Magnets — Question 6

Question 6

The attraction of coaxial magnets has a fourth-power distance law

Two small permanent magnets have moments $m_1\hat x$ and $m_2\hat x$ and lie on the x axis separated by r, much larger than either magnet. Use $B_{1,\mathrm{axis}}=\mu_0m_1/(2\pi r^3)$. Hold their orientations fixed.

1. Find their interaction energy and radial force on magnet 2.

2. Find the quasistatic external work to increase the separation from r0 to 2r0 and the final-to-initial force-magnitude ratio.

Question 6 - Solution

The attraction of coaxial magnets has a fourth-power distance law

(a) Method and calculation

$$U(r)=-m_2B_1=-\frac{\mu_0m_1m_2}{2\pi r^3}.$$$$\boxed{F_r=-dU/dr=-\frac{3\mu_0m_1m_2}{2\pi r^4}}.$$

(b) Interpretation and further calculation

$$W_{\mathrm{ext}}=U(2r_0)-U(r_0)=\boxed{\frac{7\mu_0m_1m_2}{16\pi r_0^3}}.$$$$\boxed{|F(2r_0)|/|F(r_0)|=1/16}.$$

Physical check

The negative radial force indicates attraction; the interaction energy is counted once, not once per dipole.