PHYSICS 241E - Pledged Problem 1 - Section 7
Due November 25, 1996

A rod with length L, mass m, and resistance R slides without friction down parallel rails of negligible resistance, as shown. The rails are connected at the bottom as shown, forming a conducting loop with the rod as the top member. The plane of the rails makes an angle with the horizontal and a uniform vertical magnetic field exists throughout the region.

a) What is the steady state terminal velocity of the rod in terms of the noted quantities and physical constants?

Constant velocity requires that . Define a coordinate system as shown. Then,

To calculate

where . This will cause a current to flow in the counter-clockwise sense as viewed from above, the magnitude of which is given by

The current through the sliding rod flows in the horizontal plane perpendicular to the vertical field so will experience a horizontal force given in the above coordinate system by

Finally, , the force exerted by the rails must be perpendiular to their surface in the absence of friction and will have whatever magnitude is required to support the sliding bar. The component will be zero as required if

Thus,

b) What is the rate of power dissipation in the resistive bar?

Alternatively, one can use Conservation of Energy, noting that the dissipated energy must come from the change in gravitational potential energy of the rod: