Question

You have a parallel plate capacitor, a spherical capacitor and a cylindrical capacitor. Each capacitor is charged by and then removed from the same battery. Consider the following situations

(i) the separation between the plates of the parallel plate capacitor is reduced

(ii) the radius of the outer spherical shell of the spherical capacitor is increased

(iii) the radius of the outer cylinder of the cylindrical capacitor is increased

Which of the following is correct?

• A. In each of these situations (i), (ii) and (iii), charge on the given capacitor remains the same and potential difference across it also remains the same
• B. In each of these situations (i), (ii) and (iii), charge on the given capacitor remains the same but potential difference, in situations (i) and (iii), decreases, and in situation (ii), increases
• C. In each of these situations (i), (ii) and (iii), charge on the given capacitor remains the same but potential difference, in situations (i), decreases, and in situations (ii) and (iii), increases
• D. Charge on the capacitor in each situation changes. It increases in all these situations but potential difference remains the same

Answer 1

Answer: (C)

Each capacitor is charged and then removed from the battery. Changing the plate separation in a parallel plate capacitor or the radius of any shell or cylinder in the spherical or the cylindrical capacitors will not change the charge on any capacitor. Each capacitor, in the disconnected state, is an isolated system.
In situation (i), since $C=\frac{\epsilon _{0} A}{d}$ , capacity will increase as $d$ is reduced. Therefore, $V=\frac{Q}{C} \, $ decreases, $Q$ being the same.
In situation (ii), $C=4\pi ϵ_{0}\frac{a b}{b - a}=4\pi ϵ_{0}\frac{a}{1 - a / b}$
$a$ and $b$ are the radii of the inner and the outer spherical shells, respectively.
As $b$ is increased, capacity will reduce and
$V=\frac{Q}{C}$ increases, $Q$ being the same.
In situation (iii), $C=2\pi \left(\epsilon \right)_{0}\frac{L}{ln \left(\frac{b}{a}\right)}$
$L$ is the length of cylinders; $a$ and $b$ are the radii of inner and outer cylinders, respectively.
As $b$ is increased, capacity will decrease and $V=\frac{Q}{C}$ increases, $Q$ being the same