# Electric potential inside a polarised conductor

Say a conductor with an initial electric potential of zero is subject to an arbitrary charge. I understand that because if this outside charge, there would be charge distribution inside the conductor, so as to make the electric field in it zero. What happens to the initial electric potential inside the conductor? Would it be greater than zero since now one side of the conductor is positively charged and another negatively?

• It depends on how you manipulate your conductor. If it is insulated from the environment, it's potential will generally change in order to conserve its charge (which I think was what you had in mind). The value and sign of the change depends crucially on the charge and the geometry of the problem. However, you can also fix the potential of a conductor, like when you ground it or apply the voltage from a battery. In this case, by definition the voltage won't change even if it is polarised, which is not contradictory as generally its charge will vary to compensate.
– LPZ
Jul 11, 2022 at 15:58

## 2 Answers

Let's be a little more precise about what we mean by a zero potential. We'll take the potential of earth to be zero, and before we bring up the charge we'll connect our conductor to earth to make its potential zero as well. Then we disconnect the conductor from earth.

Now we bring up the external charge, and as you say it will polarise the conductor. The question is whether the potential of the conductor has been changed, and the simple way to test this is to connect it to earth again and see if any charge flows between earth and the conductor. If no charge flows the potential of the conductor must be unchanged, and if charge flows the potential must have changed.

And if we tried this we would find that charge does flow between earth and the conductor as soon as we connect them. If we bring up a positive charge and connect the conductor to earth we'll find electrons flow from earth onto the conductor to give it a net negative charge. Likewise if we bring up a negative charge we'll find electrons flow off the conductor to earth giving the conductor a net positive charge. Either way bringing the external charge close to the conductor does change its potential relative to earth.

Actually calculating the change in the potential would be hard, and if would depend on the size and shape of the conductor. However our thought experiment makes it clear that the potential does change.

By this question, I am guessing that you are wondering how physics textbooks and such claim that the potential difference inside of a conductor is zero, even though for the charges to move to either side, there must have been some potential difference inside the conductor the first place!

This is a good question, and the key insight is that the properties of conductors (charge only occurs on the surface, potential inside is constant, etc...) are only well-defined in the electrostatic regime. When a charged object is brought close to a conductor, there actually is a potential difference inside the conductor initially! However, recall that conductors are made up of free charges which rapidly flow across that potential difference and reach equilibrium. This all occurs in an extremely short amount of time, and as long as you look at the equilibrium situation, there really is constant potential in a conductor.

Another way to think about this is by contradiction. Suppose that there was a potential difference inside the conductor. In that case, charges would naturally move down that potential difference to a lower energy position and thereby remove the potential difference!

In electrostatics, you are only dealing with the situation after everything has moved to its equilibrium position inside the conductor because it all happens so quickly.

More directly to your question, the potential difference caused by the external charge and the potential of the charges on your conductor's surface cancel out perfectly to produce constant potential inside the conductor.

• The potential is constant inside the conductor but it does not have to be zero. All we require is that $\nabla V = 0$. Or did you mean to say the electric field is zero inside the conductor? Jul 11, 2022 at 17:56
• You are correct. thank you for noticing!
– ER06
Jul 11, 2022 at 18:01