# What will be the current in a zero resistance wire if apply some finite potential across it ends?

Well ,I know that current can flow through a zero resistance wire if potential difference across its ends is 0 because once electrons are set into motion by application of electric field for a moment in that 0 resistance wire , they keep on flowing . But ,I was thinking what will happen to the current if we apply potential difference across its ends ? According to OHM's law if V is finite and R->0 then I should approach to infinity .Does this really happens or ohm's law fail in this case ? If this is the case as per OHM's law then why don't we apply potential across a superconductor and then it could produce enough current for the world .

• How do you propose to apply this potential difference?
– nasu
Commented May 31, 2021 at 20:51

The Ohm's Law is valid for a steady state condition. If we apply a voltage between the ends of any wire, the current is initially zero, and takes some (very small) time to reach the steady state $$I = \frac{V}{R}$$. While it is increasing, voltage is also proportional to the rate of growth, $$V = L \frac{dI}{dt} + RI$$ where $$L$$ is the inductance of the wire. The solution is:
$$I = \frac{V}{R}(1 + e^{\frac{-Rt}{L}})$$
When $$R > > L$$ the exponential term fades quickly, converging to the Ohm's Law.
When $$R = 0$$, the differential equation is simply: $$V = L \frac{dI}{dt}$$ with the solution:$$I = \frac{Vt}{L}$$