Question

Please help with both problems. Im really lost on this chapter

Group Problem 1. A conducting bar with mass m and resistance R slides on frictionless conducting rails, separated by a distance 1 from each other, in a region that has a static uniform magnetic field B directed into the page (see figure below). An external agent is pushing the bar, maintaing its motion to the right with constant velocity ū. At time t = 0, the agent abruptly stops to push the bar, and the bar contin- ues forward while being slowed down by the magnetic force. Find the speed of the bar as a function of time. Group Problem 2: Magnetic Brakes. You have a summer job at a company that develops systems to safely lower large loads down ramps. Your team is investigating a magnetic system by modeling it in the laboratory. The safety system is a conducting bar that slides on two parallel conducting rails that run down the ramp (similar to the one in the previous problem). The bar is perpendicular to the rails and is in contact with them. At the bottom of the ramp, the two rails are connected together. The bar slides down the rails through a uniform vertical magnetic field. The magnetic field is supposed to cause the bar to slide down the ramp at a constant velocity even when friction between the bar and the rails is negligible. Before setting up the laboratory model, your task is to calculate the constant velocity of the bar (sliding down the ramp on rails in a vertical magnetic field) as a function of the mass of the bar, the strength of the magnetic field, the angle of the ramp from the horizontal, the length of the bar (which is the same as the distance between the rails), and the resistance of the bar. Assume that all of the other conductors in the system have a much smaller resistance than the bar. After finishing this calculation you arrange for a test, using a B field of 0.1 T and rails tilted 30 degrees up from the horizontal. The combined mass of the bar and the loaded box is 400 kg, and the length of the conducting bar is 1 meter. You would like to limit the speed to 1 m/s. Is this design reasonable? Will it work, or should you adjust some parameters? Assume the bar is made of copper. The resistivity of copper is 1.68 x 10-12.m at ambient temperature, and you can choose from bars with cross-sections lcm x 1cm or 2cm x 2cm.

Answer 1

The bar generates an emf due to change in flux in the loop and due to this emf, a current develops in the loop in such a way that it opposes the motion of the bar (Faraday's law).

Problem 1 @ B Let the speed of bar at any particular tinne t be on. Current at that time be I Now, E= Blu I= { = Blu I Now, net force = FB = BXIX l. - Bx Blu xl length=l Resistance =R 2 T Ř Ř Net retardation of bor= F in = 8424 = a a=-du MR . e B?l? u = -du (du = -B² 42 (dt MR dtv u mR MR - 821 24/mR U = ve me