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1. Induction driven oscillation: (total 30 points) Due to Faraday's induction, a current system acquires an “inertia” that limits any current change. Just like it is difficult for a massive object to change its state of motion, it is difficult for a system of a large inductance to change its current flow. In this problem, we study the consequence of this induction driven “inertia”. A rectangular conductor loop is placed in the xy plane and is positioned in alignment with the x and y axes. It has a mass m and a side length h along the y direction. The conductor loop has negligible resistance. It is is connected with an inductor of inductance L on the right, but the other segments of the loop have negligible contributions to the self-inductance. As shown in Figure 1, the loop is partially inserted into a rectangular region of uniform, constant external magnetic field B = -B, 2 (into the plane), and that portion has a length l. The rectangular loop is free to move along the x direction as a rigid body, without any friction. [The conductor loop never entirely exits the region of external magnetic field.] 1 Х I x В x 个 X h Х X Ilus Х ! X т. Is

(a) (5 points) Show that the total magnetic flux through the conductor loop is conserved. Note that the total magnetic flux is the sum of two contributions: (1) the external magnetic field; (2) the 1 loop's own magnetic field when a current flows through the inductor. (b) If the loop has a velocity in the x direction, i.e. a nonzero dl(t)/dt, a motional electromotive force will induce a current I (which in general can be time varying I = I(t)). The current is in turn subject to a Lorentz force due to the external magnetic field, which alters the motion of the conductor loop along the x direction. (10 points) To quantify this feedback mechanism, derive a differential equation for I(t), and show that I(t) exhibits harmonic oscillation. Find the angular frequency of the oscillation. [Note: You should choose the sign of I in accord with the current direction indicated in Figure 1.] (c) (10 points) Suppose that initially the loop is free of current 7(0) = 0, and is stationary with an inserted portion (0) = lo into the region of external magnetic field. At t = 0, an impulsive push is imparted to the loop so that it acquires an initial velocity uo moving to the left. Find the position of the loop as a function of time, I = l(t), and the time varying current, I = I(t), for t> 0.