Inductors and Inductance — Question 8

Question 8

Geometric scaling changes inductance differently when turns or turn density is fixed

An ideal long solenoid has $L=\mu_0N^2A/\ell$. Every linear dimension is enlarged by factor s while the material permeability is unchanged.

1. Find L scaling when N is fixed.

2. Find L scaling when the turns per unit length n is fixed instead.

Question 8 - Solution

Geometric scaling changes inductance differently when turns or turn density is fixed

(a) Method and calculation

$$A\prime=s^2A,\quad\ell\prime=s\ell,\quad N\prime=N.$$$$\boxed{L\prime/L=s}\quad\text{at fixed }N.$$

(b) Interpretation and further calculation

$$n=N/\ell\text{ fixed}\Rightarrow N\prime=sN.$$$$\boxed{L\prime/L=s^3}\quad\text{at fixed }n.$$

Physical check

The two scaling laws correspond to different winding constraints; specifying only the size change is insufficient.