Chapter 29 Problem 33

Problem

A conducting rod rests on two long frictionless parallel rails in a magnetic field ${\vec {B}}$ ( $\perp$ to the rails and rod)

(a) If the rails are horizontal and the rod is given an initial push, will the rod travel at constant speed even though a magnetic field is present?

(b) Suppose at $t=0$, when the rod has speed $v=v_{0}$ , the two rails are connected electrically by a wire from point a to point b. Assuming the rod has resistance $R$ and the rails have negligible resistance, determine the speed of the rod as a function of time. Discuss your answer.

Solution

(a)

yes, the rod travels at constant speed.

Changing flux would cause an emf across the rod, but no current can flow. Without the current, there is no force to oppose the motion of the rod.

(b)

${\mathcal {E}}=-{\frac {d\Phi _{B}}{dt}}=-{\frac {BdA}{dt}}=-{\frac {Blvdt}{dt}}=-Blv$

$I={\frac {\mathcal {E}}{R}}=-{\frac {Blv}{R}}$

$F=ma=m{\frac {dv}{dt}}=IlB=-{\frac {B^{2}l^{2}}{R}}v$

$\rightarrow {\frac {dv}{v}}=-{\frac {B^{2}l^{2}}{mR}}dt$

$\int _{v_{0}}^{v}{\frac {dv^{\prime }}{v^{\prime }}}=-{\frac {B^{2}l^{2}}{mR}}\int _{0}^{t}dt^{\prime }$ $\rightarrow \ln {\frac {v}{v_{0}}}=-{\frac {B^{2}l^{2}}{mR}}t$

$v(t)=v_{0}e^{-{\frac {B^{2}l^{2}}{mR}}t}$