A) \[\left( K+3 \right)\frac{C}{4}\]
B) \[\left( K+2 \right)\frac{C}{4}\]
C) \[\left( K+1 \right)\frac{C}{4}\]
D) \[\frac{KC}{4}\]
Correct Answer: A
Solution :
The condenser with air as the dielectric has capacitance \[{{C}_{1}}=\frac{{{\varepsilon }_{0}}}{d}\,\left( \frac{3A}{4} \right)=\frac{3{{\varepsilon }_{0}}A}{4d}\] Similarly, the condenser with K as the dielectric constant has capacitance \[{{C}_{2}}=\frac{{{\varepsilon }_{0}}K}{d}\,\,\left( \frac{A}{4} \right)=\frac{{{\varepsilon }_{0}}A\,K}{4d}\] Since, \[{{C}_{1}}\]and \[{{C}_{2}}\] are in parallel \[{{C}_{net}}={{C}_{1}}+{{C}_{2}}\] \[=\frac{3\,{{\varepsilon }_{0}}A}{4d}+\frac{{{\varepsilon }_{0}}A\,K}{4d}\] \[=\frac{{{\varepsilon }_{0}}A}{d}\left[ \frac{3}{4}+\frac{K}{4} \right]\] \[=\frac{C}{4}\,(K+3)\]
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