PHY531 Problem Set 2. Due September 20, 2001.

1. The work required to bring a point charge $q$ up to the centers, which are a distance $b$ apart, of two thin parallel conducting coaxial rings, each of radius $a$, is $W_1$ and $W_2$, respectively. Show that the charges on the rings are
   

   $$Q_{1,2} = \frac{a }{b^2 q} (a^2+b^2)^{1/2}[ (a^2+b^2)^{1/2} W_{1,2}-a W_{2,1}]$$ (1)

   
   
   .

2. Two parallel coaxial circular rings of radii $a$ and $b$ carry uniformly distributed charges $Q_1$ and $Q_2$. The distance between their planes is $c$. Show that the force between them is
   

   $$F = \frac{c k^3 Q_1 Q_2}{4 \pi (ab)^{3/2}} \left ( \frac{E(k... ...k^2} \right ) \ \ {\rm where}\ \ k^2 = \frac{4ab}{c^2+(a+b)^2}$$ (2)

   
   
   and $E(k)$ is a complete elliptic integral of the second kind of modulus $k$.

   The complete elliptic integrals with $m=k^2$ are defined in Abramowitz and Stegun chapter 17 as
   

   $$E(k) = \int_0^{\pi/2} d\alpha \sqrt{1-k^2 \sin^2 \alpha}$$

   $$K(k) = \int_0^{\pi/2} d\alpha \frac{1}{\sqrt{1-k^2 \sin^2 \alpha}} .$$ (3)

   
   
   Complete reduction techniques are found there. You may find the derivative formulas
   

   $$\frac{dE(k)}{dk} = \frac{1}{k} (E(k)-K(k))$$

   $$\frac{dK(k)}{dk} = \frac{1}{k} \left (\frac{E(k)}{1-k^2}-K(k) \right )$$ (4)

   
   
   useful.

3. A grounded conducting sphere of radius $a$ has its center on the axis of a charged circular ring with total charge $Q$. Any radius vector of length $c$ from its center to the ring makes an angle $\alpha$ with the axis. Show that the force sucking the sphere into the ring is
   

   $$\frac{Q^2 E(k) (c^2-a^2) k^3 \cos \alpha}{4 \pi c^2 a^2 \sin^3 \alpha (1-k^2)}$$ (5)

   
   
   where
   

   $$k^2 = \frac{4a^2 c^2 \sin^2 \alpha}{a^4+c^4 - 2 a^2 c^2 \cos 2\alpha} .$$ (6)

   
   
   and $E(k)$ a complete elliptic integral of the second kind.

4. Jackson Problem 1.23. Part c is not optional. Jackson's ``accurate'' numerical values should be changed to $\Phi_1 = 48.86$, $\Phi_2=47.15$, $\Phi_3 = 38.25$, and $\Phi_4 = 19.75$.

5. Jackson Problem 2.3

6. Jackson Problem 2.5