It is proven only when there is a potential difference current flows in that branch between two nodes, then how we say that the current flows in a short circuit where the voltage on that branch is 0?
It is proven only when there is a potential difference current flows in that branch between two nodes ...
No, it's not.
If there's a resistor between those two nodes, then yes.
If there's zero resistance, then whatever current that there is flowing will generate no voltage. Just think of a superconductor.
If there's an inductor between the two nodes, then you can have a current flowing, with the voltage related to how fast the current is changing, not to the actual value of the current.
In a real and practical circuit, there is no such thing as 'zero resistance', ergo no such thing as 'infinite current', which is what one might expect from the math.
Besides unavoidable wire resistance, there is also a 'source impedance'... Real voltage sources always have some internal resistance. In an actual short circuit, the source impedance would be what limits the current to something less than infinity.
As a simple example - In an alkaline battery, there's about 1/4 to 1 ohm resistance, meaning from a typical AA, C, D etc you're limited to a few amps even if you short the terminals with a superconducting wire. After some time, the battery starts to get hot. That's caused by the power being dissipated over the internal resistance.
... how we say that the current flows in a short circuit where the voltage on that branch is 0?
Figure 1. There is a short-circuit between A and B.
I can think of two ways of looking at the problem.
- A and B are the same node of the circuit. Current through that node is determined by the other elements of the circuit.
- If current didn't flow from A to B then VA-B would rise to a non-zero value and the current would have to flow.