Question

A conducting bar of mass m is placed on two long conducting rails a distance l apart. The rails are inclined at an angle theta with respect to the horizontal, andthe bar is able to slide on the rails with negligible friction. The bar and rails are in a uniform and constant magnetic field of magnitude B orientedperpendicular to the incline. A resistor of resistance R connects the upper ends of the rails and completes the circuit as shown. The bar is released from rest atthe top of the incline.

(e) Express the speed of the bar as a function of time t from the time it is released at t = 0.

Answer 1

voltage generated due to rod of lenght l moving with a velocity v in a magnetic field
from free body diagram of rod,
force on the rod with current i and length l= ilB.
m(dv/dt)= mgsinθ - i lB;
i= (vBl)/R;
m(dv/dt) = mgsinθ - (vB²l²)/R,
let B²l² /Rm=k,
then,
dv/dt + kv=gsinθ,
integrating factor = e^kt.
solving this differential equation we get,
e^kt v=(gsinθ/k)e^kt -(gsinθ)/k
hence
v= [(gsinθ)/k](1-e^-kt). ans.

Answer 2

Answer 3

initial condition should be v=0 at t=0, not v=v0.