Suppose the volume X occupied by a capacitor is A*d where A is the total area of the dielectric layers and d is the thickness of each dielectric layer (this assumes that the metal plates are infinitely thin). Assume that the maximum electric field that can be supported by the dielectric before it breaks down is EMAX . Also, recall that €=€r€0 where €0= 8.85 • 10^(-12)F/m.

1. Derive an equation for maximum energy that can be stored in a capacitor, WMAX , in terms of X, €0, €r, and EMAX. Show and explain all the steps you used to derive this from the fundamental equations for capacitors. Show the units for each term.

2. Determine WMAX for a capacitor which occupies a volume X = 1cm^3 where EMAX = 100 V/mm and €r= 2. Be careful to use consistent units.

3. Suppose you have a choice of using one of two new dielectric materials:
• Dielectric (a) has EMAX = 100 V/mm and €r= 4

• Dielectric (b) has EMAX = 200 V/mm and €r= 2

Which would you use to achieve a higher energy storage density (in joules/m^3) and why.

A capacitor can be designed with two large plates and a thin dielectric between them but that is not a common approach because it would make for an impractical shape. One commonly-used approach is a multi-layer capacitor constructed with a stack of thin metal plates separated by a dielectric layer, with the plates alternately connected to the two capacitor terminals. This structure can be thought of as a set of parallel capacitors, each of which is created by one of the dielectric layers situated between opposing metal plates.