#### Question

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

#### Solution

The total area of the flat plate is *A*. The width of each stair is the same. Therefore, the area of the surface of each stair facing the flat plate is the same, that is, `A/3` .

From the figure, it can be observed that the capacitor assembly is made up of three capacitors. The three capacitors are connected in parallel.

For capacitor *C*_{1}, the area of the plates is `A/3` and the separation between the plates is *d*.

For capacitor *C*_{2}, the area of the plates is `A/3` and the separation between the plates is (*d + b*).

For capacitor *C*_{3}, the area of the plates is `A/3` and the separation between the plates is (*d *+ 2*b*).

Therefore ,

`C_1 = (∈_0A)/(3d)`

`C_2 = (∈_0A)/(3(d+b)`

`C_3 = (∈_0A)/(3(d+2b)`

As the three capacitors are in parallel combination,

`C = C_1 + C_2 + C_3`

⇒ `C = (∈_0A)/(3d) + (∈_0A)/(3(d+b)) + (∈_0A)/(3(d+2b)`

⇒ `C = (∈_0A)/3 ((3d^2 + 6bd + 2b^2))/(d(d+b)(d+2b))`