Magnets — Question 1

Question 1

Cutting a uniformly magnetized bar preserves the sum of its moments

A uniformly magnetized bar has length L, area A, and magnetization $M\hat x$. It is cut across its length into pieces L/3 and 2L/3, without changing M.

1. Find each magnetic dipole moment and their vector sum.

2. In the equivalent pole model, use pole strength p=MA to compare the new end poles and explain why no isolated pole is produced.

Question 1 - Solution

Cutting a uniformly magnetized bar preserves the sum of its moments

(a) Method and calculation

$$m=MV=MAL,\qquad m_1=MA(L/3),\quad m_2=MA(2L/3).$$$$\boxed{\mathbf m_1+\mathbf m_2=MAL\hat x=\mathbf m}.$$

(b) Interpretation and further calculation

$$p_1=p_2=MA,\quad m_i=p_iL_i.$$$$\boxed{\text{Each piece has two ends with opposite equivalent poles.}}$$

Physical check

The pole model describes the external dipole field; magnetic flux still has zero net outward flux through any closed surface.