Electric Field Due to a Sphere and Thin Shell

Q.

The flux associated with a spherical surface is $104{\mathrm{Nm}}^2{\mathrm{C}}^{-1}$. What will be the flux, if the radius of the spherical shell is doubled?

Q. A charge $+q$ is kept inside the cavity present in a solid conducting sphere, as shown in figure.


If the sphere is given a charge $Q$, then total charge appearing on outer surface of sphere will be:

1. $Q-q$
2. $-(q+Q)$
3. $Q$
4. $Q+q$

Q.

A sphere of radius R has a uniform distribution of electric charge in its volume. At a distance x from its centre, for x < R , the electric field is directly proportional to

1. $\frac{1}{x^2}$

2. $\frac{1}{x}$

3. $x$

4. $x^2$

Q. A sphere of radius $R$ has a uniform distribution of electric charge in its volume. At a distance $x(<R)$ from its centre, the electric field is directly proportional to

1. $x$
2. $x^2$
3. $\frac{1}{x}$
4. $\frac{1}{x^2}$

Q. A conducting shell of inner radius $R_1$ and outer radius $R_2$ is given a charge $+Q$. A point charge $q_1$ is placed inside the shell and $q_2$ is placed outside the shell.


Then for various locations of $q_1$ and $q_2$, match the following entries of column$-\text{I}$ with the entries of column$-\text{II}$.

Column-$\text{I}$	Column-$\text{II}$
$\left(\text{i}\right).$ If $q_1$ is at center and $q_2=0$ , then $\overrightarrow{E}$ at center of shell due to charge on outer surface of shell is	$\left(\text{a}\right).\frac{q_1}{4\pi {\epsilon }_0{r}^2}$
$\left(\text{ii}\right).$ If $q_1$ is at center and $q_2$ is at distance $r$ from the center, then $\overrightarrow{E}$ at a point distant $r_2$ ($>r$) from the center of the shell due to outer surface charge is	$\left(\text{b}\right).$ Zero
$\left(\text{iii}\right).$ If $q_1$ is not at center and $q_2=0$ , then $\overrightarrow{E}$ at point $P$ ($P$ is at distance $r$ from $q_1$ due to charge of the inner surface of shell) is	$\left(\text{c}\right).$ Cant be determined

1. $\text{i}-\text{b},\text{ii}-\text{c},\text{iii}-\text{a}$
2. $\text{i}-\text{a},\text{ii}-\text{b},\text{iii}-\text{c}$
3. $\text{i}-\text{b},\text{ii}-\text{a},\text{iii}-\text{c}$
4. $\text{i}-\text{b},\text{ii}-\text{b},\text{iii}-\text{c}$

Q. Electric field in the space is given as $\overrightarrow{E}=2x\hat{i}+y\hat{j}$, then charge enclosed inside cube of side length $2m$ as shown in figure will be

1. $12{ϵ}_0$
2. $24{ϵ}_0$
3. $-6{ϵ}_0$
4. $8{ϵ}_0$

Q. Charges $Q,2Q$ and $-Q$ are given to three concentric conducting spherical shells $A,B$ and $C$ respectively as shown in figure. The ratio of charges on the inner and outer surfaces of shell $C$ will be

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

Q. A spherical charge distribution with volume charge density varying as $\rho \left(r\right)={\rho }_0[\frac{5}{4}-\frac{r}{R}]$, up to $r=R$ and $\rho \left(r\right)=0$ for $r>R$. Here, $r$ is the distance from the centre of sphere. The electric field at a distance $r_o(r_o<R)$ from the centre of sphere will be

1. $\frac{{\rho }_0{r}_o}{3{ϵ}_0}[\frac{5}{4}-\frac{r_o}{R}]$
2. $\frac{4\pi {\rho }_0{r}_o}{3{ϵ}_0}[\frac{5}{3}-\frac{r_o}{R}]$
3. $\frac{{\rho }_0{r}_o}{4{ϵ}_0}[\frac{5}{3}-\frac{r_o}{R}]$
4. $\frac{4{\rho }_0{r}_o}{3{ϵ}_0}[\frac{5}{4}-\frac{r_o}{R}]$

