Question

A pair of conducting, parallel, frictionless rails is mounted on an insulating platform. The distance between the rails is L=0.30 m. A light conducting bar can slide on the rails without friction. When the conducting bar is at x = 0, the enclosed area of the loop is 0.03 m². There is zero resistance in the conducting bar or rails, the rails are connected on one end by a R=1.5 resistor. A uniform magnetic field of B -0.25 T is normal to the loop and pointing out of the plane. At r = 0, the conducting bar is at x = 10 m and is pushed to the left at constant velocity at a constant speed of v = 12 m/s. (Note: The figure is not to scale) a) Sketch the direction of the induced current b) Calculate the potential (voltage) across the 1.5 resistor, c) Find the magnitude of the induced current d) Calculate the power dissipation in the 1.5 resistor e) Calculate the magnitude of force that must be applied to move the bar at constant speed 1.50 1 X=0 x = 10 m

Answer 1

T> R 1.5 x-lom HO m Whenever there is a change in magnetic flux linked to a loop due to change in magnetic field, area of loop or orientation of loop with respect to magnetic field direction then emf is induced in the loop. According to Faraday's law, the emf E induced is the negative change in the magnetic flux per unit time. døm dt The direction in which the induced emf drives current around a wire loop can be found through the negative sign. It is determined using Lenz's law. According to Lenz's Law, the direction of the induced emf drives current in the loop to always oppose the change in magnetic flux that causes the emf. When the bar moves into the magnetic field a current is induced in the loop in such a way that induced current opposes the decrease in magnetic flux. So it produces the induced field which opposes the change in the flux caused by moving bar and to oppose this, current in the loop must induce counterclockwise. Therefore, induced current circulates counterclockwise in the loop. As the bar moves into the field, area of the loop inside the magnetic field decreases and magnetic flux linked to rectangular loop decreases thus according to Faraday's law emf is induced in the loop. When bar enters with velocity

in magnetic field, then in time dt bar enters in the magnetic field a distance, dx = vdt Now, calculate the magnetic flux through this loop. dom = BdA here, dA is the area of the infinitesimal rectangular element. dA = Lvdt Substitute Lvdt for dA in døm BdA LBvdt døm dem = BvL dt Now, time rate of change of flux is, døm - = BvL dt

Now, magnitude of induced emf, do dt E = BvL This induced emf drives current in the loop whose magnitude is given by, & R BvL R Here, R is the resistance of the loop. magnitude of induced current ByL i = R (0.25) x (1.2) x (0.3) 1.5 i = 0.06A Potential across resistor V = iR V = (0.06) (1.5) V = 0.09V Power dissipated in the resistor P=iR P = (0.062 x (1.5) P = 5.4 x 10-3W Due to this induced current, force acting on the loop which is given by

7-12) Here, 1 - is the length of the bar L the force is determined using is taken in the direction of the current and direction right hand rule and will be in the direction of LXB Which is towards right. Therefore an equal and opposite force must be applied to move the bar at constant velocity. magnitude of force applied F = ILB F = (0.06) (0.3) (0.25) F = 4.5 x 10-3N