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 difcult for a massive object to change its state of motion, it is difcult for a system of a large inductance to change its current ow. 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 11 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 selfinductance. As shown in Figure 1, the loop is partially inserted into a rectangular region of uniform, constant external magnetic eld B 2 Bo 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 eld] Figure 1: A rectangular conductor loop inserted into an external uniform magnetic eld. (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 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 / (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 I(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 = 1(t), and the time varying current, I = I(t), for t > 0. (d) (5 points) Following the same setup of Part (c), let us consider a small modification that the conductor loop has a resistance R. Because of Joule heating at the resistor, the system is dissipative, and after a sufficiently long time the conductor loop comes to a full stop and the current dies off. Find the total amount of energy dissipated as heat at the resistor