A parallel plate capacitor is made of two dielectric blocks in series. One of the blocks has thickness $d_{1}$ and dielectric constant $K_{1}$ and the other has thickness $d_{2}$ and dielectric constant $K_{2}$ as shown in figure. This arrangement can be thought as a dielectric slab of thickness d (= $d_{1} + d_{2}$ ) and effective dielectric constant K. The K is

Question

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Answer 1

The capacities of two individual condensers are
$C_{1} = \frac{K _{1} ε _{0} A}{d _{1}}; C_{2} = \frac{K _{2} ε _{0} A}{d _{2}}$
The arrangement is equivalent to two capacitors joined in series

$∴$ Equivalent capacitance, $\frac{1}{C _{e q}} = \frac{1}{C _{1}} + \frac{1}{C _{2}} = \frac{d _{1}}{K _{1} ε _{0} A} + \frac{d _{2}}{K _{2} ε _{0} A}$

$= \frac{1}{ε _{0} A} [\frac{d _{1}}{K _{1}} + \frac{d _{2}}{K _{2}}] = \frac{1}{ε _{0} A} [\frac{K _{2} d _{1} + K _{1} d _{2}}{K _{1} + K _{2}}]$

or $C_{e q} = ε_{0} A (\frac{K _{1} + K _{2}}{K _{2} d _{1} + K _{1} d _{2}}) . . . . (i)$

Also $C_{e q} = \frac{K ε _{0} A}{d _{1} + d _{2}}$
From (i) and (ii) $ε_{0} A (\frac{K _{1} + K _{2}}{K _{2} d _{1} + K _{1} d _{2}}) = ε_{0} A (\frac{K}{d _{1} + d _{2}})$

$∴ K = \frac{K _{1} K _{2} ( d _{1} + d _{2} )}{K _{1} d _{2} + K _{2} d _{1}}$

Answer 2

The capacities of two individual condensers are

C_{1} = \frac{K _{1} ε _{0} A}{d _{1}}; C_{2} = \frac{K _{2} ε _{0} A}{d _{2}}

The arrangement is equivalent to two capacitors joined in series

∴

Equivalent capacitance,

\frac{1}{C _{e q}} = \frac{1}{C _{1}} + \frac{1}{C _{2}} = \frac{d _{1}}{K _{1} ε _{0} A} + \frac{d _{2}}{K _{2} ε _{0} A}

= \frac{1}{ε _{0} A} [\frac{d _{1}}{K _{1}} + \frac{d _{2}}{K _{2}}] = \frac{1}{ε _{0} A} [\frac{K _{2} d _{1} + K _{1} d _{2}}{K _{1} + K _{2}}]

or

C_{e q} = ε_{0} A (\frac{K _{1} + K _{2}}{K _{2} d _{1} + K _{1} d _{2}}) . . . . (i)

Also

C_{e q} = \frac{K ε _{0} A}{d _{1} + d _{2}}

From (i) and (ii)

ε_{0} A (\frac{K _{1} + K _{2}}{K _{2} d _{1} + K _{1} d _{2}}) = ε_{0} A (\frac{K}{d _{1} + d _{2}})

∴ K = \frac{K _{1} K _{2} ( d _{1} + d _{2} )}{K _{1} d _{2} + K _{2} d _{1}}