Question

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

• A. $\frac{{\epsilon }_{0}A(3d^2+6bd+2b^2)}{3d(d+b)(d+2b)}$
• B. $\frac{{\epsilon }_{0}A(3d^2-6bd-2b^2)}{3d(d-b)(d-2b)}$
• C. $\frac{{\epsilon }_{0}A(2d^2-bd+2b^2)}{3d(d+b)(d+2b)}$
• D. $\frac{{\epsilon }_{0}A(d^2-bd-b^2)}{3d(d+b)(d-2b)}$

Answer 1

The correct option is A $\frac{{\epsilon }_{0}A(3d^2+6bd+2b^2)}{3d(d+b)(d+2b)}$

As we know, in parallel plate capacitor, if dielectric divides the area then parts formed will be in parallel as shown below.


$C=C_1+C_2+C_3$

$=\frac{{\epsilon }_0\frac{A}{3}}{d}+\frac{{\epsilon }_0\frac{A}{3}}{(d+b)}+\frac{{\epsilon }_0\frac{A}{3}}{(d+2b)}$

$=\frac{{\epsilon }_{0}A\left[\right(d+b\left)\right(d+2b)+d(d+2b)+d(d+b\left)\right]}{3d(d+b)(d+3b)}$

$=\frac{{\epsilon }_{0}A[d^2+2bd+bd+2b^2+d^2+2bd+d^2+db]}{3d(d+b)(d+3b)}$

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

<!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> Hence, (a) is the correct answer.