Gauss's Law

Q. Electric charge is uniformly distributed along a long straight wire of radius 1mm. The charge per cm length of the wire is Q coulomb. Another cylindrical surface of radius 50 cm and length 1m symmetrically encloses the wire as shown in the figure. The total electric flux passing through the cylindrical surface is

1. $\frac{Q}{{ϵ}_0}$
2. $\frac{100Q}{{ϵ}_0}$
3. $\frac{10Q}{\left(\pi {ϵ}_0\right)}$
4. $\frac{100Q}{\left(\pi {ϵ}_0\right)}$

Q.

Why is the electric field uniform in parallel plates?

Q.
The electric flux for Gaussian surface A that enclose the charged particles in free space is $(given{q}_1=-14nC,q_2=78.85nC,q_3=-56nC)$

1. ${10}^{3}N{m}^2{c}^{-1}$
2. ${10}^{3}C{N}^{-1}{m}^{-2}$
3. $6.32\times {10}^{3}N{m}^2{C}^{-1}$
4. $6.32\times {10}^{3}C{N}^{-1}{m}^{-2}$

Q. The value of flux due to q through square plate is

1. $\frac{q}{6{ϵ}_o}$
2. $\frac{q}{6{ϵ}_o}$
3. $\frac{q}{12{ϵ}_o}$
4. $\frac{q}{24{ϵ}_o}$

Q. Which of the following graphs correctly represents the variation of $\overrightarrow{E}$ vs $r$ for two point charges $+q$ and $-2q$ kept some distance apart along the line joining these two charges?

1. 
2. 
3. 
4. 

Q.

An electric dipole is placed at the center of an imaginary sphere. Which of the following option is/are correct?

1. The electric field is zero at every point of the sphere

2. The flux of the electric field at every point of the sphere is zero

3. The electric potential is zero at every point of the sphere

4. The electric potential is zero on a circle on the surface

Q. Figure shows an imaginary cube of side $a$. A uniformly charged rod of length $a$ moves towards right at a constant speed $v$. At $t=0$, the right end of the rod just touches the left face of the cube. Which of the following represents the correct graph between electric flux passing through the cube with time?

1. 
2. 
3. 
4. 

Q. A point charge +$q$ is placed on the axis of a closed cylinder of radius $R$ and height $\frac{25R}{12}$ as shown. If electric flux coming out form the curved surface of cylinder is $\frac{xq}{10{\in }_0}$, then calculate $x$.

Q. A long solid non-conducting cylinder of radius $10\text{cm}$ contains a uniformly distributed charge of density $10{\text{nC/m}}^3$. Find the magnitude of electric field at a point $\text{P}$ inside the cylindrical volume at a distance $5\text{cm}$ from its axis.
(Take ${ϵ}_o=9\times {10}^{-12}{\text{C}}^2/{\text{N-m}}^2$)

1. $27.8\text{N/C}$
2. $17.8\text{N/C}$
3. $47.8\text{N/C}$
4. $37.8\text{N/C}$

Q. Which of the following curves between electric field $\left(E\right)$ at a distance $\left(x\right)$ from an infinite long uniformly charged sheet is correct?
[Assume $x$ to be very small as compared to dimension of sheet]

1. 
2. 
3. 
4. 

Q. A cube has sides of length $l=0.2\text{m}$. It is placed with one corner at the origin as shown in figure. The electric field is uniform and given by $\overrightarrow{E}=(2.5\hat{i}-4.2\hat{j})\text{N/C}$. Find the electric flux through the entire cube.

1. Zero
2. $10{\text{N-m}}^2\text{/C}$
3. $15{\text{N-m}}^2\text{/C}$
4. $5{\text{N-m}}^2\text{/C}$

Q.

Is a cube a Gaussian Surface?

Q. A charge $Q$ is placed at the centre of a imaginary hemispherical surface. Using symmetry arguments and Gauss's law, find the flux of the electric field due to this charge through the curved surface of the hemisphere as shown in the figure.

1. $\frac{Q}{3{\epsilon }_0}$
2. $\frac{2Q}{{\epsilon }_0}$
3. $\frac{Q}{2{\epsilon }_0}$
4. $\frac{Q}{{\epsilon }_0}$

Q. The intensity of an electric field depends only on the coordinates $x$ and $y$ as follows:
$E=\frac{a(x\hat{i}+y\hat{j})}{x^2+y^2}$
where, $a$ is a constant and $\hat{i}$ and $\hat{j}$ are the unit vectors of the $x$ and $y-$ axes. Find the charge within a sphere of radius $R$ with the centre at the origin.

Q.

A spherical volume contains a uniformly distributed charge of volume density $2.0\times {10}^{-4}C{m}^{-3}$. Find the strength of electric field at a point inside the volume at a distance $4.0cm$ from the centre.

1. $3\times {10}^4\frac{N}{C}$

2. $3\times {10}^5\frac{N}{C}$

3. $3\times {10}^6\frac{N}{C}$

4. $3\times {10}^7\frac{N}{C}$

Q. A positively charged sphere of radius $r_0$ carries a volume charge density $\rho$ as shown in figure. A spherical cavity of radius $\frac{r_0}{2}$ is then scooped out and left empty. $C_1$ is the centre of the sphere and $C_2$ is the centre of the cavity. What is the direction and magnitude of the electric field at point B?

1. $\frac{\rho r_0}{6{ϵ}_0}$, towards left
2. $\frac{\rho r_0}{6{ϵ}_0}$, towards right
3. $\frac{17\rho r_0}{54{ϵ}_0}$, towards right
4. $\frac{17\rho r_0}{54{ϵ}_0}$, towards left

Q. $q_1,q_2,q_{3}and{q}_4$ are point charges located at points as shown in the figure and S is spherical Gaussian surface of radius R. Which of the following is true according to the Gauss’s law.

