A Current-Carrying Coil as a Magnetic Dipole — Question 5

Question 5

Canceling the dipole moment of concentric coils leaves a faster-decaying axial field

Two concentric coplanar N-turn coils have radii R and 2R. Their moments oppose, with inner current I along the +z moment sense. Use the exact axial coil field and $(1+u)^{-3/2}=1-3u/2+O(u^2)$. Use vacuum permeability $\mu_0\simeq4\pi\times10^{-7}\,\mathrm{T\,m/A}$ where numerical values are needed.

1. Choose the outer current magnitude I2 to cancel total dipole moment and find the central field.

2. Derive the leading nonzero axial field for z much larger than 2R.

Question 5 - Solution

Canceling the dipole moment of concentric coils leaves a faster-decaying axial field

(a) Method and calculation

$$NI\pi R^2-NI_2\pi(2R)^2=0\Rightarrow\boxed{I_2=I/4}.$$$$B_z(0)=\frac{\mu_0NI}{2R}-\frac{\mu_0N(I/4)}{4R}=\boxed{\frac{7\mu_0NI}{16R}}.$$

(b) Interpretation and further calculation

$$B_z=\frac{\mu_0NIR^2}{2z^3}\left[\left(1+\frac{R^2}{z^2}\right)^{-3/2}-\left(1+\frac{4R^2}{z^2}\right)^{-3/2}\right].$$$$\boxed{B_z\simeq\frac{9\mu_0NIR^4}{4z^5}}\quad(z\gg2R).$$

Physical check

The dipole term cancels, but the fields are not identical functions of position. Coplanar symmetry also removes the intervening axial inverse-fourth-power term.