Solenoids and Toroids — Question 1

Question 1

Stretching a long solenoid changes its field through the turn density

A tightly wound air-core solenoid has N=1200 turns uniformly distributed over length $\ell=.600$ m. Its radius is much smaller than its length and its current is I=.800 A. Neglect end effects at the center. Use vacuum permeability $\mu_0\simeq4\pi\times10^{-7}\,\mathrm{T\,m/A}$ where numerical values are needed.

1. Derive the interior field from an Ampère rectangle and evaluate it.

2. The solenoid is stretched to 1.50 times its length with N fixed. Find the field at fixed current and the current required to restore the initial field.

Question 1 - Solution

Stretching a long solenoid changes its field through the turn density

(a) Method and calculation

$$B L=\mu_0(nL)I,\quad n=N/\ell\Rightarrow\boxed{B=\mu_0NI/\ell}.$$$$B=(4\pi\times10^{-7})(1200)(.800)/.600=\boxed{2.01062\,\mathrm{mT}}.$$

(b) Interpretation and further calculation

$$B\prime/B=(N/1.50\ell)/(N/\ell)=2/3\Rightarrow B\prime=1.34041\,\mathrm{mT}.$$$$\boxed{I\prime=1.50I=1.20\,\mathrm A}\quad\text{restores }nI.$$

Physical check

The radius does not enter the ideal long-solenoid result, but it still controls whether the end-effect approximation is valid.