Capacitance of a Random Conductor

Q. $\text{Statement}1:$ A parallel plate capacitor is charged by a battery of voltage $V$. The battery is then disconnected. If the space between the plates is filled with a dielectric, the energy stored in the capacitor will decrease.

$\text{Statement}2:$ The capacitance of a capacitor increases due to the introduction of a dielectric between the plates.

1. Statement 1 is correct, statement 2 is incorrect.
2. Statement 1 is incorrect, statement 2 is correct.
3. Both the statements are incorrect.
4. Both the statements are correct.

Q. Two charges $q_1$ and $q_2$ are placed $30\text{cm}$ apart as shown in the figure. A third charge $q_3$ is moved along the arc of a circle of radius $40\text{cm}$ from $C$ to $D$. The change in potential energy of the system is $\frac{x{q}_3}{4\pi {ϵ}_0}$ where $x$ is

1. $6q_2$
2. $6q_1$
3. $8q_2$
   
4. $8q_1$

Q.

A conducting sphere of radius 10 cm is charged to $10\mu C$. Another uncharged sphere of radius 20 cm is allowed to touch it for some time. After that if the spheres are separated then surface density of charges, on the spheres will be in the ratio of:

1. 1:3
2. 3:1
3. 2:1
4. 1:1

Q.

A conducting sphere of radius 10 cm is charged to $10\mu C$. Another uncharged sphere of radius 20 cm is allowed to touch it for some time. After that if the spheres are separated then surface density of charges, on the spheres will be in the ratio of:

1. 1:3
2. 3:1
3. 2:1
4. 1:1

Q. mention the methods to increase the capacitance of a parallel plate capacitor

Q. An uncharged capacitor of $50\mu \text{F}$ is connected to a battery of $10\text{V}$. What amount of energy supplied by the battery is lost as heat while charging the capacitor ?

1. $5\text{mJ}$
2. $10\text{mJ}$
3. $1.25\text{mJ}$
4. $2.5\text{mJ}$

Q.

If n drops, each of capacitance C, coalesce to form a single big drop, the capacitance of the big drop will be

1. $\frac{V}{n^{2/3}}$
   

2. $\frac{V}{n^{1/3}}$
   

3. $V{n}^{1/3}$
   

4. $V{n}^{2/3}$

Q.

A spherical drop of capacitance$1\mu F$ is broken into eight drops of equal radius. Then, the capacitance of each small drop is ......

1. $\frac{1}{8}\mu F$
2. $8\mu F$
3. $\frac{1}{2}\mu F$
4. $\frac{1}{4}\mu F$