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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 = -Bo ĉ (into the plane), and that portion has a length 1. 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.] Х I X Box 个 Х Х hlx X X y ん山 eller L X X IX m Figure 1: A rectangular conductor loop inserted into an external uniform magnetic field.

(b) Since the geometry of the setup is spherical symmetric, and the initial charge distribution is spherical symmetric, physical quantities must respect spherical symmetry at all times t > 0. Additionally, the system is reflection invariant about any plane that contains the shell center. (5 points) Use symmetry arguments to determine the direction of the electric field at any t > 0. Find out which of the spherical coordinates the electric field magnitude depends on. (c) (5 points) Use symmetry arguments to show that the magnetic field in fact vanishes everywhere and at all times. (d) (5 points) Show that in this setup, at any given time t > 0, we can apply Coulomb's law and/or Gauss's law to find the electric field from the instantaneous charge distribution.