to do this because scalar quantities are easier to deal with. We are able to do this because the curl of electric field vanishes.

Note: If $$\nabla \times \textbf{E} = 0$$, then $$\textbf{E} = -\nabla V$$, i.e. $$E$$ can be written as the gradient of some scalar function. This follows from applications of Stoke's theorem and the fundamental theorem of gradients.

You see, we have the equation $$\nabla\times {\bf E} = 0$$.
Now the curl of a function being zero implies that the function may be constructed as the gradient of some other scalar function.
And that scalar function is what is called the potential, the electric potential, $${\bf E}= -\nabla V.$$
Moving another charge through this potential causes a buildup of potential energy in that system.

So it seems to me that potential is more fundamental. Even in batteries, we keep the potential of the source fixed, so that current i.e. field in the circuit may stay unchanged.

• Let me question the answer^^: You demonstrate quite rightly that given the irrotational nature of the field there is a one-to-one relation between field and potential. How do you jump to the conclusion that the potential is more fundamental? Oct 28, 2022 at 6:28
• It's easier to measure the scaler potential thing, but more physical is the field. Right? Oct 28, 2022 at 6:50