Magnetic Force on a Current-Carrying Wire — Question 9

Question 9

Summing carrier forces explains why wire force follows conventional current

A neutral wire segment has cross-sectional area A and directed length $L\hat x$. Its mobile carriers have density n, signed charge q and drift velocity $v_d\hat x$. A uniform transverse B acts externally. The lattice has no mean drift.

1. Sum the magnetic forces on the carriers and rewrite the result using conventional current I.

2. Compare the force for positive carriers drifting right and negative carriers drifting left when both produce the same positive I. Explain why electrical neutrality does not force the magnetic sum to vanish.

Question 9 - Solution

Summing carrier forces explains why wire force follows conventional current

(a) Method and calculation

$$N=nAL,\quad\vec F=Nqv_d\hat x\times\vec B.$$$$I=nqAv_d\Rightarrow\vec F=IL\hat x\times\vec B.$$

(b) Interpretation and further calculation

$$q>0,v_d>0\text{ or }q<0,v_d<0:\quad qv_d>0.$$$$\rho_{\mathrm{total}}=0\quad\text{does not imply}\quad\sum_iq_i\vec v_i=0.$$Internal forces transmit the carrier momentum change to the wire as a whole.

Physical check

Magnetic force weights charge by velocity; stationary compensating lattice charge does not cancel the carrier magnetic force.