Electric Potential Due to a Solid Sphere of Charge

Q. Two charged spherical conductors of radii $R_1$ and $R_2$ are connected by a wire. Then the ratio of surface charge densities of the spheres $\left({\sigma }_1/{\sigma }_2\right)$ is

1. $\frac{R_1^2}{R_2^2}$
2. $\frac{R_1}{R_2}$
3. $\frac{R_2}{R_1}$
4. $\sqrt{\left(\frac{R_1}{R_2}\right)}$

Q. Eight small drops, each of radius $r$ and having same charge $q$ are combined to form a big drop. The ratio of potential of the bigger drop to the smaller drop is

1. $8:1$
2. $4:1$
3. $2:1$
4. $1:8$

Q. Four metallic plates are placed as shown in the figure. Plate $2$ is given a charge $Q$ whereas all other plates are uncharged. Plates $1$ and $4$ are joined together. All plates have same area. The charge appearing on the right side of plate $3$ is

1. zero
2. $+\frac{Q}{4}$
3. $-\frac{3Q}{4}$
4. $\frac{+Q}{2}$

Q. Two spherical conductors of capacitance $3\mu \text{F}$ and $5\mu \text{F}$ are charged to potentials of $300\text{Volt}$ and $500\text{Volt}$. The spheres are then connected, then the final charge on $3\mu \text{F}$ and $5\mu \text{F}$ capacitors respectively will be:

1. $900\mu \text{C},2500\mu \text{C}$
2. $1200\mu \text{C},2200\mu \text{C}$
3. $1275\mu \text{C},2125\mu \text{C}$
4. $1000\mu \text{C},2400\mu \text{C}$

Q. Four large identical metal plates are placed as shown in the figure. Plate $2$ is given a charge $Q$. All other plates are neutral. Now Plate $1$ and $4$ are earthed. Area of each plates is $A$. The charge appearing on the right side of plate $3$ is

1. $\frac{Q}{2}$
2. $\frac{Q}{4}$
3. $\frac{Q}{8}$
4. $Q$

Q. $1000$ identical drops of mercury are charged to the same potential at $10\text{V}$. All these are now combined to make one large drop. The potential of the large drop is

1. $1000\text{V}$
2. $90\text{V}$
3. $100\text{V}$
4. $10\text{V}$

Q. Two spherical conductors $\text{A}$ and $\text{B}$ of radii $R$ and $2R$ respectively are each given a charge $Q$. When they are connected by a metallic wire. The charge will

1. Flow from $\text{A}$ to $\text{B}$
2. Flow from $\text{B}$ to $\text{A}$
3. Remain stationary on conductor
4. None of these

Q. A large hollow metallic sphere $A$ (of radius $R$ ) is positively charged to a potential of $100\text{V}$ and a small sphere $B$ (of radius $\frac{R}{5}$) is also positively charged to a potential of $100\text{V}$. Now $B$ is placed inside $A$ and they are connected by a wire. The final potential of $A$ will be:

1. $200\text{V}$
2. $150\text{V}$
3. $120\text{V}$
4. none of above

Q. Two metal spheres are very far apart but are connected by a thin wire as shown-


If the combined charge of both the sphere is $Q$, then the common potential will be

1. $\frac{kQ}{r_1-r_2}$
   
2. $\frac{-kQ}{r_1+r_2}$
   
3. $\frac{kQ}{r_1+r_2}$
   
4. $\frac{-kQ}{r_1{r}_2}$
   

Q. Three identical large metal plates of area $A$ are separated by distances $d$ and $2d$ from each other, as shown in the figure. The top metal plate is uncharged, while other metal plates have charges $+Q$ and $–Q$. Top and bottom metal plates are connected by switch $S$ through a resistor of unknown resistance. What is the energy (in $\text{mJ}$) dissipated in the resistor when the switch is closed and steady state is achieved?

Given: $\frac{{ϵ}_{0}A}{d}=6\mu \text{F},Q=60\mu \text{C}$

Q. Two metal spheres (radii $r_1,r_2$ with $r_1<r_2$) are very far apart but are connected by a thin wire. If their combined charge is $Q$, then what is their common Potential?

1. $\frac{kQ}{(r_1+r_2)}$
2. $\frac{kQ}{(r_1-r_2)}$
3. $\frac{-kQ}{r_1{r}_2}$
4. $\frac{-kQ}{(r_1+r_2)}$

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

Q. Two spherical conductors of capacitance $3.0\mu \text{F}$ and $5.0\mu \text{F}$ are charged to potentials of $300\text{volt}$ and $500\text{volt}$. The two are connected, resulting in redistribution of charges. Then the final potential is (in volt) -

Q. Two metal spheres of capacitance $C_1$ and $C_2$ carry some charges. They are put in contact and then separated. The final charge $Q_1$ and $Q_2$ on them will satisfy

1. $\frac{C_1}{C_2}>\frac{Q_1}{Q_2}$
2. $\frac{C_1}{C_2}<\frac{Q_1}{Q_2}$
3. $\frac{C_1}{C_2}=\frac{Q_1}{Q_2}$
4. $\frac{C_1}{C_2}=\frac{Q_2}{Q_1}$

Q. A small sphere of radius r1 and charge q1 is enclosed by a sphericalshell of radius r2 and charge q2. Show that if q1 is positive, chargewill necessarily flow from the sphere to the shell (when the two areconnected by a wire) no matter what the charge q2 on the shell is.

