Electric Field Due to a Line of Charge Not on Its Axis

Q. The electric field in a region is given by $\overrightarrow{E}=\frac{E_{0}x}{l}\hat{i}$. Find the charge contained inside a cubical volume bounded by the surfaces $x=0,x=a,y=0,y=a,$$z=0$ and $z=a$.
Take
$E_0=5\times {10}^3\text{N/C},l=2\text{cm,}$
$a=1\text{cm},{\epsilon }_0=8.86\times {10}^{-12}{\text{C}}^2/{\text{Nm}}^2$

1. $1.1\times {10}^{-12}\text{C}$
2. $2.2\times {10}^{-12}\text{C}$
3. $0.6\times {10}^{-14}\text{C}$
4. None of the above

Q.

An infinitely long uniform line charge distribution of charge per unit length $\lambda$ lies parallel to the y-axis in the y-z plane at $z=\frac{\sqrt{3}}{2}a$. If the magnitude of the flux of the electric field through the rectangular surface ABCD lying in the x-y plane with its centre at the origin is $\frac{\lambda L}{n{\epsilon }_0}$ (${\epsilon }_0=$ permittivity of free space), then the value of $n$ is

Q. Find the net electric field due to a finite wire of linear charge density $+\lambda \text{C/m}$ at a point $\text{P}$ located at a perpendicular distance $d$ as shown in the figure.

$[K=\frac{1}{4\pi {\epsilon }_0}]$

1. $\sqrt{2}\frac{K\lambda }{d}$
2. $2\sqrt{2}\frac{K\lambda }{d}$
3. $\frac{K\lambda }{\sqrt{2}d}$
4. $\frac{1}{2\sqrt{2}}\frac{K\lambda }{d}$

Q. The electric field in a region of space is given by $\overrightarrow{E}=5\hat{i}+2\hat{j}{\text{NC}}^{-1}$. The electric flux due to this field through an area $2{\text{m}}^2$ lying in the $yz$- plane in S.I units is

1. $10\sqrt{2}$
2. $10$
3. $20\sqrt{2}$
4. $20$

Q.

A long cylindrical wire carries a positive charge of linear density $2\times {10}^{-8}C{m}^{-1}$ . An electron revolves around it in a circular path under the influence of the attractive electrostatic force. Find the kinetic energy of the electron . Note that it is independent of the radius.

Q. If the constant electric field in the figure has a magnitude $E_0$, calculate the total electric flux through the paraboloidal surface.

1. $E_0\pi R^2$
2. $2E_0\pi R^2$
3. $2E_0\pi R$
4. $\frac{E_0}{2\pi R}$

Q. The electric field in a region is given by $\overrightarrow{E}=\alpha x\hat{i}$. Here, $\alpha$ is a positive constant of proper dimension. Find the total flux passing through a cube bounded by the coordinates, $x=l,x=2l,y=0,y=l,z=0,z=l.$

1. $\alpha l^3$
2. $2\alpha l^3$
3. $3\alpha l^3$
4. $4\alpha l^3$

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

Q. A square frame of edge $5\text{cm}$ is placed with its positive normal making an angle of ${60}^{\circ }$ with a uniform electric field of $200\text{N/C}$. Find the electric flux through the surface bounded by the frame.

1. $0.15{\text{N-m}}^2/\text{C}$
2. $0.25{\text{N-m}}^2/\text{C}$
3. $0.35{\text{N-m}}^2/\text{C}$
4. $0.45{\text{N-m}}^2/\text{C}$

Q.

Find the magnitude of the electric field at a point 4 cm away from a line charge od density $2\times 10-11C{m}^{-1}$

Q. As shown in the figure, a point charge $q_1=+1\times {10}^{-6}C$ is placed at the origin in x-y plane and another point charge $q_2=+3\times {10}^{-6}$ is placed at the co-ordinate (10, 0). In that case, which of the following graph(s) shows most correctly the eletric field vector in $E_x$ in x-direction?

1. 
2. 
3. 
4. 

Q. Four charges are arranged at the corners of a square $ABCD$, as shown in the adjoining figure. The force on the charge kept at the centre $O$ is

1. zero
2. along the diagonal $BD$
3. perpendicular to side $AB$
4. along the diagonal $AC$

Q. A charge $q$ is placed at point $\text{O}$ in a cavity in a spherical uncharged conductor. Point S is outside the conductor. If $q$ is displaced from $\text{O}$ towards $\text{S}$ such that it is still inside the cavity, then electric field at $\text{S}$ will

1. increase
2. decrease
3. first increase and then decrease
4. remains unchanged

Q. The figure shows two charges of equal magnitude $q$ separated by a distance $2a$. As we move away from the charge situated at $x=0$ towards the charge situated at $x=2a$, which of the following graphs shows the correct behaviour of electric field?

1. 
2. 
3. 
4. 

Q. Consider Gauss's law $\oint \overrightarrow{E}.\overrightarrow{ds}=\frac{q}{{\in }_0}$

Then, for the situation shown in figure at the Gaussian surface

1. $\overrightarrow{E}$ due to $q_2$ would be zero
2. $\overrightarrow{E}$ due to both $q_1$ and $q_2$ would be non-zero
3. $\varphi$ due to both $q_1$ and $q_2$ would be non-zero
4. $\varphi$ due to $q_2$ would be zero

Q.

A 10 -cm long rod carries a charge of $+50\mu C$ distributed uniformly along its length. Find the magnitude of the electric field at a point 10 cm from from both the ends of the rod.

