The inductance $L$ of a long solenoid of length $\ell$, cross-section area $A$, and turns per length $n$ is given by: $$ L = \mu_0 n^2 \ell A $$ where $\mu_0$ is the magnetic constant.
I am currently attempting to derive this from Ampere's law and Faraday's law. Using Ampere's law, I have been able to show that the magnitude of the magnetic field inside the solenoid is given by $|B| = \mu_0 n I$. Faraday's law states $$ \mathcal{E} = -\frac{d}{dt}\int_S B \cdot dA $$ where $S$ is a surface having a (closed curve) $C$ as its boundary, and $\mathcal{E}$ is the induced EMF along $C$. I define $C$ to follow the wire along a spiraling path along the solenoid, and then along a simple path back around to the starting point. In this way, the resulting EMF along this curve should correspond to the voltage across the solenoid. However, I am having a hard time picturing the induced surface $S$ and assessing the flux integral rigorously.
Intuitively, one could argue that $S$ is some sort of spiralling surface, which is pierced repeatedly by the magnetic field inside the solenoid, one time for each loop. In this case, the flux integral becomes $(B A) n \ell$, and we get $L = v/\frac{dI}{dt} = (\frac{d}{dt} BA n \ell)/(\frac{dI}{dt}) = \mu_0 n^2 \ell A$, as desired. However, this argument is not as clear or formal as I would like it to be.
How can one derive this result more rigorously? Is there even a well-defined surface induced by $C$, or does a different closed curve $C$ need to be selected?
