Question

Three parallel plate air capacitors are connected in parallel. Each capacitor has plate area A/3 and separation between the plates is d, 2d and 3d respectively. The equivalent capacity of the combination is (\(ε_0\)-permittivity of free space)

• A. \(\frac{9ε_0A}{17d}\)
• B. \(\frac{11ε_0A}{17d}\)
• C. \(\frac{11ε_0A}{18d}\)
• D. \(\frac{9ε_0A}{14d}\)

Hint:

Capacitance is eps0 A / d for each; in parallel they add.

Answer 1

The Correct Option is C

Step 1: Factor out the common terms:
$C_{eq}=\dfrac{\varepsilon_0A}{3}\left(\dfrac1d+\dfrac1{2d}+\dfrac1{3d}\right)=\dfrac{\varepsilon_0A}{3d}\left(1+\dfrac12+\dfrac13\right)$.

Step 2: Sum the bracket:
$1+\dfrac12+\dfrac13=\dfrac{11}{6}$.

Step 3: Result:
$C_{eq}=\dfrac{\varepsilon_0A}{3d}\times\dfrac{11}{6}=\dfrac{11\varepsilon_0A}{18d}$. Option C.

Final Answer:
Parallel capacitances add: (1/3 + 1/6 + 1/9) eps0 A/d = 11 eps0 A/(18 d). \[ \boxed{\text{(C) }\dfrac{11\varepsilon_0A}{18d}} \]