9HD, Physics Midterm 1, October 26, 2009
You are permitted to use one 8 /1/2 x 11 sheet of paper with formulas. You will not need your
calculator. MAXIMUM SCORE IS 100 POINTS. SHOW ALL WORK FOR PARTIAL CREDIT.
r
)
,
r
Useful formulas: In spherical coordinates, for spherical symmetry, (r , , ) = r
1 2
r
r 2 r r
1. Fill in the blanks.
(a) (1 point)The direction of the electric field is defined as the direction that a _____________________
test charge would move.
r
(b). (4 points) If a point charge moves at constant velocity v = v o x into a region of constant electric field,
r
E = E o z , the path of the charge will be a ________________________________. How would the path
and 2 (r , , ) =
change if the sign of the charge were to be reversed?__________________________________________
2. (20 pts) Consider a coaxial cable, which has a solid inner conductor of radius a, an outer cylindrical
conducting shell of radius b, and a length L, where L>>b,a..
r
(a) Use Gausss Law to derive the E field between the two conductors, if there is a linear charge density of
+ on the inner conductor and - on the outer conductor.
(b) Now find the capacitance of the cable.
(c) Find the energy stored in the field between the two conductors, and show that this energy is of the
Q2
form
, where C is the capacitance of the cable.
2C
3. (40 pts) Consider a spherical charge distribution with radius a and volume charge density (r)= o r2,
where o is a positive constant and r is the distance from the center of the sphere.
(a) Find the electric field and the potential, both outside and inside the spherical charge distribution.
Assume that potential is zero at infinity. [Hint: First find the electric field and potential outside the
spherical charge distribution. When you find the potential inside the spherical charge distribution, note
that the potential is a continuous function of r for all values of r, i.e., (r) is continuous at r=a.]
(b) Show that your potential satisfies equation, Laplaces 2 = 0 , for r>a. What equation should the potential
satisfy for r a? [Note: You do NOT need to show that your potential actually satisfies this equation for r a.]
(c) Suppose an electron is released from rest at infinity and is attracted to the spherical charge
distribution. What is the kinetic energy of the electron when it reaches the surface of the sphere?
4. (15 pts) Consider a line charge + which lies on the x-axis between the origin and x=L. with finite
length L. Write an integral expression for the electric field at the position (0,0,z) in Cartesian
coordinates.(Do NOT assume that z <<L.)
5. (10 pts) Consider a sheet of conductor folded into a corner so that one part is in the x-y plane and the
other part is in the y-z plane. The conductor is grounded. A point charge +Q is located at Cartesian
coordinates (a,0,b). (A cross-sectional diagram of the conductors in the x-z plane, with y=0, is shown
below in the figure. Explain how you would find the potential at a point (x,y,z) in the region where both
x 0 and z 0. (Note that you do NOT need to write an explicit mathematical formula for the potential.)
.
+Q at
(a,0,b)
z-axis
b
a
x-axis
6. (10 pts) (a) Consider perfect conductor which is regular tetrahedron (polyhedron composed of four
equilateral triangular faces, three of which meet at each vertex --see figure). There is a total charge of +5Q
on the tetrahedron. A charge of +Q is located off-center inside a spherical cavity inside the tetrahedron.
(a) What is the induced charge on the inside of the spherical cavity? Make some statements about the
charge distribution on the inside of the cavity.
(b) What is the induced charge on the outside of the tetrahedron?
(c) What is the electric field inside the body of the tetrahedral conductor? What can you say about the
potential in the body of the conductor?
(d) If a Gaussian surface completely surrounds the tetrahedron, what is the total electric flux going
through that surface?
+Q