Question 354: Electrostatics & Capacitance

Question

A parallel plate capacitor with square plates is filled with four dielecytrics of dielectrics constants K
1
, K
2
, K
3
, K
4
arranged as shown in the figure. The effective dielectric constant K will be :
![](https://app-content.cdn.examgoal.net/fly/@width/image/1l8p77ao2/4c06eef8-b9d0-4709-8f90-8eca5e513047/1c089a20-4123-11ed-8a36-9f4a474c8481/file-1l8p77ao3.png?format=png)

Accepted Answer

Here,  ${C_1} = {{{K_1}{\varepsilon _0}.L.{L \over 2}} \over {{d \over 2}}} = {{{K_1}{\varepsilon _0}{L^2}} \over d}$
${C_2}{{{K_2}{\varepsilon _0}.L.{L \over 2}} \over {{d \over 2}}} = {{{K_2}{\varepsilon _0}{L^2}} \over d}$
${C_3} = {{{K_3}{\varepsilon _0}{L^2}} \over d}$
${C_4} = {{{K_4}{\varepsilon _0}{L^2}} \over d}$
Here  C
1
C
2
are in series and C
3
, C
4
are in series.
Equivalent capacitance point A and B is,
C
eq
$=$ ${{{C_1}{C_2}} \over {{C_1} + {C_2}}} + {{{C_3}{C_4}} \over {{C_3} + {C_4}}}$
$ = {C_{12}} + {C_{34}}$
${C_{12}}\,\, = \,\,{{{{{K_1}{\varepsilon _0}{L^2}} \over d} 	imes {{{K_2}{\varepsilon _0}{L^2}} \over d}} \over {{{{K_1}{\varepsilon _0}{L^2}} \over d} + {{{K_2}{\varepsilon _0}{L^2}} \over d}}}$
$ = \,\,{{{K_1}{K_2}} \over {{K_1} + {K_2}}} 	imes {{{\varepsilon _0}{L^2}} \over d}$
Similarly,
${C_{34}} = \,\,{{{K_3}{K_4}} \over {{K_3} + {K_4}}} 	imes {{{\varepsilon _0}{L^2}} \over d}$
$ 	herefore $  C
eq
$=$  $\left[ {{{{K_1}{K_2}} \over {{K_1} + {K_2}}} + {{{K_3}{K_4}} \over {{K_3} + {K_4}}}} ight]{{{\varepsilon _0}{L^2}} \over d}$
$ \Rightarrow $  ${{K{\varepsilon _0}{L^2}} \over d}$  $ = \left[ {{{{K_1}{K_2}} \over {{K_1} + {K_2}}} + {{{K_3}{K_4}} \over {{K_3} + {K_4}}}} ight]{{{\varepsilon _0}{L^2}} \over d}$
$ \Rightarrow $  $K = {{{K_1}{K_2}} \over {{K_1} + {K_2}}} + {{{K_3}{K_4}} \over {{K_3} + {K_4}}}$
![](https://app-content.cdn.examgoal.net/fly/@width/image/1l8p783x6/9aaa2160-762b-467b-9b6f-77cb2867b5bd/32a183a0-4123-11ed-8a36-9f4a474c8481/file-1l8p783x7.png?format=png)

Answer

$K = {{{K_1}{K_2}} \over {{K_1} + {K_2}}} + {{{K_3}{K_4}} \over {{K_3} + {K_4}}}$

Answer

$K = {{\left( {{K_1} + {K_2}} \right)\left( {{K_3} + {K_4}} \right)} \over {2\left( {{K_1} + {K_2} + {K_3} + {K_4}} \right)}}$

Answer

$K = {{\left( {{K_1} + {K_2}} \right)\left( {{K_3} + {K_4}} \right)} \over {{K_1} + {K_2} + {K_3} + {K_4}}}$

Answer

$K = {{\left( {{K_1} + {K_4}} \right)\left( {{K_2} + {K_3}} \right)} \over {2\left( {{K_1} + {K_2} + {K_3} + {K_4}} \right)}}$

Electrostatics & Capacitance

A parallel plate capacitor with square plates is filled with four dielecytrics of dielectrics constants K 1 , K 2 , K 3 , K 4 arranged as shown in the figure. The effective dielectric constant K will be :

Choose the correct answer: