PHY531 Problem Set 3. Due October 2, 2001.

1. A complex function $f(z)$ of a complex variable $z = x + i y$ where $x$ and $y$ are real, and
   

   $$f(z) \equiv F(x,y) + i G(x,y)$$ (1)

   
   
   is said to be analytic in a region if the derivative defined by the limit
   

   $$\frac{df(z)}{dz} \equiv \lim_{h \rightarrow 0} \frac{f(z+h)-f(z)}{h}$$ (2)

   
   
   is independent of how the complex variable $h$ is taken to zero.

   a.

   Show that $f(z)$ is analytic in a region if and only if
   

   $$i \frac{\partial f(x+iy)}{\partial x} = \frac{\partial f(x+iy)}{\partial y} .$$ (3)

   
   

   b.

   Show that if $f(z)$ is analytic in a region, both $F$ and $G$ satisfy the two dimensional Laplace's equation there.

   c.

   Show that if both $f(z)$ and $g(z)$ are analytic functions in the appropriate regions, that $f(g(z))$ is also analytic.

   d.

   Show that the function
   

   $$\ln(z) \equiv \frac{1}{2} \ln (x^2+y^2) + i {\rm atan2}(y,x)$$ (4)

   
   

   is analytic except at the origin and along the negative $x$ axis. Here ${\rm atan2}(y,x) \equiv {\rm arg}(z)$ is the quadrant correct arctangent that goes from $-\pi$ to $\pi$ and has a discontinuity along the negative $x$ axis.

   What simple physical situations lead to electric or magnetic fields given by the negative gradient of the functions $F$ and $G$ corresponding to the complex function $\ln(z)$?

   Many two-dimensional electrostatic problems can be solved by conformal mapping which uses the properties proved above to transform the boundaries of a known solution to other solutions.

2. A region of space in circular cylindrical coordinates is defined by
   

   $$0 \leq \phi \leq \alpha$$

   $$0 \leq \rho \leq a$$

   $$0 \leq z \leq L$$ (5)

   
   
   where $0 < \alpha \leq 2\pi$. The potential on all surfaces is zero except the surface $\rho =a$ which is held at a fixed potential $V$. Calculate an expansion of the potential everywhere inside the region in terms of modified Bessel functions.

   Show that near $z=L/2$, the charge density on the walls near $\rho=0$ has the form of Jackson Eq. 2.75 as expected.

3. A long conducting circular cylinder of radius $a$ has its axis along $z$. At $z=0$ the cylinder has a small slit cut into it so the portion along $z > 0$ is insulated from the portion along $z<0$. A battery is connected across the gap so that the potential of the upper half of the cylinder differs from the lower half by $V$. Calculate a series expansion of the potential everywhere. Calculate the electric field along the axis and make an accurate plot of $-a E_z(z)/V$ versus $z/a$.

4. Repeat problem 3 for the case where the cylinder has a square cross section of side $2d$.

   A conformal mapping of the inside of a square to a circle leads to an elliptic function transformation. One result is that away from the corners, the square of side $2d$ behaves approximately like a circle of radius,
   

   $$a^* = \frac{2}{K\left (\frac{1}{\sqrt 2} \right )} d ,$$ (6)

   
   
   where $K$ is a complete elliptic integral of the first kind,
   

   $$K(k) = \int_0^{\pi/2} d\alpha \frac{1}{\sqrt{1-k^2 \sin^2 \alpha}} .$$ (7)

   
   

   Plot the $-a E$ versus $z/a$ for $d=a$ and $a^*=a$ on the same graph as for problem 3 for comparison.