Charge vs Potential in Spherical Conductor

Q. A system comprises of a solid non conducting sphere of charge $Q$ and radius $R$. A point charge $q$ is placed at a distance $r$ from the centre of sphere. Find the total energy of the system
$[k=\frac{1}{4\pi {\epsilon }_0}]$

1. $\frac{3k{Q}^2}{5R}+\frac{kQq}{R}$
2. $\frac{3k{Q}^2}{5R}$
3. $\frac{3k{Q}^2}{5R}+\frac{kQq}{r}$
4. $\frac{kQq}{r}$

Q. 4.) On loading a metal wire of cross section of 10 power -6 m square and length 2m by a mass of 210kg it extends by 16mm and suddenly broke from the point of support. If density of that metal is 8000 kg per m cube and its specific heat is 420J per kg per K the rise in temperature of wire is

Q. A charge $q$ is distributed over two concentric hollow conducting sphere of radii $r$ and $R(>r)$ such that their surface charge densities are equal. The potential at their common centre is

1. $V_c=\frac{q}{4\pi {ϵ}_0}\left(\frac{r+R}{r^2+R^2}\right)$
2. Zero
3. $\frac{q}{4\pi {ϵ}_0}(\frac{1}{r}+\frac{1}{R})$
4. $\frac{q}{4\pi {ϵ}_0}\frac{(r+R)}{(r^2+R^2{)}^2}$

Q. The radii of two charged metal spheres are 5 cm and 10 cm both having same charge 75 mC. If they are connected by a wire, the quantity of charge transferred through the wire is

1. 25 mC
2. 50 mC
3. 75 mC
4. Zero

Q. A charge $q$ is distributed over two concentric hollow conducting sphere of radii $r$ and $R(>r)$ such that their surface charge densities are equal. The potential at their common centre is

1. $\frac{q}{4\pi {ϵ}_0}(\frac{1}{r}+\frac{1}{R})$
2. Zero
3. $V_c=\frac{q}{4\pi {ϵ}_0}\left(\frac{r+R}{r^2+R^2}\right)$
4. $\frac{q}{4\pi {ϵ}_0}\frac{(r+R)}{(r^2+R^2{)}^2}$

Q.

The insulated metal spheres of radii R₁ and R₂ having charges Q₁ and Q₂ respectively are connected to each other. There is

1. No change in the energy of the system

2. An increase in the energy of the system

3. Always a decrease in the energy of the system

4. A decrease in the energy of the system unless Q₁R₂=Q₂R₁

Q. A large conducting sphere of radius $r$, having a charge $Q$ is placed in contact with a small neutral conducting sphere of radius $r^{\prime }$ and is then separated. The charge on smaller sphere will now be:

1. $\frac{Q{r}^{\prime }}{r^{\prime }+r}$
2. $\frac{Q(r-r^{\prime })}{r^{\prime }}$
3. $\frac{Q(r+r^{\prime })}{r}$
4. $\frac{Qr}{r^{\prime }+r}$

Q. If two conducting spheres are separately charged and then brought in contact, then

1. The total energy of the two spheres is conserved

2. The total charge on the two spheres is conserved

3. Both the total energy and charge are conserved

4. The final potential is always the mean of the original potentials of the two spheres

Q. Two concentric spherical shells are separated by vacuum. The inner shell has total change $Q$ and radius $r$. The outer shell has total charge $-Q$ and radius $2r$. The potential energy stored in the capacitor is

1. $\frac{Q^2}{16\pi {\epsilon }_{0}r}$
2. $\frac{Q^2}{2\pi {\epsilon }_{0}r}$
3. $\frac{4Q^2}{7\pi {\epsilon }_{0}r}$
4. $\frac{Q^2}{8\pi {\epsilon }_{0}r}$

Q. Two metallic spheres of radii $1\text{cm}$ and $3\text{cm}$ are given charges of $-1\times {10}^{-2}\text{C}$ and $5\times {10}^{-2}\text{C}$ respectively. If these are connected by a conducting wire (potential on them becomes equal), the final charge on bigger sphere is

1. $1\times {10}^{-2}\text{C}$
2. $3\times {10}^{-2}\text{C}$
3. $2\times {10}^{-2}\text{C}$
4. $4\times {10}^{-2}\text{C}$

Q. A metallic sphere has a charge of 10 µC. A unit negative charge is brought from A to B both 100 cm away from the sphere but A being east of it while B being on west. The net work done is
(a) Zero (b) 2/10 joule (c) –2/10 joule (d) –1/10 joule

Q. Two uniformly charged spherical drops at potential V coalesce to form a larger drop.If capacity of each smaller drop is C then find capacity and potential of larger drop.