1. Using the definition of capacitance and the definitions of electric field and potential, show that the capacitance of a parallel-plate capacito ris C = ε0 A / d, where A is the plate area and d is the plate separation.
2. Show that if the capacitor's gap is filled with a dielectric of relative permittivity (dielectric constant) K, the capacitance is C = K ε0 A / d. What are some values of K for real substances?
3. You are given a series RC circuit, in an initial state of being fully charged, i.e. zero current flowing through the resistor and a voltage V(0) across the capacitor (and, of course, the resistor). Show that this voltage decays exponentially as
V(t) = V(0) exp(-t/ τ)
where τ=RC. Show that the slope of the log of this curve plotted versus time t is -1/RC. What is the intercept of this graph?
4. What time constants are expected if the parallel-plate capacitor has a separation of either 0.1 mm, 0.5 mm, or 2.5 mm, is of circular cross-section with diameter 20 cm, letting R = 10 kΩ, 1 kΩ, or 100 Ω? (Use the known value of ε0 for this problem.)
5. Consider and draw a new circuit that has an additional capacitor with capacitance Cscope in parallel with the capacitor from the original RC circuit. What is the total effective capacitance of these parallel capacitors in terms of d, e, A, and Cscope?