Magnetic Force on a Current-Carrying Wire — Question 10

Question 10

A moving current-carrying rod exchanges mechanical power with the maintained current source

A rod of length L along +y carries maintained current I upward and moves with speed u along +x in a uniform field $B\hat z$. Use the supplied motional-emf relation $\mathcal E_{\mathrm{mot}}=\int(\vec u\times\vec B)\cdot d\vec\ell$, integrated upward along the rod.

1. Find the rod force and the mechanical power delivered by the magnetic interaction.

2. Find the signed motional emf and verify the relation between mechanical power and the electrical power needed to oppose this emf.

Question 10 - Solution

A moving current-carrying rod exchanges mechanical power with the maintained current source

(a) Method and calculation

$$\vec F=IL\hat y\times B\hat z=ILB\hat x.$$$$P_{\mathrm{mech}}=\vec F\cdot\vec u=ILBu.$$

(b) Interpretation and further calculation

$$\vec u\times\vec B=-uB\hat y\Rightarrow\mathcal E_{\mathrm{mot}}=-uBL.$$$$P_{\mathrm{mech}}=-I\mathcal E_{\mathrm{mot}}=IuBL.$$The maintained-current source supplies the converted energy, in addition to any resistive losses.

Physical check

There is no contradiction with zero magnetic work on an individual charge: the full circuit and source supply the mechanical energy.