Physics 416G : Midterm 1
October 23, 2015
Please write all the intermediate steps for your benefit!
Closed book! No calculators, no smart phones, no other electronics are allowed.
All the necessary formulas are given in a separate sheet.
1
Problem 1 (20 point)
In Electrostatics, the potential for a point charge Q located at the origin of our coordinate system is
given by
φ(~r) =
1 Q
.
4π0 r
a) Using the potential, compute the electric field.
~ · E.
~
b) Using the electric field, compute the divergence of electric field ∇
Hint: Think carefully about the answer in b). If you think the answer is not correct, please redo the
computation in a different way to find a correct answer.
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Problem 2 (40 point)
Consider a long coaxial metal cylinders (see the Figure). The inner cylinder carries a uniform surface
charge density σ0 on the surface (radius a). The outer cylinder is electrically neutral.
a) Find the electric field in each of the four regions: (i) inside the inner cylinder (ρ < a), (ii) between
the cylinders (a < ρ < b), (iii) inside the outer cylindrical shell (b < ρ < c), (iv) outside the cable
(ρ > c). ρ is the radial coordinate, in the cylinderical coordinates (ρ, ϕ, z). Plot the magnitude of the
electric field as a function of ρ.
b) Find the induced charge density σb , and σc in terms of σ0 .
Now we ground the outer shell of the outer cylinder by connecting a wire to the ground. Then
disconnect the wire.
c) Find the potential difference between a point on the axis ρ = a and a point on the inner shell ρ = b
of the outer cylinder.
d) Compute the capacitance and the electrostatic energy stored in the system by considering a long
coaxial cylinder of unit length.
Hint: Note that it is not necessary to choose a particular reference point to solve d).
See next page
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Problem 3 (40 point)
An uncharged metal sphere of radius R is placed in an otherwise uniform electric field E = E0 ẑ, where
ẑ is the unit vector along direction of the magnetic field, from the positive charges to negative charges
in the left picture. The field will push positive charge to the “northern” surface of the sphere, leaving
a negative charge on the “southern” surface. This induced charge, in turn, distorts the field in the
neighborhood of the sphere.
a) List the boundary conditions you can use to solve the problem.
b) Compute the potential both inside the sphere and outside the sphere (inside the parallel plate).
c) Compute the induced charge at the surface of the sphere. This can be evaluated by the difference
(discontinuity) of the electric field between inside and outside of the sphere.
d) In the right figure, it is suggested to solve the same problem in a different way, by putting some
image charges and satisfying the boundary conditions. What is the nature of the image charges and
their strength?
Hint : The following formula may be beneficial if you know what you are looking for.
A) The general potential for charge distribution with Azimuthal symmetry can be written as
φ(r, θ) =
∞ X
l=0
Bl
Al r + l+1
r
l
Pl (cos θ) .
B) The multipole expansion of the potential for general charge distribution ρ(~r0 ) is given by
1
r̂i r̂j Qij
Q r̂i pi
φ(~r) =
+ 2 +
+
·
·
·
,
4π0 r
r
r3
Z
Z
Z
1
Q = d3 x0 ρ(~r0 ) ,
p~ = d3 x0~r0 ρ(~r0 ) ,
Qij =
d3 x0 [3ri0 rj0 − δij r02 ]ρ(~r0 ) .
2