Physics 408 Exam 1

1. a) Draw a picture for the equation being sure to label the relevant vectors.

b) Calculate using the equation in a) above. Be sure to justify your steps and comment on the resulting equation by explaining both sides.

c) Write the resulting equation in b) in integral form showing the step involving somebody’s theorem.

d) Prove the vector identity . (hint: replace the vector field in the divergence theorem with , where is an arbitrary constant vector.)

e) Using the expression for the electric field of a point charge at the origin, calculate the curl of . Do we need to worry about a possible contribution from the point ? Justify your reasoning by making use of the identity .

f) Can you generalize your result (using words only) in e) to the more general electric field written in a) above?

g) What are the differential equations in differential form (Maxwell’s equations) describing the electric field in electrostatics?

2. a) Make a direct calculation of the electric field at the origin, due to a quarter sphere (radius R) with total charge Q uniformly distributed throughout the volume. The full sphere would be centered on the origin. The quarter sphere intersects the x-y plane in a semi-circle with positive y coordinates and intersects the x-z plane in a semi-circle with positive z coordinates.

b) Calculate the electric potential at the origin for the same charge configuration in a).

c) What components of the electric field can be determined from the potential calculated in b)? Justify your work and explain your result.

3. a) Using the relation for a continuous charge distribution, , show that the energy to create a point charge is infinite.

b) To remedy this deficiency give the point charge structure by assuming the charge is spherical with radius R and has a charge density given by . Calculate the electric field inside and outside the charge distribution. Be sure to carefully justify your work.

c) Again using , calculate the energy needed to create a charge with the structure given above in b).