2. In free space, let D = 8xyz*a, + 6x²z*ã, + 16x²yz'a, pC/m². a. Find the total electric flux passing through the rectangular surface z = 2, 0

2. Consider an insulating spherical shell of inner radius a and outer radius b. a. If the shell has a net charge Q uniformly distributed over its volume, find the vector electric field in all regions of space (r b) as a function of r. b. Now assume that the shell has a non-uniform charge density given by r2 p(r) = Po ab What is the net charge of the shell? c. For the charge distribution in part (b), find the vector electric field in all regions of space (r b) as a function of r.

2. Find the electric field at a distance z above the center of a ring of radius a that has a uniform linear charge density A. Compute (only) the Er and Ez components of a cylindrical coordinate system.

1. A thin sheet of sides 2a and 2b lying in the xy plane and centered at the origin has a uniform charge density o (see figure). a. Using the result of Example 1 in the Chapter 23 slides, show that the electric field on the z-axis is given by of ab E = 4k,o arctan | zva² + b² + z². To get full credit you must show an appropriate differential of charge dq and explain any symmetry arguments you might have use to arrive to this result. b. What is the field in the limit a and b going to infinity? Compare your answer with the result of the What If? in Example 3 in the Chapter 23 slides. c. Using the results of part (a), find an integral expression for the electric field at an arbitrary point on the x-axis created by a cube of side 2a centered at the origin with uniform volume charge density p. You do not need to solve this integral.

2. A continuous line charge distribution with uniform line nc charge density of +5 is stretched infinitely across the m horizontal axis. What is the electric field at point (0,10)?

5. A thick, conducting, metal shell of inner radius R1 = 0.5m and outer radius R2 = 1.5m has a net charge of (-10C). Also, a (-3C) point charge is at the origin (center). Find the surface charge density on the inner surface of the shell. 6. Same setup as problem 5. Find the magnitude of the electric field at 4 meters from the center of the shell. 7. Same setup as problem 5. Find the magnitude of the electric field at 1.1 meters from the center of the shell.

8. A solid sphere with radius a = 2cm in encased in a spherical shell with inner radius a = 2cm and outer radius b = 4cm. (A cross section is shown on the right). The sphere has charge density Psphere = 1µC and the shell has charge density Pshell = 2µC. Let r measure the dis- tance from the center of the sphere. Answer the following questions: (Note: Your answer can contain a, b, A, B, and r the radial distance you are looking at.) (a) What shape of Gaussian surface will you use to solve this problem? (b) What is the electric field at r = 3cm? a Psphere Pshell b