Q. A non-conducting spherical ball of radius $R$ contains a spherically symmetric charge with volume charge density $\rho =k{r}^n$, where $r$ is the distance from the center of the ball and $n$ is a constant. The value of $n$ such that the electric field inside the ball is directly proportional to square of distance from the centre is

Q. Half of the space between the plates of parallel plate capacitor is filled with a medium of dielectric constant $K$ parallel to the plates. If initially the capacitance is $C$, then the new capacitance will be

1. $\frac{KC}{K+1}$
   
2. $\frac{(K+1)C}{2}$
   
3. $\frac{2KC}{K+1}$
   
4. $KC$

Q. The electrostatic potential $\left({\varphi }_r\right)$ of a spherical symmetric system kept at origin is shown in the adjacent figure, and given as


${\varphi }_r=\frac{q}{4\pi {ϵ}_{0}r}...(r⩾R_0)$,
${\varphi }_r=\frac{q}{4\pi {ϵ}_0{R}_0}...(r\le R_0)$.
Which of the following options is/are correct?

1. For spherical region $r<R_0,$ the total electrostatic energy stored is zero
2. Within $r=2R_0,$ the total charge is $q$.
3. There will be no charge anywhere except at $r=R_0.$
4. The electric field is discontinuous at $r=R_0.$

Q.

The ratio of electric field at a point A and at point B due to the uniformly charged spherical conductor as shown, is

1. 2:5

2. 1:3

3. 0

4. 1:1

Q. A conducting shell is kept in uniform electric field $E_0\hat{i}$ as shown in figure. P is point very close to shell. Choose the correct option(s).

1. Field due to induced charges at P is towards arrow B
2. Field due to induced charges at P is towards arrow A
3. Net field at P is towards arrow B
4. Field due to induced charges at center of the shell is zero.

Q. A solid metallic sphere has a charge + 3Q. concentric with this sphere is a conducting spherical shell having charge –Q. The radius of the sphere is a and that of the spherical shell is b. (b > a). What is the electric field at a distance R (a < R < b) from the centre.

1. $\frac{Q}{2\pi {ϵ}_{0}R}$
2. $\frac{3Q}{2\pi {ϵ}_{0}R}$
3. $\frac{3Q}{4\pi {ϵ}_0{R}^2}$
4. $\frac{4Q}{2\pi {ϵ}_0{R}^2}$

Q. A metal ball of radius $R$ has a charge $Q$ and capacitance $C$.The charge is changed from $q$ to $2q$. The capacitance changes to

1. $\frac{C}{2}$
2. $2C$
3. $4C$
4. $C$

Q. The electric field in a region is radially outward with magnitude $E=A{\gamma }_0$ . The charge contained in a sphere of radius ${\gamma }_0$ centered at the origin is

1. $\frac{1}{4\pi {\epsilon }_0}A{\gamma }_0^3$
2. $4\pi {\epsilon }_{0}A{\gamma }_0^3$
3. $\frac{4\pi {\epsilon }_{0}A}{{\gamma }_0}$
4. $\frac{1}{4\pi {\epsilon }_0}\frac{A}{{\gamma }_0^3}$

Q.

A sphere of radius R has a uniform distribution of electric charge in its volume. At a distance x from its centre, for x < R , the electric field is directly proportional to

1. $\frac{1}{x^2}$

2. $\frac{1}{x}$

3. $x$

4. $x^2$

Q.

Two small spheres each carrying a charge $q$ are placed $r$ metre apart. If one of the spheres is taken around the other one in a circular path of radius $r$, the work done will be equal to

1. 

2. 

3. 

4. Zero

Q. Electric field in the space is given as $\overrightarrow{E}=2x\hat{i}+y\hat{j}$, then charge enclosed inside cube of side length $2m$ as shown in figure will be

1. $12{ϵ}_0$
2. $24{ϵ}_0$
3. $-6{ϵ}_0$
4. $8{ϵ}_0$