A Current-Carrying Coil as a Magnetic Dipole — Question 7

Question 7

Opposite coil currents produce a field gradient whose optimal spacing is finite

Two identical N-turn coils of radius R lie at z=-s and +s. Their equal current magnitudes I give the left coil moment along +z and the right coil moment along -z. Thus $B_z(z)=C[f(z+s)-f(z-s)]$, with $C=\mu_0NIR^2/2$ and $f(u)=(R^2+u^2)^{-3/2}$. Use vacuum permeability $\mu_0\simeq4\pi\times10^{-7}\,\mathrm{T\,m/A}$ where numerical values are needed.

1. Find the axial field at the midpoint and the gradient $G=dB_z/dz$ there.

2. For fixed R, N and I, find s that maximizes the gradient magnitude.

Question 7 - Solution

Opposite coil currents produce a field gradient whose optimal spacing is finite

(a) Method and calculation

$$B_z(0)=0,\quad f\prime(u)=-\frac{3u}{(R^2+u^2)^{5/2}}.$$$$\boxed{G=2Cf\prime(s)=-\frac{3\mu_0NIR^2s}{(R^2+s^2)^{5/2}}}.$$

(b) Interpretation and further calculation

$$\frac d{ds}\frac{s}{(R^2+s^2)^{5/2}}=\frac{R^2-4s^2}{(R^2+s^2)^{7/2}}=0\Rightarrow\boxed{s=R/2}.$$$$\boxed{|G|_{\max}=\frac32\left(\frac45\right)^{5/2}\frac{\mu_0NI}{R^2}}.$$

Physical check

Here the same separation as a Helmholtz pair maximizes a gradient, because the current senses are opposite. The midpoint field itself is zero.