Q. A metal sphere $\text{A}$ of radius $r_1$ charged to a potential ${\varphi }_1$ is enveloped by a thin walled conducting spherical shell $\text{B}$ of radius $r_2$. Then potential ${\varphi }_2$ of the sphere $\text{A}$ after it is connected to the shell $\text{B}$ by a thin conducting wire will be:

1. ${\varphi }_1(\frac{r_1}{r_2})$
2. ${\varphi }_1\left(\frac{r_2}{r_1}\right)$
3. ${\varphi }_1(1-\frac{r_2}{r_1})$
4. ${\varphi }_1\left(\frac{r_1{r}_2}{r_1+r_2}\right)$

Q. Two concentric conducting spheres of radii $R$ and $2R$ are carrying charges $Q$ and $2Q$ respectively. If the charge on inner sphere is doubled, the potential difference between the two spheres will :

1. become two times
2. become four times
3. be halved
4. remain same

Q. $512$ identical drops of mercury are charged to a potential of $2\text{V}$ each. The drops are then joined to form a bigger drop. The potential of this big drop is $\text{V}$

Q. The given graph shows variation (with distance r from centre) of
\( \theta \Lambda \)

1. Electric potential of a uniformly charged hollow sphere
2. Electric potential of a uniformly charged non-conducting solid sphere
3. Electric field of a uniformly charged nonconducting sphere
4. Electric field of a uniformly charged conducting sphere

Q. $512$ identical drops of mercury are charged to a potential of $2\text{V}$ each. The drops are then joined to form a bigger drop. The potential of this big drop is $\text{V}$

Q. A small conducting sphere of radius $r$ is lying concentrically with a bigger hollow conducting sphere of radius $R$. The bigger and smaller spheres are charged with $Q$ and $q$ respectively $(Q>q)$. The two spheres are insulated from one another. The potential difference between the spheres will be :

1. $\frac{1}{4\pi {ϵ}_0}(\frac{q}{r}-\frac{Q}{R})$
2. $\frac{1}{4\pi {ϵ}_0}(\frac{q}{r}-\frac{q}{R})$
3. $\frac{1}{4\pi {ϵ}_0}(\frac{q}{r}+\frac{Q}{R})$
4. $\frac{1}{4\pi {ϵ}_0}(\frac{q}{R}-\frac{Q}{r})$

Q. Two conducting spheres having radii 10 cm and 20 cm. One sphere has given charge of $150\mu C$ and it is connected by a wire to another sphere. Their common potential will be:

1. $9\times {10}^6$ volt
2. $4.5\times {10}^6$ volt
3. $1.8\times {10}^6$ volt
4. $1.35\times {10}^9$ volt

Q. Figure shows two conducting thin concentric shells of radii $r$ and $3r$. The outer shell carries $q$ while inner shell is neutral and is connected to earth by a switch $S$. Find the charge that will flow from earth to inner shell after the switch $S$ is closed.

1. $-\frac{q}{3}$
2. $-\frac{q}{2}$
3. $\frac{q}{3}$
4. $\frac{2q}{3}$

Q. An insulated sphere of radius R has a volume charge density $\rho$ . The variation of its potential with respect to the distance from the centre is best represented by

1. 
2. 
3. 
4. 

Q. A uniformly charged sphere is placed as shown. Find the amount of work done by an external agent to bring a 2C charge (Very slowly without acceleration) from infinity to the point P? Take infinity as the reference

1. $1.8\times {10}^{12}J$
2. $1.8\times {10}^{11}J$
3. $18\times {10}^{12}J$
4. $1.8\times {10}^{14}J$

Q. A uniformly charged sphere is placed as shown. Find the amount of work done by an external agent to bring a 2C charge (Very slowly without acceleration) from infinity to the point P? Take infinity as the reference

1. $1.8\times {10}^{12}J$
2. $1.8\times {10}^{11}J$
3. $18\times {10}^{12}J$
4. $1.8\times {10}^{14}J$

Q. Correct curve of potential $\left(V\right)$ versus distance $\left(r\right)$ from centre of two charged spherical shells is

1. 
2. 
3. 
4. 

Q. An insulated sphere of radius R has a volume charge density $\rho$ . The variation of its potential with respect to the distance from the centre is best represented by

1. 
2. 
3. 
4. 

Q. Two metal spheres (radii $r_1,r_2$ with $r_1<r_2$) are very far apart but are connected by a thin wire. If their combined charge is Q, then what is their common potential?

1. $-kQ/r_1{r}_2$
2. $-kQ/(r_1+r_2)$
3. $kQ/(r_1+r_2)$
4. $kQ/(r_1-r_2)$

Q. A copper sphere is suspended in an evacuated chamber maintained at $300$ K. The sphere is maintained at a constant temperature of $500$ K by heating it electrically. A total of $300$ W of electric power is needed to do it. When half of the surface of the surface of the copper sphere is completely blackened, $600$ W is needed to maintain the same temperature of the sphere. Calculate the emissivity of copper.