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Homework #4

1. You drop a magnet through a loop of wire.

a. Sketch a plot of the magnetic field as a function of time through the loop.
b. Sketch a plot of the emf as a function of time in the loop.

The first part should be easy, just imagine dropping a magnet. The second part requires Faraday’s
law. Recall that the emf is the derivative of the magnetic flux with respect to time. This is
mathematically similar to how velocity is the derivative of position with respect to time.

2. Assuming that raindrops are spherical can you show that a
rainbow is located around 42o around a line from the sun to you
(in other words, if you look at the shadow of your head on the
floor the rainbow is a circle centered at this location but at an
angle of 42o out).

Look at the rays that are traced through the spherical droplet in
the diagram to the right. Note that they “bunch up” at a
particular angle (rays 7-9 in the diagram). In other words, the
angle of deviation (between the light coming in and the light
reflecting back out) doesn’t change much at the angle at which
the rainbow is seen. Start by finding an expression for the angle
of deviation as a function of incident angle.

Next take its derivative with respect to the incident angle and
put it equal to zero (wolfram alpha can handle this, or you can
plot it instead). Then find the angle of deviation at which this
occurs. This should be around 180o – 42o or 138o.

3. A thin layer of oil has a wonderful metallic rainbow color as it glistens of the surface of the Gulf of
Mexica, having leaked from the Deepwater Horizon Oil Spill and been responsible for the death of the
8,332 species living in the region, including more than 1,200 fish, 200 birds, 1,400 mollusks, 1,500
crustaceans, 4 sea turtles and 29 marine mammals. If the thickness of the oil film is 1 m and the
refractive index of the oil is 1.2, calculate the colors that appear the most brightest when viewing the oil
film from above.

This is very similar to the problem in the notes at the end of cycle #1 (but with dead fish). Just
remember to consider the phase shifts and the change in wavelength in the oil film.

Homework #5

1. Consider a system consisting of 3 charges. A 1 C charge is located at x = 0, a 2 C charge is
located at x = 2 cm and -1 C charge is located at x = 5 cm.

a. Obtain an expression and sketch a plot of electric potential as a function of x
b. Obtain an expression and sketch a plot of electric field as a function of x.

This is just a system of three point charges. The electric field and electric potential can be added
together for the three charges to get the total electric field and electric potential for the entire system.
However, note that only one of the charges is located at the origin. Therefore, the r in the equations
(distance from point charge) is going to have to be modified as you may be closer or further from the
charge depending on where it is located! Test your expressions. What if you are located at the same
location as the charge? Does the term for this charge give you 0 for the distance away from the
charge?

2. Imagine there is a charged sphere, where the charge is uniformly distributed over its volume. The
electric field inside this sphere would ordinarily be given by Gauss’ law because the sphere is
spherically symmetric. However, what if we broke this symmetry? What if we introduced a spherical
cavity inside the sphere which is off-center?

Find the electric field inside this spherical cavity.

This is really easy, once you figure it out! The only hint I’ll give is that
although it appears to be related to Gauss law, check out the section on
superposition (adding stuff together) as it may help.

3. In the lecture we used the superposition principle to investigate the electric field along the axis of a
uniformly charged ring. Could we use the same principle to find the electric field along a line
perpendicular to the plane, and going through the center, of a uniformly charged square.

How does the square geometry affect the math?

Plot the electric field along the axis of a uniformly charged ring, and the electric field through the
center of a uniformly charged square on the same plot.

This is similar to the problem in the notes, but here the integration may require wolfram alpha as the
distance isn’t constant anymore. However, due to symmetry we can still expect the electric field to be
entirely in the x-direction. Furthermore, we don’t necessarily have to integrate the entire square just a
part of it as if we broke the square into 8 parts (each side broken in two) then each part would
contribute equally to the electric field. Good luck!

Homework #6

1. In homework #1 you were asked to consider two atomic nuclei as if they were point charges of
magnitude 1.6×10-19 C and mass of 1.66×10-27 kg. In particular you tried to determine how fast the
nuclei would have to be traveling to get to within 1×10-15 m of each other. Can you do this in an easier
way now?

Without using energy this is a much more challenging problem. However, now that you have been
introduced to potential energy it should be very easy!

2. Consider a cell which has a 7.5 nm thick membrane
and a membrane surface area of 5 x 10-9 m2. The
relative permittivity of the insulating cell membrane is
8.7 and the membrane has a resting potential of 60 mV.
Note that this electric potential arises due to the
combination of ion pumps and ion channels in the cell
membrane, which separate charge from the inside and
outside of the membrane.

a. What is the magnitude of the charge on the inner and
outer surfaces of the cell membrane?
b. How many ions does this charge correspond to?

The first part can be solved very easily by treating the cell membrane as a parallel-plate capacitor.
The second part is just converting the charge in coulombs to the number of ions (each with a charge of
1.6×10-19 C).

3. Imagine that in order to protect your house from building up positive charge and attracting a
lightning strike you install a dissipative lightning rod. During a thunderstorm the corona discharge from
your dissipative lightning rod into the surround air has a current of 1×10-4 A.

a. If the thunderstorm lasts for an hour, and the discharge is constant during this time, how much
electric charge flows out of the lightning rod?
b. How many electrons flow into the lightning rod from the surrounding air?

From the definition of what current is you should be able to solve part a with no trouble. Similar to the
last problem, you then have to write this charge in terms of the number of electrons (each with a
charge of 1.6×10-19 C).