Physics 7701: Problem Set #5

Here are some hints, suggestions, and comments on the assignment.

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  1. 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 algebraically. [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.]
  2. 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.
  3. Density of uniform rod with delta functions.
    1. 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.
    2. Similar, but now you have to simplify the delta functions as in the example from class (using the properties of a delta function).
    3. Again, the same, but more work to do simplifying because spherical coordinates.
  4. Delta functions of functions. These are straightforward applications of the standard formula. Find the zeros and plug in.
  5. 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.
  6. Fourier representation of δ functions.
    1. The Fourier series for a step function in (-L,L) can be found in Lea Chapter 4.
    2. 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?
    3. Even if your results look different, check whether they might be the same (Mathematica is a good place to start!).
  7. 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.

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Physics 7701: Assignment #5 hints.
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