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Quiz 1, Physics 4002, October 11, 2023

Instructions:  This is a open book and notes exam. You may use your electronic devices if you have used them to take notes for the class, but you are not allowed to browse the internet in an  attempt to find a solution. Do each problem on the sheets of paper provided, with only one problem on each sheet (the problems will be separated for grading).  If you have used more than one sheet of paper for a problem, staple together those sheets.  Make sure your name and ID number are on every sheet, so they can be identified if they become separated.

1. (30 pts.) Consider a sphere of radius R that carries a charge density ρ = kr, where kis a constant.

(a) Find the total charge Q on the sphere

(b) Using Gauss’s Law, determine the electric field in all space for both r < R and r > R, expressing it in terms of Q.

(c) Determine the potential at the origin, assuming that the potential is zero at large distances.

2. (30 pts.) Consider a conducting sphere of radius R that is charged to a potential V0  (let V= 0 at infinity).

(a) What is the potential and electric field outside the sphere (r > R)?

(b) Determine the surface charge density and the total charge on the sphere. (c) What is the capacitance of the sphere?

(d) Determine the energy of this configuration by integrating ε0 E2  /2  over all space.  Show that this energy is equal to CV02  /2 .

3.  (40 pts.) Consider a spherical shell of radius R that has an imposed potential given by V(θ) = V0 cos θ .

(a) Determine the potential inside and outside of the sphere in terms of a Legendre polynomial expansion.

(b) Calculate the radial electric field inside and outside of the sphere and find the surface charge density.

(c) What is the total charge on the spherical shell?