Question

An uncharged parallel-plate capacitor having a dielectric of constant $k$ is connected to a similar air-filled capacitor charged to a potential $V$. The two share the charge and the common potential is $V^{\prime }$. The dielectric constant $k$ is :

• A. $\frac{V-V^{\prime }}{V^{\prime }}$
• B. $\frac{V^{\prime }-V}{V^{\prime }}$
• C. $\frac{V^{\prime }-V}{V^{\prime }+V}$
• D. $\frac{V^{\prime }-V}{V}$

Answer 1

Answer: (A)

$\frac{V-V^{\prime }}{V^{\prime }}$

If the capacitance of air filled capacitor is $C_1=C$ then the capacitance of dielectric filled capacitor is $C_2=kC$.
Here the charge on air filled capacitor is $Q=CV$ and this charge is shared to the two capacitance. Now the charge on air filled capacitor is $Q_1=CV/2$ and charge on dielectric filled capacitor is $CV/2$.
Common potential , $V^{\prime }=\frac{Q_1+Q_2}{C_1+C_2}=\frac{CV/2+CV/2}{C+kC}=\frac{CV}{C+kC}$
or $\frac{V^{\prime }}{V}=\frac{1}{1+k}$
or $1+k=\frac{V}{V^{\prime }}$
or $k=\frac{V}{V^{\prime }}-1=\frac{V-V^{\prime }}{V^{\prime }}$