Inductors and Inductance — Question 1

Question 1

A solenoid inductance follows from flux linkage rather than flux through one turn

An ideal long air-core solenoid has N=800 turns, cross-sectional area $A=2.00\times10^{-4}$ square metre and length ell=.400 m. Use $B=\mu_0NI/\ell$ and $\mu_0\simeq4\pi\times10^{-7}$ T m/A.

1. Derive and evaluate its inductance.

2. Find the change if N doubles at fixed geometry, and distinguish linkage from single-turn flux.

Question 1 - Solution

A solenoid inductance follows from flux linkage rather than flux through one turn

(a) Method and calculation

$$\Phi=BA=\mu_0NIA/\ell,\quad\Lambda=N\Phi=\mu_0N^2IA/\ell.$$$$\boxed{L=\Lambda/I=\mu_0N^2A/\ell=.402124\,\mathrm{mH}}.$$

(b) Interpretation and further calculation

$$N\to2N:\quad\Phi\to2\Phi,\quad\Lambda\to4\Lambda\quad(I\text{ fixed}).$$$$\boxed{L\prime=4L=1.60850\,\mathrm{mH}}.$$

Physical check

Both the field per turn and the number of linked turns increase when turns are added.