Magnets — Question 7

Question 7

Parallel side-by-side dipoles repel despite having parallel moments

Two small magnets have parallel fixed moments along +y and are separated by r along +x. At the second magnet the equatorial field of the first is $\mathbf B_1=-\mu_0m_1\hat y/(4\pi r^3)$.

1. Find their interaction energy and radial force.

2. Compare the force magnitude with that of the same magnets in the coaxial parallel arrangement; then reverse only m2 and determine the new force sign.

Question 7 - Solution

Parallel side-by-side dipoles repel despite having parallel moments

(a) Method and calculation

$$U=-\mathbf m_2\cdot\mathbf B_1=\frac{\mu_0m_1m_2}{4\pi r^3}.$$$$\boxed{F_r=+\frac{3\mu_0m_1m_2}{4\pi r^4}}\quad\text{(repulsion)}.$$

(b) Interpretation and further calculation

$$\boxed{|F_{\mathrm{side}}|/|F_{\mathrm{axial}}|=1/2}.$$$$\mathbf m_2\to-\mathbf m_2\Rightarrow U\to-U,\quad\boxed{F_r\to-F_r}\quad\text{(attraction)}.$$

Physical check

Parallel moments alone do not determine attraction; the separation direction also enters the dipole field.