If there was zero potential difference between any two points in a circuit would current still flow?

Question

Suppose there is a wire of zero resistance.So,by Ohms law it must have 0 potential difference between any two points across the resistor,now one connects the wire to a battery which has non-zero potential difference,would the current still flow through the resistor?

Answer 1

The answer is yes and no.

If a wire has zero voltage across it, and also zero resistance, then by the voltage-current relation

$$V = IR$$

it is consistent that $I = 0$, but it is also equally consistent that $I$ has any nonzero value. Basically, a zero-voltage zero-resistance wire can carry any current one likes. And this part isn't just fantasy: we do have such "zero-resistance wires", they're called superconductors. Once a current is started in one, it continues even with no applied voltage!

But you are asking about something else: you are asking about applying a nonzero voltage across a zero-resistance wire and then pointing out how that since $R = 0$, then $V = 0$ on some wire segment in between the terminals despite $V \ne 0$ across the whole wire.

The problem there is there's a math catch: what's $I$ when you dropped nonzero $V$ across it? The answer is ...

$$I = \frac{V}{R} = \frac{V}{0}$$

for $V \ne 0$. Oops.

So, in fact, your current $I$ is undefined in this situation, and so also is your "voltage across a small segment of wire" - you can't just multiply by $0$ in $V = IR$ and get $0$ out and have everything work.

What does this mean? It means that real-life wires are not modellable by this model under these circumstances.

(But didn't I say zero-resistance wires exist? Yes, they do; but because they're real-life things and not simplistic models, if you just drop voltage across them, other effects intercede to ensure you end up with a defined current. Undefined current - the other half of this - is "fantasy".)

Answer 2

Yes, the current would flow, its value being determined by the internal resistance of the battery.

Answer 3

Zero resistance only implies zero potential difference in steady state.

If you apply a potential difference across a zero resistance component (such as attaching a wire directly to the battery poles) then charge will start flowing and accelerate and accelerate and accelerate... The electric force due to the potential difference is unrestricted due to no resistance in the component, so it will never stop accelerating and will never reach steady state.

In theory, the current will thus forever increase, forever accelerate. In practice, the wire will burn out at some point.

Answer 4

The fact that there is a current flowing makes the potential zero. This current will be limited by the inner resistance of the battery. No current means potential difference.
What about the potential of the battery? How can this be non-zero while the potential difference between the poles is zero? Well, as said, this difference is zero due to the current. A current flowing means that electrons will be pushed to the other side of the battery. This is done by the inner potential of the battery. So, while the poles have zero potential difference (due to the current), the inner parts of the battery will make the current flow.
As soon as the current diminishes a potential will built up to get the electrons going again.

See also this question.