Proof of the formula for the flux density inside a long solenoid

Although most of the formulae for magnetic fields have been quoted without proof, it is interesting to read through one example. This one uses Ampere's law.

Consider a closed path PQRS around a long thin solenoid (Figure 1). The field outside is negligible compared with that inside, and if the solenoid is thin we can also ignore sections QR and SF. This leaves only the length inside the solenoid RS.

The field due to each element of the solenoid is parallel to RS and so:

line integral = ∫Bcos dθ dI = B x RS
But RS = L, the length of the solenoid, and if the solenoid has N turns and carries a current I we have:

BL = μ_oNI and so: