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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
    \begin{displaymath}
f(z) \equiv F(x,y) + i G(x,y)
\end{displaymath} (1)

    is said to be analytic in a region if the derivative defined by the limit
    \begin{displaymath}
\frac{df(z)}{dz} \equiv \lim_{h \rightarrow 0} \frac{f(z+h)-f(z)}{h}
\end{displaymath} (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
    \begin{displaymath}
i \frac{\partial f(x+iy)}{\partial x} = \frac{\partial f(x+iy)}{\partial y}
 .
\end{displaymath} (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
    \begin{displaymath}
\ln(z) \equiv \frac{1}{2} \ln (x^2+y^2) + i {\rm atan2}(y,x)
\end{displaymath} (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
    $\displaystyle 0 \leq \phi \leq \alpha$      
    $\displaystyle 0 \leq \rho \leq a$      
    $\displaystyle 0 \leq z \leq L$     (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,

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

    where $K$ is a complete elliptic integral of the first kind,
    \begin{displaymath}
K(k) = \int_0^{\pi/2} d\alpha \frac{1}{\sqrt{1-k^2 \sin^2 \alpha}}
 .
\end{displaymath} (7)

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


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