Question 286: Electrostats & Capacitance

Question

A parallel plate capacitor with square plates is filled with four dielecytrics of dielectrics constants K1, K2, K3, K4 arranged as shown in the figure. The effective dielectric constant K will be :

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
$$=$$  $$\left[ {{{{K_1}{K_2}} \over {{K_1} + {K_2}}} + {{{K_3}{K_4}} \over {{K_3} + {K_4}}}} ight]{{{\varepsilon _0}{L^2}} \over d}$$
$$ \Rightarrow $$  $${{K{\varepsilon _0}{L^2}} \over d}$$  $$ = \left[ {{{{K_1}{K_2}} \over {{K_1} + {K_2}}} + {{{K_3}{K_4}} \over {{K_3} + {K_4}}}} ight]{{{\varepsilon _0}{L^2}} \over d}$$
$$ \Rightarrow $$  $$K = {{{K_1}{K_2}} \over {{K_1} + {K_2}}} + {{{K_3}{K_4}} \over {{K_3} + {K_4}}}$$

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

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

Electrostats & Capacitance

A parallel plate capacitor with square plates is filled with four dielecytrics of dielectrics constants K1, K2, K3, K4 arranged as shown in the figure. The effective dielectric constant K will be :

Choose the correct answer: