Physics 7701: Problem Set #5
Here are some hints, suggestions, and comments on the assignment.
Recent changes to this page:
- 26-Sep-2013 --- Original version.
- LRC circuit problem.
This is very similar to the damped, periodically driven harmonic
oscillator problem from the last problem set. The only essential
difference is the actual form of the driving force, which was f(t)
there and is now the half-rectified sine wave EMF(t).
Again, the first step is to analyze EMF(t) as an exponential Fourier
series (why exponential?). Then the coefficients for Q(t)
(which are simply related to the voltage across the capacitor) follow
[Note: The solution to Lea Example 4.3, which
is similar this problem, is not correct (see the
Lea errata on the 7701 info page). A corrected version of the
example is available here.]
- Delta sequences.
You will have to make some assumptions about the function f(x)
(extended to f(z)) when checking whether the "sifting property"
of the delta function is provided by the delta sequence in the
n goes to infinity limit. First make the simplest assumption,
that f(z) is analytic, and solve the relevant integral by
contour integration. Then look at your solution and determine
whether it is possible to add singularities somewhere in the
complex plane and not change the result.
- Density of uniform rod with delta functions.
- Use both delta functions and theta functions to write
the mass density with an overall constant. Determine the
constant by integrating the density over the volume, which should be
equal to the total mass M, but also equal to the constant
times the length l.
- Similar, but now you have to simplify the delta
functions as in the example from class (using the properties
of a delta function).
- Again, the same, but more work to do simplifying because
- Delta functions of functions.
These are straightforward applications of the standard formula.
Find the zeros and plug in.
- Impulse on string.
A string problem like in the last problem set, but with a different
initial condition. Solve with a Fourier series.
Use Example 6.3 in Lea as a guide.
- Fourier representation of δ functions.
- The Fourier series for a step function in (-L,L) can
be found in Lea Chapter 4.
- Calculate the coefficients of the Fourier series as usual.
The delta sequence functions are even about the origin, so do
you expect cosines or sines in the expansion?
- Even if your results look different, check whether they
might be the same (Mathematica is a good place to start!).
- Solving the diffusion equation.
Follow example 3.15 in Lea to set up the problem and find
the general solution.
Then it's just a matter of enforcing the boundary condition
with the T2 temperature and projecting
out (with appropriate integrals!) the coefficients.
Physics 7701: Assignment #5 hints.
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