If we assume the solenoid to be long, then we can assume the magnetic field inside to be uniform. We can apply Ampere's Law to the loop
(1)
If there are
turns of wire per metre of the solenoid them the loop
will enclose
turns of wire, each carrying a current I, so will enclose a total current of
The right hand side of (1) is then
As for the left hand side, outside the coil the magnetic field is zero so this part of the path contributes nothing to the integral. Inside the coil, the magnetic field is a uniform
so for the coil of length
the left hand integral is
Hence
cancels from both sides to give