A Current-Carrying Coil as a Magnetic Dipole — Question 9

Question 9

Spatially separated opposite dipoles retain an inverse-fourth-power far field

Two small coils are modeled as point dipoles: $+\mu\hat z$ at z=+a and $-\mu\hat z$ at z=-a. Consider z>a on the axis, much farther from each coil than its radius. Use the signed axial point-dipole field $B_z=\mu_0\mu_z/(2\pi r^3)$. Use vacuum permeability $\mu_0\simeq4\pi\times10^{-7}\,\mathrm{T\,m/A}$ where numerical values are needed.

1. Write the net axial field and derive its leading far-field form for z much larger than a.

2. Compare its decay with that of one dipole and explain why zero total moment does not imply zero field everywhere.

Question 9 - Solution

Spatially separated opposite dipoles retain an inverse-fourth-power far field

(a) Method and calculation

$$B_z(z)=\frac{\mu_0\mu}{2\pi}\left[\frac1{(z-a)^3}-\frac1{(z+a)^3}\right].$$$$\frac1{(z\mp a)^3}=\frac1{z^3}\left(1\pm\frac{3a}{z}+O(a^2/z^2)\right)\Rightarrow\boxed{B_z\simeq\frac{3\mu_0\mu a}{\pi z^4}}.$$

(b) Interpretation and further calculation

$$B_{\mathrm{single}}=\frac{\mu_0\mu}{2\pi z^3},\quad\boxed{B_z/B_{\mathrm{single}}\simeq6a/z}.$$$$\vec\mu_{\mathrm{total}}=0,\quad\text{but}\quad(z-a)^{-3}\ne(z+a)^{-3}.$$The leading dipole term cancels; unequal source distances leave a higher-order field.

Physical check

Separated dipoles lack the coplanar symmetry of concentric opposed coils, so their first surviving axial far-field term has a different power.