A Current-Carrying Coil as a Magnetic Dipole — Question 4

Question 4

Canceling the central field of concentric coils leaves a large far-field moment

Two concentric coplanar coils each have N turns, with radii R and 2R. The inner coil has current I and moment along +z. The outer coil carries an adjustable positive current magnitude I2 with moment along -z. Use vacuum permeability $\mu_0\simeq4\pi\times10^{-7}\,\mathrm{T\,m/A}$ where numerical values are needed.

1. Choose I2 to cancel the field at the common center.

2. Find the resulting total dipole moment and the leading field far away on the positive z axis.

Question 4 - Solution

Canceling the central field of concentric coils leaves a large far-field moment

(a) Method and calculation

$$B_z(0)=\frac{\mu_0NI}{2R}-\frac{\mu_0NI_2}{4R}=0.$$$$\boxed{I_2=2I}.$$

(b) Interpretation and further calculation

$$\vec\mu_{\mathrm{total}}=N\pi R^2I\hat z-N\pi(2R)^2(2I)\hat z=\boxed{-7N\pi R^2I\hat z}.$$$$\boxed{\vec B(z)\simeq-\frac{7\mu_0NIR^2}{2z^3}\hat z\quad(z\gg2R)}.$$

Physical check

Central field weights turns by inverse radius, whereas dipole moment weights them by radius squared; canceling one need not cancel the other.