question_answer

A capacitor having capacitance with air, is filled with two dielectrics as shown. How many times capacitance will increase?

A) \[12\]                                   

B) \[6\]

C)  \[8/3\]                                

D)  \[3\]

Correct Answer: B

Solution :

Key Idea: After filling with dielectrics the two capacitors will be in parallel order. As shown, the two capacitors are connected in parallel. Initially the capacitance of capacitor                                  \[C=\frac{{{\varepsilon }_{0}}A}{d}\] where \[A\] is area of each plate and \[d\] is the separation between the plates. After filling with dielectrics, we have two capacitors of capacitance.          \[{{C}_{1}}=\frac{{{K}_{1}}{{\varepsilon }_{0}}(A/2)}{d}=\frac{8}{2}\frac{{{\varepsilon }_{0}}A}{d}=\frac{4{{\varepsilon }_{0}}A}{d}=4C\] and  \[{{C}_{2}}=\frac{{{K}_{2}}{{\varepsilon }_{0}}(A/2)}{d}=\frac{4}{2}\frac{{{\varepsilon }_{0}}A}{d}=\frac{2{{\varepsilon }_{0}}A}{d}=2\,\,C\] Hence, their equivalent capacitance                 \[{{C}_{eq}}={{C}_{1}}+{{C}_{2}}\]                        \[=4C+2C=6C\] \[i.e.,\] new capacitance will be six times of the original.