Two long parallel guide wires of zero resistance, separated by a distance 1, are

Question

Two long parallel guide wires of zero resistance, separated by a distance 1, are connected to a potential difference Delta V. A bar of ins m and resistance R makes contact with these wires, and can slide along them without friction, always remaining perpendicular to the wires. An externally applied uniform magnetic field B is perpendicular to the plane of the bar and the two wires. Obtain an expression for the velocity of the bar as a function of time, assuming that it starts from rest at time t = 0.

Explanation / Answer

from the definiton of force

F = ma

m is mass and a is acccleration = dv/dt

so

F = m dV/dt --------------------1

from the force on a wire in a magnetic field

F = I L B sin theta

there

theta = 90 so that sin 90 = 1

thus

F = i L B --------------------2

from 1 and 2

m dV/dt = i L B

dV/dt = i LB/m

dV = i LB/m dt

V(t) = i L B/m dt

also from ohms law V = i R

i = V/R

v(t) = L B Vdt /(m R)

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