Homework Set #6, Physics 3320, Fall 2011 Due Wednesday, Oct 12, 2011 (start of class) 1. Go to the course website and find the link to the upcoming paper assignment. Or, go right to
http://www.colorado.edu/physics/phys3320/phys3320_fa11/paper_info.html
Read the assignment, browse the American Journal of Physics website, and pick an article you
would like to read. (You aren’t committed to it yet, but I want you to pick one) For this homework,
just tell us what article you picked, in the form
Author, First, Am. J. Phys <volume>, page number, (year)
And then write the title of the article “Full Title of the article here”.
Then, please download a pdf of your article from AJP, and upload it to our D2L site (There is a tab
labeled “DROPBOX” at the top, between Discussions and Quizzes, where you can upload the file.)
I am NOT yet asking for a summary, (you don’t even have to have read the full paper yet, just find
one you’re interested in!) Do read the abstract, though, lest you discover later you’ve chosen an
article that’s not interesting to you.
2. Consider (again!) our standard coax cable as an “infinite” length wire of radius a surrounded by a
thin conducting cylinder, coaxial with the wire, with inner radius b and outer radius c.
Again, assume a<<b and c-b << b (thin shell and wire), as show in the figure.
a) In the last HW set, you found the self-inductance per length of the coax. Now let’s investigate the
induced E field for a time dependent current I(t) = I0 cos ω t , which flows along the wire and a
corresponding current I(t) which flows in the opposite direction on the outer cylinder.
Find E(s,t) in the “coax region” (a<s<b) where s is the usual radial coordinate and the current on the
wire is I0 in the +z direction at t = 0.
(Assume that we have the quasi-static situation in which the currents are identical in magnitude at
each moment in time, and the changes in current are sufficiently slow. Also, assume that the
magnitude of E →0 as s→∞. I claim this implies E=0 all the way in to the coax, can you argue why?
This may turn out to be a useful “boundary condition”)
b) Find the displacement current density Jd in the “coax region” (a<s<b) for this E, and integrate it
to get the total displacement current Id .
Using physically reasonable numbers for a real coax (say a=1 mm, and b=1 cm), determine the
frequency ω for which Id equals just 1% of I0. Briefly, comment. (E.g, What sort of frequency is
this, and do we need to worry about the displacement current story for lower or higher frequencies
than this? What does this tell you about our usual “quasistatic” approximation?)
3. We now want to investigate energy flow in the same cylindrical coax cable defined above.
However, for now, let’s just look at fields constant in time, not varying in time. Assume that
constant current I flows in the +z direction on the inner wire and that total current I flows in the
opposite direction in the shell. Also assume that there is a constant voltage difference V between the
wire and the shell, as shown (top of next page)
a) For this steady current and voltage case: Find E and B everywhere in space. You may assume that
the coax cable (wire plus shell) is neutral. (Hint: This isn’t bad - look back at HW #2, where you
already solved a closely related problem!)
b) Calculate the Poynting vector S everywhere. The magnitude of S gives the energy flux density
and represents the power per area moving through space. Does its direction make sense for the coax?
Integrate this flux through the cross sectional area of the coax to find the power transported down the
coax line. Does your answer make sense relative to the circuit maintaining the current and voltage?
Briefly, comment!
4. Section 7.3.4 of Griffiths describes how Maxwell’s equations would be changed in the present of
magnetic monopole charges (and magnetic monopole currents.)
a) Griffiths states what the mathematical equation for “conservation of magnetic current” looks like.
(Eq. 7.44) Derive it for yourself, starting from the “full” set of equations (Eq. 7.43)
b) In Chapter 8.1.2, Griffiths derives the conservation of energy formula (Eq. 8.14). He starts from
Eq 8.6, which tells you “work done by the fields on charges”. But if there were magnetic monopoles,
the Lorentz force law would have to be modified to include the forces on magnetic monopoles.
Make a guess (the simplest possible!) as to what that modified Lorentz law would look like, and then
work through in parallel to Griffiths section 8.1.2, keeping track of any new bits due to magnetic
monopoles, to derive the modification to the conservation of energy equation (Eq 8.14) due to
“magnetic charges”. Briefly, comment!
5. Consider the electric field E(x,y,z,t) = E0cos(k(x-ct)) ŷ (where k and c are known constants).
a) What is the charge density ρ(x,y,z,t) which this E field implies? What are the units of k and c?
b) Come up with the simplest possible B field which satisfies Faraday’s law, given this E field.
Then, CHECK that the remaining free-space Maxwell’s equations are satisfied, if you make the
correct choice for the constant “c”. (What is this required “c”? What is the required magnitude of
the B field? What is the current density J(x,y,z,t) which this E and B field imply? )
c) What is the Poynting vector, S, associated with these fields? Describe in words what this E and B
field look like. Can you “interpret” them physically? What does the constant “k” tell you? Does this
set of E and B provide a valid, self-consistent, physically possible solution to Maxwell’s equations?
What’s the physics here?