MATH 406: Problem Set 13

due Wednesday, April 27, 2005

1. 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 $E_0$ .

   
   
   
   

2. 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.

   
   
   
   

3. Find the Fourier transform $\hat{u}(\omega)$ for the following functions of time $t$ :
      a) $$\,u(t) = \frac{5}{1+t^2}$$   b) $$\,u(t) = \frac{1}{20 + 8t +t^2} \,$$   (F 5.1.5)$$$$

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