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. If the area of each stair is \( \frac{A}{3}\) and the height is d, the capacitance of the arrangement is: 

 

(1) \( \frac{11 \varepsilon_{0} \mathrm{A}}{18 \mathrm{d}}\)

(2) \(\frac{13 \varepsilon_{0} \mathrm{A}}{17 \mathrm{d}}\)

(3) \(\frac{11 \varepsilon_{0} \mathrm{A}}{20 \mathrm{d}}\)

(4)\( \frac{18 \varepsilon_{0} \mathrm{A}}{11 \mathrm{d}}\)   

Answer 1

Correct option is : (1) \(\frac{11 \varepsilon_{0} \mathrm{A}}{18 \mathrm{d}}\)    

All capacitor are in parallel combination. 

Also effective area is common area only

\(\Rightarrow \mathrm{C}_{\mathrm{eq}}=\mathrm{C}_{1}+\mathrm{C}_{2}+\mathrm{C}_{3}\)

\(\Rightarrow \mathrm{C}_{\mathrm{eq}}=\frac{\mathrm{A} \varepsilon_{0}}{3 \mathrm{d}}+\frac{\mathrm{A} \varepsilon_{0}}{3(2 \mathrm{d})}+\frac{\mathrm{A} \varepsilon_{0}}{3(3 \mathrm{d})}\)

\(\Rightarrow \mathrm{C}_{\text {eq }}=\frac{\mathrm{A} \varepsilon_{0}}{3}\left(\frac{11}{6 \mathrm{d}}\right)\)

\(\Rightarrow \mathrm{C}_{\mathrm{eq}}=\frac{11 \mathrm{A} \varepsilon_{0}}{18 \mathrm{d}}\)