question_answer

A parallel plate capacitor of area ?A? plate separation ?d? is filled with two dielectrics as shown. What is the capacitance of the arrangement?

A) \[\frac{3K{{\varepsilon }_{0}}A}{4d}\]

B) \[\frac{4K{{\varepsilon }_{0}}A}{4d}\]

C) \[\frac{\left( K+1 \right){{\varepsilon }_{0}}A}{2d}\]

D) \[\frac{K\left( K+3 \right){{\varepsilon }_{0}}A}{2\left( K+1 \right)d}\]

Correct Answer: D

Solution :

[d] \[{{c}_{1}}=\frac{\left( A/2 \right){{\varepsilon }_{0}}}{d/2}=\frac{A{{\varepsilon }_{0}}}{d},\] \[{{c}_{2}}=K\frac{A{{\varepsilon }_{0}}}{d},{{c}_{3}}=K\frac{A{{\varepsilon }_{0}}}{2d}\] \[\therefore {{c}_{eq}}=\frac{{{c}_{1}}\times {{c}_{2}}}{{{c}_{1}}+{{c}_{2}}}+{{c}_{3}}=\frac{\left( 3+K \right)KA{{\varepsilon }_{0}}}{2d\left( K+1 \right)}\] (\[\therefore {{C}_{1}}\]and \[{{C}_{2}}\]are in series and resultant of these two in parallel with \[{{C}_{3}}\])