Two circular plates each of radius ' r ' are kept parallel to each other distance ' d ' apart. The capacitance of the capacitor formed is ' C 1 '. If the radius of each

Question

Two circular plates each of radius ' r ' are kept parallel to each other distance ' d ' apart. The capacitance of the capacitor formed is ' C 1 ​ '. If the radius of each of the plates is increased to 3 ​ times the earlier radius and their distance of separation decreased to half the initial value, the capacitance now becomes ' C 2 ​ '. The ratio C 1 ​ : C 2 ​ is

TestHub · Accepted Answer

The capacitance of a parallel plate capacitor is given by $C = \frac{ϵ _{0} A}{d}$

For circular plates of radius $r$ , the area is $A = π r^{2}$

Initial capacitance is $C_{1} = \frac{ϵ _{0} π r ^{2}}{d}$

When the radius is increased to $3 r$ , the new area is $A^{'} = π (3 r)^{2} = 3 π r^{2}$

The new distance of separation is $d^{'} = \frac{d}{2}$

The new capacitance is $C_{2} = \frac{ϵ _{0} ( 3 π r ^{2} )}{d /2} = \frac{6 ϵ _{0} π r ^{2}}{d}$

Taking the ratio of $C_{1}$ and $C_{2}$ :

$\frac{C _{1}}{C _{2}} = \frac{\frac{ϵ _{0} π r ^{2}}{d}}{\frac{6 ϵ _{0} π r ^{2}}{d}} = \frac{1}{6}$

The ratio $C_{1} : C_{2}$ is $1 : 6$

Answer: $1 : 6$

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