A parallel-plate capacitor has plate area S and the spacebetween the plates is filled with a dielectric whose permittivityvaries linearly from e1 at one plate (y =0) to e2 at theother plate (y = d). As usual neglecting edge effects, find the capacitance as a formula containing S,d, e1, ande2. Test that yourformula gives the correct well-known result for the capacitance ofa parallel-plate capacitor with uniform dielectric when you put
e1 = e2.
(Hint: First, find an algebraic expression for permittivityas a function of y. Is it linear, and does it give thecorrect permittivity if you put y = 0 and y =d into it in turn? If not, your permittivity formula iswrong. For the last part, you may find that you have to evaluate0/0. To do this, either use l'Hôpital's rule, or lete2 = e1 + D, find the capacitance, and then let D tend to zero. You might need to remember, or lookup, one of the well-known power-series formulas.)