A Circulating Charged Particle — Question 10

Question 10

Linear drag shrinks the speed of a magnetic orbit while the magnetic force remains work-free

A nonrelativistic particle moves perpendicular to uniform B and also experiences isotropic drag $\vec F_d=-b\vec v$, with b positive. Initially its speed is v0. Ignore every other force and define $\omega=|q|B/m$.

1. Derive its speed and kinetic energy as functions of time.

2. Find its total path length before asymptotic rest and the heat transferred to the drag medium; explain which force supplies the energy loss.

Question 10 - Solution

Linear drag shrinks the speed of a magnetic orbit while the magnetic force remains work-free

(a) Method and calculation

$$\frac{dK}{dt}=\vec v\cdot(q\vec v\times\vec B-b\vec v)=-bv^2=-\frac{2b}{m}K.$$$$K(t)=\tfrac12mv_0^2e^{-2bt/m},\quad v(t)=v_0e^{-bt/m}.$$

(b) Interpretation and further calculation

$$s_{\mathrm{total}}=\int_0^\infty v_0e^{-bt/m}\,dt=mv_0/b.$$$$Q_{\mathrm{medium}}=K(0)-K(\infty)=\tfrac12mv_0^2,\quad W_B=0.$$

Physical check

A shrinking orbit does not imply magnetic work when an additional dissipative force is present.