Final Exam Physics 106 May 18, 2011
Do this exam by yourself. No calculators or books are needed or permitted. However, a
formula sheet (four pages of US letter size paper) is allowed. Problems are on BOTH SIDES
of this page. Please explain your reasoning. Draw sketches and free body diagrams. Even if
you don’t know how to solve the problem, state in words what you think is happening and why.
Express all answers in terms of the variables defined in the problem.
1. [20 points] A thick flat plastic sheet with a dielectric constant εis placed in the homogeneous
electric field ~
E0, so that the axis perpendicular to the surface of the plastic sheet makes an
angle αwith ~
E0. Draw a sketch of the electric field lines outside and inside of the plastic
sheet. What is the magnitude of the electric field inside it?
2. [20 points] An infinite flat metal plate of thickness 2hcarries the current density j.
(a) [10 points] What is the orientation and magnitude of ~
Bat a distance xfrom the midplane
of the plate? Consider both inside and outside of the plate, and sketch B(x).
(b) [10 points] The same infinite flat metal plate contains a cylindrical hole of radius h,
with its axis lying in the midplane. The current density jis running parallel to the
cylindrical hole, as shown in the figure. Let the xaxis start at the axis of the hole. What
is B(x)?
3. [20 points] A cone of height Hand radius of base Ris permanently magnetized with uniform
magnetization M. The vector ~
Mis collinear with the axis of the cone. Let that axis be z, and
the apex of the cone the coordinate origin.
What is the magnetic field along the zaxis, at the distance hfrom the apex, ~
B(h)? (Here
h+zis larger than the thickness of the cone at the distance zfrom its apex.)
[Hint: model
the cone as a stack of infinitesimal disks.]
4. [20 points]
(a) [10 points] Compute the total resistance of the resistor network shown in the figure.
(b) [10 points] For what value of r(expressed as a function of R) the total resistance of the
whole network will also equal r?
(c) [EXTRA CREDIT = 10 points] What is the resistance of the infinite network shown in
the figure?
[Hint: if one cell from the network is removed, its resistance is still
r
.]
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