Chapter 30, Problem 20P

You are working in a factory that produces long bars of copper with a square cross section. In one section of the production process, the bars must slide down an inclined plane of angle θ. It has been found that the bars travel with too high a speed and become dented or bent when they arrive at the bottom of the plane and must be discarded. In order to prevent this waste, you devise a way to deliver the bars at the bottom of the plane at a lower speed. You replace the inclined plane with a pair of parallel metal rails, shown in Figure P30.19, separated by a distance ℓ. The smooth bars of mass m will slide down the smooth rails, with the length of the bar always perpendicular to the rails. The rails are immersed in a magnetic field of magnitude B, and a resistor of resistance R is connected between the upper ends of the rails. Determine the magnetic field necessary in your device so that the bars will arrive at the bottom of the plane with a maximum speed v_max.