(a) Two spheres have radii a and b, and their centers are a distance d apart. Show that the capacitance of this system is C =  4?ε0 1 a +  1 b −  2 d  provided that d is large compared with a and b. Suggestion: Because the spheres are far apart, assume the potential of each equals the sum of the potentials due to each sphere. (b) Show that as d approaches infinity, the above result reduces to that of two spherical capacitors in series.

A parallel-plate capacitor has circular plates of 6.21 cm radius and 1.18 mm separation. (a) Calculate the capacitance. (b) What charge will appear on the plates if a potential difference of 164 V is applied?

A parallel-plate capacitor has a capacitance of 2.3 µF and a plate separation of 1.7 mm. (a) What is the maximum potential difference between the plates such that dielectric breakdown of the air between the plates does not occur? (Use Emax = 3 MV/m.) kv (b) How much charge is stored at this maximum potential difference? mc

Consider a parallel-plate capacitor with a plate area of A = 8.50 cm². The separation between the plates is di = 3.00 mm (the space between the plates is filled with air). The plates of the capacitor are charged by a 6.00 V battery, i.e., the potential difference between the plates is V₂ = 6.00 V. The plates are then disconnected from the battery and pulled apart (without discharge) to a sepa- ration of df = 8.00 mm. In the following, neglecting any fringing effects. (a) Will the new potential difference between the plates be larger, smaller, or the same compared to the initial potential difference of Vi = 6.00 V? Explain. (Hint: Note that the charge will not change when the plates are pulled apart. Why is that?) (b) Find the potential difference Vf between the plates after the plates have been pulled to their new, larger separation df. (c) Find the electrostatic energy stored in the capacitor before and after the plates are pulled apart. (d) To separate the plates, you will need to do…