Q. The magnetic field at all points within the cylindrical region whose cross-section is indicated in the accompanying figure starts increasing at a constant rate $\beta$. The induced electric field $E$ satisfies


$E\propto r^p(r<R)$ and $E\propto r^{-q}(r>R)$
where, $r$ is the distance from the axis of the region. Find the value of $p+q$.(Integer only)

Q.

A uniformly charged rod (with total charge Q) and length l applies a force on other charge Q¹placed on its axis (i.e. along the rod) at a distance 1000l from its centre can be approximated as $\frac{KQ{Q}^1}{(1000l{)}^2}$ The reason it came out to be so is because

1. For such large distances, the rod can be considered as a point charge

2. charges are conserved

3. charge is quantized

4. charges are invariant in nature

Q. A long cylindrical wire have linear density $+2\times {10}^{-8}\text{C/m}$. An electron revolves around it in a circular path under the influence of the attractive electrostatic force. Find the kinetic energy of the electron.

1. $1\times {10}^{-17}\text{J}$
2. $2\times {10}^{-17}\text{J}$
3. $3\times {10}^{-17}\text{J}$
4. $4\times {10}^{-17}\text{J}$

Q. The electric field $10cm$ from a long wire is $2.4kN/C$. If wire carries uniform charge, what will be the field strength at $40cm$ from the wire?

1. $150N/C$
2. $75N/C$
3. $0.6kN/C$
4. $4.8kN/C$

Q. A uniformly charged semicircular arc of radius $R$ has a linear charge density $\lambda$ . The electric field at its centre is :

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

Q. Curent $i$ is flowing through a wire of non-uniform cross-section as shown. Match the following two columns.


Column-1 Column-2

$\text{(a)}$ Current density	$\text{(p)}$ is more at $1$
$\text{(b)}$ Electric field	$\text{(q)}$ is more at $2$
$\text{(c)}$ Resistance per unit length	$\text{(r)}$ is same at both sections $1$ and $2$
$\text{(d)}$ Potential difference per unit length	$\text{(s)}$ Data insufficient

1. $\text{a}\to \text{p};\text{b}\to \text{q};\text{c}\to \text{r};\text{d}\to \text{s}$
2. $\text{a}\to \text{p};\text{b}\to \text{p};\text{c}\to \text{q};\text{d}\to \text{q}$
3. $\text{a}\to \text{p};\text{b}\to \text{q};\text{c}\to \text{r};\text{d}\to \text{r}$
4. $\text{a}\to \text{p};\text{b}\to \text{p};\text{c}\to \text{p};\text{d}\to \text{p}$

Q. Two charges of 6 $\mu C$ and 24 $\mu C$ are placed 12 cm apart. The intensity of the electric field is zero, at a point

1. 4 cm from the smaller charge and the point inside
2. 4 cm from the smaller charge and the point outside
3. 4 cm from the bigger charge towards the smaller charge
4. 12 cm from both charges

Q. Net electric field at point P due to straight long wire having linear charge density λ is

रेखीय आवेश घनत्व λ वाले सीधे लम्बे तार के कारण बिंदु P पर नेट विद्युत क्षेत्र है

1. $\frac{\lambda }{\pi {\epsilon }_{0}r}$
2. $\frac{\lambda }{2\pi {\epsilon }_{0}r}$
3. Zero
   
   शून्य
4. $\frac{\lambda }{4\pi {\epsilon }_{0}r}$

Q.

A charge Q is distributed uniformly along a rod of lengths L. The electric field at a distance ‘d’ along its axis from its closer end will be

1. 

2. 

3. 

4. None of these

Q. Obtain the formula for the electric field due to a long thin wire ofuniform linear charge density E without using Gauss’s law. [Hint:Use Coulomb’s law directly and evaluate the necessary integral.]

Q. In a region of space, there is an electric field of ${10}^3{\text{NC}}^{-1}$ towards west. A body having deficient of ${10}^6$ electrons is lying in this region. The magnitude and direction of force acting on the body will be

1. $1.6\times {10}^{-13}\text{N}\text{(towards west)}$
2. $1.6\times {10}^{-13}\text{N}\text{(towards west)}$
3. $1.6\times {10}^{-10}\text{N}\text{(towards east)}$
4. $1.6\times {10}^{-10}\text{N}\text{(towards west)}$

Q. Find the net electric field due to a finite wire of linear charge density $+\lambda \text{C/m}$ at a point $\text{P}$ located at a perpendicular distance $d$ as shown in the figure.

$[K=\frac{1}{4\pi {\epsilon }_0}]$

1. $\sqrt{2}\frac{K\lambda }{d}$
2. $2\sqrt{2}\frac{K\lambda }{d}$
3. $\frac{K\lambda }{\sqrt{2}d}$
4. $\frac{1}{2\sqrt{2}}\frac{K\lambda }{d}$

Q. Two charges $+q$ and $-q$ are placed at a distance $b$ apart as shown in the figure. The electric field at a point $P$ on the perpendicular bisector as shown is :

1. along vector $\underset{A}{\to }$
2. along vector $\underset{B}{\to }$
3. along vector $\underset{C}{\to }$
4. zero

Q. An electron is rotating around an infinite positive linear charge in a circle of radius 0.1 m, if the linear charge density is $1\mu C/m,$ then the velocity of electron in m/s will be

1. $0.562\times {10}^7$
2. $5.62\times {10}^7$
3. $562\times {10}^7$
4. $0.0562\times {10}^7$