Question

A capacitor is made of two plates. The first plate is flat plate of area $A$ and second plate having a stair like structure as shown. The area of each stair is $\frac{A}{3}$ and height is $a$. Find the capacitance of the assembly if $C=$$\frac{{\epsilon }_{0}A}{a}$.

• A. $\frac{11}{18}C$
  
• B. $\frac{17}{18}C$
  
• C. $\frac{13}{11}C$
  
• D. $\frac{11}{13}C$
  

Answer 1

The correct option is A $\frac{11}{18}C$


From the given figure it is clear that, the given arrangement behaves as parallel combination of three individual capacitors.

Since area $A$, and seperation between the plates are given. So, we can get individual capacitances of all three plates as follows,

$C_1=\frac{{\epsilon }_0\times \frac{A}{3}}{a}=\frac{{\epsilon }_{0}A}{3a}$ ; $C_2=\frac{{\epsilon }_0\times \frac{A}{3}}{2a}=\frac{{\epsilon }_{0}A}{6a}$ ; $C_3=\frac{{\epsilon }_0\times \frac{A}{3}}{3a}=\frac{{\epsilon }_{0}A}{9a}$

Therefore, the equivalent capacitance or net capacitance,

$C_{eq}=C_1+C_2+C_3$

$\Rightarrow C_{eq}=\frac{{\epsilon }_{0}A}{a}$ $(\frac{1}{3}+\frac{1}{6}+\frac{1}{9})$

$\Rightarrow C_{eq}=\frac{11}{18}C$ $[\because C=\frac{{\epsilon }_{0}A}{a}]$

Hence, option (a) is the correct answer.