You may have heard in lecture that whenever the magnetic flux through a loop of wire changes, there is an induced EMF. There are many ways this can happen – moving a bar magnet near a stationary loop of wire, turning a loop of wire through a static magnetic field, etc. – but we will focus on a specific type of inductance in this lab: the self-inductance in a loop of wire caused by varying the current through the wire.

Mathematically, the EMF induced in a loop of wire { $\mathcal{E}$ { $\mathcal{E}$ is given by

{ $\mathcal{E} = -N\dfrac{d\varphi}{dt} = -L\dfrac{dI}{dt}$ { $\mathcal{E} = -N\dfrac{d\varphi}{dt} = -L\dfrac{dI}{dt}$ ,

(1)

where

• { $N$ { $N$  is the total number of loops (also called turns) on the coil; * { $\varphi$ { $\varphi$  is the flux through the coil; * { $L$ { $L$  is the coefficient of self-inductance; and * { $I$ { $I$  is the current through the coil.

In this experiment, we will be using a long hollow coil of wire – called a solenoid – as our model inductor.

Have I heard of a different kind of solenoid?

Possibly. The term is used for a long, coiled cylinder of wire in the context of Physics. Such coils are often parts of electromagnets that move objects linearly, and people have subsequently started calling such devices solenoids as well. In particular, what is called a 'starter solenoid' in cars is a device that is used to make the physical connection for current to a vehicle's starter motor.

The flux at the center of a long solenoid is given by

{ $\varphi = BA = \mu_0nAI$ { $\varphi = BA = \mu_0nAI$

(2)

where

• { $B$ { $B$  is the magnitude of the magnetic field; * { $A$ { $A$  is the cross-sectional area of coil; * { $\mu_0 = 4\pi \times 10^{-7} \mathrm{N/A^2}$ { $\mu_0 = 4\pi \times 10^{-7} \mathrm{N/A^2}$  is the permeability of free space; and * { $n$ { $n$  is the number of terns per unit length of the coil (which, if we're working in SI units, will be in turns per meter).

Differentiating Eq. (2) and substituting into Eq. (1), we have

{ $\mathcal{E} = -\mu_0nNA\dfrac{dI}{dt}$ { $\mathcal{E} = -\mu_0nNA\dfrac{dI}{dt}$ ,

(3)

and therefore

{ $L_{\infty} = \mu_0nNA$ { $L_{\infty} = \mu_0nNA$ .

(4)

Note that this self-inductance holds for an infinite solenoid where the flux is constant along the length. In reality, most solenoids are finite, and we must therefore include a correction factor { $K$ { $K$  which accounts for the fact that magnetic flux is smaller at the ends of the solenoid than in the middle:

{ $L = \mu_0nNAK.$ { $L = \mu_0nNAK.$

(5)

This factor depends on the ratio of the solenoid’s length({ $\ell$ { $\ell$ ) to diameter({ $D$ { $D$ ), { $\ell/D$ { $\ell/D$ . Calculation of { $K$ { $K$  is somewhat complex, so we will simply tabulate and plot the results. (See Table 1 and Fig. 1.)

{ $\ell/D$ { $\ell/D$