Question

A capacitor is made of a flat plate of area A and a sector plate having a stair-like structure as shown in figure (31-E9). The width of each stair is a and the height b. Find the capacitance of the assembly.

Answer 1

$C_1=\frac{{\in }_{0}A}{3d},C_2=\frac{{\in }_{0}A}{(d+b)}$

$C_3=\frac{{\in }_{0}A}{3(d+2b)}$

The surface area at each is equal to $\frac{A}{3}.$

Then the three capacitor is parallel combination.

$C=C_1+C_2+C_3$

$=\frac{{\in }_{0}A}{3d}+\frac{{\in }_{0}A}{3(d+b)}+\frac{{\in }_{0}A}{3(d+2b)}$

$=\frac{{\in }_{0}A}{3}\frac{(3d^2+6bd+2b^2)}{d(d+b)(d+b)}$