Capacitors can be used as "electric charge counters." Consider an initially uncharged capacitor of capacitance C with its bottom plate grounded and its top plate connected to a source of electrons.

(a) If N electrons flow onto the capacitor's top plate, show that the resulting potential difference V across the capacitor is directly proportional to N.

(b) Assume the voltage-measuring device can accurately resolve voltage changes of about 1mV. What value of C would be necessary to resolve the arrival of an individual electron?

(c) Using modern semiconductor technology, a micron-size capacitor can be constructed with parallel conducting plates separated by an insulator of dielectric constant K = 3 and thickness d = 100 nm. What side ℓ should the square plates have (in μm)?