(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. (e) (5 points) Show that the volumetric charge density at a 0. Also show that the total amount of charge on the inner surface Cin (t) at r = a decreases following an exponential time dependence: Q(t) = qet/T. (1) Find out the characteristic time constant 7 in terms of the parameters of the problem. [Hint: Use Maxwell's equation and Ohm's law. Also consider applying Gauss's law to relate electric field and charge at any given time. ] (f) (5 points) Find out the current density inside the conductor for t > 0. Then calculate the total amount of Joule heat production as the conductor settles down to electrostatic equilibrium at 1- 00 . (g) (5 points) The energy density stored in the electric field is up = (1/2) & E. Calculate the total energy stored in the field, both at the initial time t = 0, and at t -> co. Does the amount of change in the field energy equal to the Joule heating you find out in Part (f)? (h) (5 points) If the shell is a perfect conductor (corresponding to the limit of an infinitely large conductivity o - co), is there any energy dissipation via Joule heating as the system settles down to electrostatic equilibrium?A spherical conducting shell has two concentric surfaces, an inner one with radius a, and an outer one with radius 17. The conductor has a uniform conductivity 0'. Its electric and magnetic susceptibilities can be neglected (i.e. no polarization and magnetization). Outside the conducting shell is free space. Initially at t = 0, there is an amount of charge q uniformly distributed on the inner surface. We would like to nd out the system's behavior at t > 0. [Notes This is not a quasi-stationary process apply the full Maxwell's equations] Figure 2: A spherical conducting shell