Solenoid

A solenoid is best understood as a coil of wire, which may or may not have a magnetic core inside. They find considerable use as inductors and in turn as transformers, which are a foundational electrical device.

Let’s define some important parameters. $N$ is the total number of turns for the entire coil. $n$ is the number of turns per unit length, and $L$ is the total length of the solenoid, such that:

$$N=nL$$

Magnetic field

We usually use Ampère’s law to find the magnetic field for an infinite solenoid. Note that the field will go through the centre of the coil, i.e., for a coil in the $\hat{z}$ direction, the field will be such that:

$$\vec{B}=B(z)\hat{z}$$

Our Amperian loop should be set as follows:¹ Since the field is only non-zero within the solenoid, the only term that contributes anything in the contour integral is the line segment parallel to the field, such that:

$${\oint }_C\vec{B}\cdot d\vec{l}=BL={\mu }_0{i}_{\text{enc}}$$

Footnotes

1. From Introduction to Electrodynamics, by David J. Griffiths. ↩