First slide

Insertion of dielectric in capacitor

Question

In the figure a capacitor is filled with dielectrics. The resultant capacitance is

Easy

A

$\frac{2 \in_{0} A}{d} [\frac{1}{k_{1}} + \frac{1}{k_{2}} + \frac{1}{k_{3}}]$
B

$\frac{\in_{0} A}{d} [\frac{1}{k_{1}} + \frac{1}{k_{2}} + \frac{1}{k_{3}}]$
C

$\frac{2 \in_{0} A}{d} [k_{1} + k_{2} + k_{3}]$
D

None of these

Solution

$C_{1} = \frac{K_{1} \in_{0} \frac{A}{2}}{(\frac{d}{2})} = \frac{K_{1} \in_{0} A}{d} C_{2} = \frac{K_{2} \in_{0} (\frac{A}{2})}{(\frac{d}{2})} = \frac{K_{2} \in_{0} A}{d} a n d C_{3} = \frac{K_{3} \in_{0} (\frac{A}{2})}{d} = \frac{K_{3} \in_{0} A}{2 d} N o w, C_{1} a n d C_{2} i n s e r i e s a n d t h a t c o m b i n a t i o n i s i n p a r a l l e l w i t h C_{3} C_{e q} = C_{3} + \frac{C_{1} C_{2}}{C_{1} + C_{2}} = (\frac{K_{3}}{2} + \frac{K_{1} K_{2}}{K_{1} + K_{2}}) \frac{\in_{0} A}{d}$

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