A Current-Carrying Coil as a Magnetic Dipole — Question 1

Question 1

The axial dipole approximation has a measurable finite-radius error

A thin circular N-turn coil of radius R lies in the xy plane. Current I gives moment $\vec\mu=NI\pi R^2\hat z$. Consider an on-axis point at z>0. Use the Biot–Savart law and pairwise cancellation of transverse components. Use vacuum permeability $\mu_0\simeq4\pi\times10^{-7}\,\mathrm{T\,m/A}$ where numerical values are needed.

1. Derive the exact axial field and its far-field dipole limit.

2. Find the minimum z/R for the dipole formula to overestimate the exact field by no more than 5.00 percent, using exact field in the error denominator.

Question 1 - Solution

The axial dipole approximation has a measurable finite-radius error

(a) Method and calculation

$$dB_z=\frac{\mu_0NI}{4\pi}\frac{R\,d\ell}{(R^2+z^2)^{3/2}},\quad\oint d\ell=2\pi R\Rightarrow\boxed{B_z=\frac{\mu_0NIR^2}{2(R^2+z^2)^{3/2}}}.$$$$z\gg R:\quad\boxed{B_{\mathrm{dip}}=\frac{\mu_0\mu}{2\pi z^3}}.$$

(b) Interpretation and further calculation

$$\epsilon=\frac{B_{\mathrm{dip}}}{B_z}-1=\left(1+\frac{R^2}{z^2}\right)^{3/2}-1\le.0500.$$$$\boxed{z/R\ge\left[(1.0500)^{2/3}-1\right]^{-1/2}}.$$

Physical check

The dipole expression preserves the moment but discards finite-radius corrections; its leading relative error is proportional to R squared over z squared.