A Current-Carrying Coil as a Magnetic Dipole — Question 3

Question 3

Two axial measurements determine an unknown coil radius before its current is found

A thin circular coil has N=100 turns and unknown radius R and current I. Its measured axial fields satisfy $B(d)/B(2d)=(5/2)^{3/2}$, where d=.0400 m, and B(d)=20.0 microtesla. Use $B(z)=\mu_0NIR^2/[2(R^2+z^2)^{3/2}]$. Use vacuum permeability $\mu_0\simeq4\pi\times10^{-7}\,\mathrm{T\,m/A}$ where numerical values are needed.

1. Find R from the ratio without knowing I.

2. Find I and state why a far-field-only model would lose the radius information in these measurements.

Question 3 - Solution

Two axial measurements determine an unknown coil radius before its current is found

(a) Method and calculation

$$s=\left[\frac{B(d)}{B(2d)}\right]^{2/3}=\frac{R^2+4d^2}{R^2+d^2}=\frac52.$$$$R^2=\frac{4-s}{s-1}d^2=d^2\Rightarrow\boxed{R=.0400\,\mathrm m}.$$

(b) Interpretation and further calculation

$$I=\frac{2B(d)(R^2+d^2)^{3/2}}{\mu_0NR^2}=\frac{4\sqrt2dB(d)}{\mu_0N}=\boxed{.0360127\,\mathrm A}.$$$$B_{\mathrm{dip}}(d)/B_{\mathrm{dip}}(2d)=8\quad\text{independent of }R.$$Only the finite-radius shape of the exact field distinguishes the coil radius from its moment.

Physical check

The measured ratio is smaller than eight, showing that these measurements are not both in the leading dipole regime.