Consider a long solenoid with $n$ turns per unit length carrying current $I$.

Apply Ampere's circuital law to a rectangular Amperian loop ABCD:
- AB is inside the solenoid (parallel to axis), length $L$.
- CD is outside (field $\approx 0$).
- BC and DA are perpendicular (no contribution).

$$\oint \vec{B} \cdot d\vec{l} = B \cdot L + 0 + 0 + 0 = BL$$

Current enclosed = number of turns in length $L$ × $I$ = $nLI$

By Ampere's law:
$$BL = \mu_0 nLI$$
$$\boxed{B = \mu_0 nI}$$

The field is uniform inside and zero outside a long solenoid.