First slide

Capacitance

Question

A parallel plate condenser with plate area A and separation d filled with dielectric as shown in fig. The dielectric constants are K₁ and K₂respectively. The capacitors will be

Easy

A

$\frac{ε_{0} A}{d} (K_{1} + K_{2})$
B

$\frac{ε_{0} A}{d} (K_{1} + K_{2}) / K_{1} K_{2}$
C

$\frac{2 ε_{0} A}{d} (K_{1} K_{2} / K_{1} + K_{2})$
D

$\frac{2 ε_{0} A}{d} (K_{1} + K_{2}) / K_{1} K_{2}$

Solution

The combination is equivalent to two capacitors in series, each with plate area A and separation d/2.
$\begin{aligned} C_{1} = \frac{K_{1} ε_{0} A}{(d / 2)} = \frac{2 K_{1} ε_{0} A}{d} \\ C_{2} = \frac{K_{2} ε_{0} A}{d / 2} = \frac{2 K_{2} ε_{0} A}{d} \end{aligned}$
Further $\frac{1}{C} = \frac{1}{C_{1}} + \frac{1}{C_{2}}$
Substitute the values of C₁ and C₂ and obtain C.

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