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Physics 415: Electromagnetic Theory I
Prof. S. Teitel stte@pas.rochester.edu ----- Fall 2002

Midterm Exam

Please show all you work. Explain your steps to maximize your chances for partial credit. Put a circle around your final answer.

Problems 1, 2, and 3 are meant as "short answer" type problems. Once you see what is needed, you should not need to do a very lengthy calculation to arrive at the answer. Problem 4 is a more involved calculation. Note that Problem 4 is worth twice the points as the other problems.

• Problem 1 [20 points total]

  a) Consider a point charge q in front of two infinite grounded planes at right angles to each other, as shown in the figure below. The intersection of the planes coincides with the z axis, and the position of the charge is given by the coordinates r=(x_o, y_o, 0). What is the force F on the charge q?

  

• Problem 2 [20 points total]

  In each of the following examples, an electric field E is produced. In each case, this field will decay as E ~ 1/rⁿ, for large r. Determine the value of n for each case, carefully explaining the reason for your answer.

  a) a spherical shell of radius R, centered on the origin, with a surface charge density of () = _ocos³(). Here _o is a constant, and is the usual spherical angular coordinate.

  b) a disk of radius R in the xy plane at z=0, centered on the origin, with surface charge density (r) = Csin(2r/R). Here r is the radial distance on the disk.

  c) four point charges located as follows:
  

  +q at r=(xo,yo,zo)	-q at r=(-xo,yo,zo)
  +q at r=(-xo,-yo,zo)	-q at r=(xo,-yo,zo)

• Problem 3 [20 points total]

  a) A circular wire loop of radius R, carrying a current I, is bent at right angles, as shown below. What is the magnetic field B at a point r = r(e_x+e_y), where r>>R?   (e_x and e_y are the unit vectors in x and y directions)

  

  b) Two circular wire loops, each of radius R, are centered about the origin and lying in the xy plane. One loop is at height z=+a and has current I circulating clockwise. The other loop is at height z=-a and has a current I circulating counter-clockwise. For large distances r>>R the magnetic field B will decay, to leading order, proportional to B ~ 1/rⁿ. What will be the value of n? You must explain your answer.

• Problem 4 [40 points total]

  Consider a spherical shell of radius R, with a fixed surface charge density given by () = _ocos²(). Here _o is a constant, and is the usual spherical angular coordinate.

  Find the electrostatic potential (r) both inside and outside the shell.