Solenoids and Toroids — Question 3

Question 3

A center-field tolerance places a quantitative limit on the long-solenoid approximation

For a thin finite solenoid, the exact center field is $B_c=\mu_0nI/\sqrt{1+(2R/\ell)^2}$. An engineer uses $B_\infty=\mu_0nI$. Define relative error as $(B_\infty-B_c)/B_c$. Use vacuum permeability $\mu_0\simeq4\pi\times10^{-7}\,\mathrm{T\,m/A}$ where numerical values are needed.

1. Find the minimum ell/R for at most 1.00 percent error.

2. For ell=10R, calculate the actual error and the multiplicative current correction that produces the target field predicted by the ideal model.

Question 3 - Solution

A center-field tolerance places a quantitative limit on the long-solenoid approximation

(a) Method and calculation

$$\epsilon=\sqrt{1+4R^2/\ell^2}-1\le.0100.$$$$\boxed{\ell/R\ge\frac2{\sqrt{(1.01)^2-1}}=14.1069}.$$

(b) Interpretation and further calculation

$$\ell=10R:\quad\epsilon=\sqrt{1.04}-1=\boxed{1.98039\%}.$$$$\boxed{I_{\mathrm{corrected}}/I_{\mathrm{ideal}}=\sqrt{1.04}=1.019804}.$$

Physical check

The ideal model overestimates the center field at fixed current, so the compensating current must be larger.