1. $\oint ({\overrightarrow{E}}_1+{\overrightarrow{E}}_2+{\overrightarrow{E}}_3).d\overrightarrow{A}=\frac{q_1+q_2+q_3}{2{ϵ}_0}$
2. $\oint ({\overrightarrow{E}}_1+{\overrightarrow{E}}_2+{\overrightarrow{E}}_3).d\overrightarrow{A}=\frac{q_1+q_2+q_3}{{ϵ}_0}$
3. $\oint ({\overrightarrow{E}}_1+{\overrightarrow{E}}_2+{\overrightarrow{E}}_3).d\overrightarrow{A}=\frac{q_1+q_2+q_3+q_4}{{ϵ}_0}$
4. $Noneoftheabove$

Q. Eight dipoles of charges of magnitude e are placed inside a cube. The total electric flux coming out of the cube will be

1. $\frac{8e}{{ϵ}_0}$
2. $\frac{16e}{{ϵ}_0}$
3. $\frac{e}{{ϵ}_0}$
4. $zero$

Q. Number of electric lines of force from 0.5 C of positive charge in a dielectric medium of dielectric constant 10 is

1. $5.65\times {10}^9$
2. $1.13\times {10}^{11}$
3. $9\times {10}^9$
4. $8.85\times {10}^{-12}$

Q.

A charge q is located above the centre of a square plate of side a, at a distance $\frac{a}{2}$. The flux of electric field through the plate Is?

1. $\frac{q}{{ϵ}_0}$

2. $\frac{q}{2{ϵ}_0}$

3. $\frac{q}{4{ϵ}_0}$

4. $\frac{q}{6{ϵ}_0}$

Q. The intensity of an electric field depends only on the coordinates $x$ and $y$ as follows:
$E=\frac{a(x\hat{i}+y\hat{j})}{x^2+y^2}$
where, $a$ is a constant and $\hat{i}$ and $\hat{j}$ are the unit vectors of the $x$ and $y-$ axes. Find the charge within a sphere of radius $R$ with the centre at the origin.

1. $-2\pi {ϵ}_{0}aR$
2. $\pi {ϵ}_{0}aR$
3. $4\pi {ϵ}_{0}aR$
4. $2\pi {ϵ}_{0}aR$

Q. The volume charge density as a function of distance $x$ from one face inside a unit cube is varying as shown in the figure. Then the total flux (in S.I. units) through the cube if $({\rho }_o=8.85\times {10}^{–12}\frac{C}{m^3})$ is :

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

Q.

Comment on the correctness of the following two statements.

Statement I – Net Electric field strength on a Gaussian surface may change if the charges inside or outside move.

Statement II – Electric flux through a Gaussian surface will not change if the charges contained inside do not cross the boundary of the assumed Gaussian surface.

1. True, True

2. True, false

3. False, True

4. False, False

Q.

The electric flux through the surface S as due to all the charges as shown in following figure is$\frac{q_1+q_3-q_2-q_4}{{ϵ}_0}$

1. True

2. false

Q.

Electric charges are distributed in a small volume. The flux of the electric field through a spherical surface of radius $10\text{cm}$ containing the total charge is $25\text{V-m}$. The flux through a concentric sphere of radius $20\text{cm}$ will be?

1. 25 V-m

2. 50 V-m

3. 100 V-m

4. 200 V-m

Q. The variation of electric field between two charges $q_1$ and $q_2$ along the line joining the charges is plotted against distance from $q_1$ (taking rightward direction of electric field as positive) as shown in the figure. Then, the correct statement is

1. $q_1$ and $q_2$ are positive and $q_1<q_2$
2. $q_1$ and $q_2$ are positive and $q_1>q_2$
3. $q_1$ and $q_2$ are negative and $\left|q_1\right|<\left|q_2\right|$
4. $q_1$ is positive and $q_2$ is negative ; $q_1<\left|q_2\right|$

Q. A point charge $q_1$ is placed inside cavity. If $q_1$ is at center of cavity , then $\overrightarrow{E}$ at a point which is at a distance $r$ from the center of cavity $(r>r_1)$ , due to the induced charge on the surface of cavity is

1. Zero.
2. $\frac{q_1}{4\pi {\epsilon }_0{r}^2}$ towards the center of cavity.
3. $\frac{q_1}{4\pi {\epsilon }_0{r}_1^2}$ away from the center of cavity.
4. $\frac{q_1}{4\pi {\epsilon }_0{r}^2}$ away from the center of cavity.

Q. The total flux associated with given cube will be, where ${}^{\prime }{a}^{\prime }$ is side of cube :–
$(\frac{1}{{ϵ}_o}=4\pi \times 9\times {10}^9)$

1. $162\pi \times {10}^{-3}\frac{{\text{Nm}}^2}{\text{C}}$
   
2. $162\pi \times {10}^3\frac{{\text{Nm}}^2}{\text{C}}$
   
3. $162\pi \times {10}^{-6}\frac{{\text{Nm}}^2}{\text{C}}$
   
4. $162\pi \times {10}^6\frac{{\text{Nm}}^2}{\text{C}}$
   

Q. The volume charge density as a function of distance $x$ from one face inside a unit cube is varying as shown in the figure. Then the total flux (in S.I. units) through the cube if $({\rho }_o=8.85\times {10}^{–12}\frac{C}{m^3})$ is :

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

Q.

Total electric flux coming out of a unit positive charge put in air is

1. 

2. 

3. 

4.