Field Inside a Solenoid

n = N/L is turns per unit length. Derived via Ampere's law on a rectangular loop. Field is uniform inside, zero outside an ideal infinite solenoid.

Setup

An ideal solenoid: $N$ total turns, length $L$, $n = N/L$ turns per unit length, carrying current $I$. Ideal means length $\gg$ radius and turns are closely wound.

Field Structure

For an ideal solenoid:

• Field inside is uniform, directed along the axis.
• Field outside is negligible (fields from adjacent turns cancel outside).

Rectangular Amperian Loop

Choose loop $abcd$ with side $ab$ of length $\ell$ inside the solenoid (parallel to axis) and side $cd$ outside:

$$\oint \vec{B} \cdot d\vec{l} = B\ell$$

Applying Ampere's Law

Turns enclosed by loop: $n\ell$. Each carries current $I$:

$$B\ell = \mu_0\,n\ell\,I$$ $$\boxed{B = \mu_0 n I}$$

Key Features

• $B$ is uniform inside — independent of position or radius.
• For a ferromagnetic core of relative permeability $\mu_r$: $B = \mu_0\mu_r nI$.

Field at the Ends

At either open end of a finite solenoid:

$$B_{\text{end}} = \frac{\mu_0 n I}{2}$$

This follows because an infinite solenoid can be split into two semi-infinite solenoids at any interior point.