Solenoid

Consider a long, closely-wound helical coil of conducting wire -- a SOLENOID -- with its diameter assumed much smaller than its length (Fig. 10.24). Experimentally (and as confirmed by the Ampere's-law derivation below), the magnetic field lines INSIDE such a solenoid, sufficiently far from its two ends, run parallel to the solenoid's own axis and are closely, uniformly spaced -- indicating a strong, essentially UNIFORM field there -- while OUTSIDE the solenoid the field lines spread out widely and are very sparse, indicating a comparatively very weak field. In the idealised limit of an INFINITELY long, perfectly and tightly wound solenoid, this qualitative picture becomes exact: the field inside is perfectly uniform, and the field outside is exactly zero everywhere.

To find the field inside such an ideal solenoid using Ampere's law, choose a RECTANGULAR Amperian loop $abcd$ (Fig. 10.25), with one side, say $ab$ (of length $L$), lying entirely INSIDE the solenoid, parallel to its axis; the opposite side $cd$ lying entirely OUTSIDE the solenoid; and the remaining two sides $bc$ and $da$ each crossing from inside to outside, PERPENDICULAR to the solenoid's axis. Evaluating $\oint\vec{B}\cdot d\vec{l}$ around this loop, side by side: along $ab$ (inside, parallel to $B$), the contribution is simply $B\cdot L$; along $cd$ (outside, where $B=0$ for an ideal solenoid), the contribution is exactly zero; and along the two perpendicular sides $bc$ and $da$, $\vec{B}$ (wherever it is nonzero, i.e. on the portion inside the solenoid) is PERPENDICULAR to $d\vec{l}$, so $\vec{B}\cdot d\vec{l}=0$ there too. So the ENTIRE line integral reduces to just the one nonzero contribution from side $ab$:

Setting this equal to $\mu_0$ times the net current ENCLOSED by the loop: if there are $n$ turns per unit length of the solenoid, each carrying current $i$, then the loop (of length $L$ along the solenoid's axis) encloses a net current of $I_{enc}=nLi$ (that is, $nL$ individual turns, each contributing current $i$). Ampere's law then gives

$BL = \mu_0(nLi) \quad\Rightarrow\quad B = \mu_0 n i.$ …