The Magnetic Dipole Moment — Question 6

Question 6

A fixed magnetic moment feels a translational force in a prescribed field gradient

A small permanent dipole is constrained to keep $\vec\mu=.0400\hat z$ ampere square metre as it moves along x. Along its path, the applied field is $\vec B=(B_0+Gx)\hat z$, with G=.300 tesla per metre. Use U=-mu dot B and ignore the dipole effect on the field source.

1. Find the force along x from the potential-energy derivative.

2. Find magnetic work and potential-energy change over a displacement delta x=.200 m.

Question 6 - Solution

A fixed magnetic moment feels a translational force in a prescribed field gradient

(a) Method and calculation

$$U(x)=-\mu(B_0+Gx),\quad F_x=-dU/dx=\mu G.$$$$F_x=(.0400)(.300)=.0120\,\mathrm N.$$

(b) Interpretation and further calculation

$$W_B=F_x\Delta x=(.0120)(.200)=2.40\times10^{-3}\,\mathrm J.$$$$\Delta U=-W_B=-2.40\times10^{-3}\,\mathrm J.$$

Physical check

An aligned fixed moment is pulled toward stronger field in this one-dimensional constrained model.