First slide

Capacitance

Question

A parallel plate condenser is filled with two dielectric as shown in fig. Area of each plate is A metre² and the separation is d metre. The dielectric constants are K¹ and K₂ respectively. Its capacitance is farad will be

Moderate

A

$\frac{ε_{0} A}{d} (K_{1} + K_{2})$
B

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

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

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

Solution

The combination is equivalent to two capacitors in parallel each with plate area A/2 and separation d. See fig.
Thus, $C_{1} = \frac{K_{1} ε_{e} (A / 2)}{d} = \frac{K_{1} ε_{0} A}{2 d}$
$C_{2} = \frac{K_{2} ε_{0} (A / 2)}{d} = \frac{K_{2} ε_{0} A}{2 d}$
The equivalent capacitance C is given by
$\begin{aligned} C = C_{1} + C_{2} = \frac{K_{1} ε_{0} A}{2 d} + \frac{K_{2} ε_{0} A}{2 d} \\ = \frac{ε_{0} A}{d} (\frac{K_{1} + K_{2}}{2}) \end{aligned}$

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