MATH 406: Problem Set 13

due Wednesday, April 27, 2005

    1. Without using any facts about electromagnetism (except that the electrostatic potential is harmonic), guess the functional dependence of the potentional $ \phi$ for a purely radial electric field, which is due to an isolated electric charge (such as an electron). Define your reasoning clearly, and specify the associated analytic function in $ \C$ .
    2. Using superposition, find the total electric potential due to this charge in a spatially constant field $ E_0$ .

    1. Show that the Zhukovsky mapping $ w = z + (A^2/z)$ has a relation with the Milne-Thompson Circle Theorem for a circle of radius $ R=A$ .
    2. (based on Fisher 4.2.15) Show that the Zhukovsky map transforms flow past a circle with $ R<A$ to flow past an ellipse with axes $ a$ and $ b < a$ (see Fisher p.282).
    3. Describe what happens when $ A \ra R$ . Why is this different from your answer to part (a)? HINT: it is.

  1. Find the Fourier transform $ \hat{u}(\omega)$ for the following functions of time $ t$ :

       a) $\displaystyle \,u(t) = \frac{5}{1+t^2}$   b) $\displaystyle \,u(t) = \frac{1}{20 + 8t +t^2} \,$   (F 5.1.5)$\displaystyle $

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