THE UNIVERSITY OF BRITISH COLUMBIA
 
Physics 108 Assignment # 5:
 
POTENTIAL & CAPACITANCE
 
Wed. 2 Feb. 2004 - finish by Wed. 9 Feb.

1. CLASSICAL RADIUS OF THE ELECTRON: You are probably familiar with Einstein's famous equation $E = m c^2$. If $m$ is the mass of an electron and $E$ is the electrostatic potential energy required to "assemble" the electron from bits of charge infinitely distant from each other into a uniform spherical shell of radius $r_0$ and net charge $e$, find the numerical value of $r_0$ in meters.¹

2. CAPACITOR WITH INSERT: Suppose we have a capacitor made of two large flat parallel plates of the same area $A$ (and the same shape), separated by an air gap of width $d$. Its capacitance is $C$. Now we slip another planar conductor of width $d/2$ (and the same area and shape) between the plates so that it is centred halfway in between. What is the capacitance $C^\prime$ of the new system of three conductors, in terms of the capacitance $C$ of the original pair and the other parameters given? (Neglect "edge effects" and any dielectric effect of air.)

3. ARRAY of CAPACITORS:
   

   The battery $B$ supplies $12$ V. The capacitances are $C_1 = 1.0$ $\mu$F, $C_2 = 2.0$ $\mu$F, $C_3 = 4.0$ $\mu$F and $C_4 = 3.0$ $\mu$F.
   1. Find the charge on each capacitor when switch $S_1$ is closed but switch $S_2$ is still open.
   2. What is the charge on each capacitor if $S_2$ is also closed?

4. THUNDERCLOUD CAPACITOR: A large thundercloud hovers over the city of Vancouver at a height of $1.0$ km. Between the cloud and the ground (both of which we may treat as parallel conducting plates, neglecting edge effects) the electric field is about $300$ V/m. The cloud has a horizontal area of $100$ km$^2$.
   1. Estimate the number of Coulombs [C] of positive charge in the cloud, assuming that the ground has the same surface density of negative charge.
   2. Estimate the number of joules [J] of energy contained in the air between the cloud and the ground.