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:

• A. \( \frac{11 \varepsilon_0 A}{18 d} \)
• B. \( \frac{13 \varepsilon_0 A}{17 d} \)
• C. \( \frac{11 \varepsilon_0 A}{20 d} \)
• D. \( \frac{18 \varepsilon_0 A}{11 d} \)

Answer 1

The Correct Option is A

Step 1: Capacitor arrangement The given system consists of three capacitors connected in parallel, each having:

Area of overlap = \( \frac{A}{3} \).

The distances between the plates are \(d\), \(2d\), and \(3d\), respectively.

Step 2: Capacitance of each section The capacitance of a parallel plate capacitor is given by:

\[ C = \frac{\epsilon_0 A}{d}. \]

For the three sections:
1. \(C_1 = \frac{\epsilon_0 A}{3d} = \frac{\epsilon_0 A}{3d}\), 
2. \(C_2 = \frac{\epsilon_0 A}{6d}\), 
3. \(C_3 = \frac{\epsilon_0 A}{9d}\).

Step 3: Total capacitance Since the capacitors are in parallel, the equivalent capacitance is:

\[ C_{\text{eq}} = C_1 + C_2 + C_3. \]

Substitute the values:

\[ C_{\text{eq}} = \frac{\epsilon_0 A}{3d} + \frac{\epsilon_0 A}{6d} + \frac{\epsilon_0 A}{9d}. \]

Take the common denominator:

\[ C_{\text{eq}} = \frac{\epsilon_0 A}{d} \left( \frac{1}{3} + \frac{1}{6} + \frac{1}{9} \right). \]

Simplify the fraction:

\[ \frac{1}{3} + \frac{1}{6} + \frac{1}{9} = \frac{6 + 3 + 2}{18} = \frac{11}{18}. \]

Thus:

\[ C_{\text{eq}} = \frac{\epsilon_0 A}{d} \cdot \frac{11}{18}. \]

Final Answer: \( C_{\text{eq}} = \frac{11 \epsilon_0 A}{18d}. \)