Question

Two long parallel horizontal rails, a distance d apart and each having a resistance $\lambda$ per unit length, are joined at one end by a resistance R. A perfectly conducting rod MN of mass m is free to slide along the rails without friction (see figure). There is a uniform magnetic field of induction B normal to the plane of the paper and directed into the paper. A variable force F is applied to the rod MN such that, as the rod moves, a constant current i flows through R. Find the velocity of the rod and the applied force F as functions of the distance x of the rod from R.v

Answer 1

Total resistance of the circuit as function of distance x from resistance R is
$R_{net}=R+2\lambda x$
Let v be velocity of rod at this instant, then motional emf induced across the rod,
e = Bvd
$\therefore Currenti=\frac{e}{R_{net}}=\frac{Bvd}{R+2\lambda x}$
$\therefore v=\frac{(R+2\lambda x)i}{Bd}$
Net force on the rod,
$F_{net}=m\frac{dv}{dt}$
$=\frac{2\lambda im}{Bd}(R+2\lambda )\cdot \frac{dx}{dt}$
$but\frac{dx}{dt}=v=\frac{(R+2\lambda x)i}{Bd}$
$F_{net}=\frac{2\lambda i^{2}m}{B^2{d}^2}(R+2\lambda x{)}^2$
This net force is equal to $F-F_m$ where $F_m$ = idB
$\therefore F=F_{net}+F_m=\frac{2\lambda i^{2}m}{B^2{d}^2}(R+2\lambda x{)}^2+idB$