Electric Potential Due to Infinite Line of Charge

Q. An infinitely long thin straight wire has uniform linear charge density of $1/3{\text{Cm}}^{-1}$. Then the magnitude of the electric field intensity at a point $18\text{cm}$ away is $x\times {10}^{11}{\text{NC}}^{-1}$. The value of $x$ is _____.

Q. A point charge $q$ held at the centre of a circle of radius $r$. Points $\text{B}$ and $\text{C}$ are two fixed points on the circumference of the circle and $\text{A}$ is a point outside the circle as shown. If $W_{AB}$ represents the work done by electric field in taking a charge $q_0$ from $\text{A}$ to $\text{B}$ and $W_{AC}$ represents the work done from $\text{A}$ to $\text{C}$, then

1. $W_{AB}>W_{AC}$
2. $W_{AB}<W_{AC}$
3. $W_{AB}=W_{AC}\ne 0$
4. $W_{AB}=W_{AC}=0$

Q. An electric dipole has the magnitude of its charge as $q$ and its dipole moment is $p$. It is placed in a uniform electric field $E$. If its dipole moment is along the direction of the field, the net force acting on it and its potential energy are respectively

1. $2qE$ and minimum
2. $qE$ and $pE$
3. Zero and minimum
4. $qE$ and maximum

Q. A thin insulating rod of mass $m$ and length $L$ is hinged at its upper end $\left(O\right)$ so that it can freely rotate in vertical plane. The linear charge density on the rod varies with distance $\left(y\right)$ measured from upper end as
$$\lambda =\{\begin{array}{cc}a{y}^2& \text{for}0\le y\le \frac{L}{2}\\ -b{y}^n& \text{for}\frac{L}{2}\le y\le L\end{array}$$ Where $a$ and $b$ both are positive constants. when a horizontal electric field $E$ is switched on the rod is found to remain stationary. Find the value of $\frac{a}{b}$. (Give integer value)

Q. A positive point charge is released from rest, at a distance $r_0$ from a positive line charge with uniform density. The speed $v$ of the point charge, as a function of instantaneous distance $r$ from the line charge, is proportional to,

1. $v\propto \left(\frac{r}{r_0}\right)$
2. $v\propto e^{+r/r_0}$
3. $v\propto \sqrt{\ln \left(\frac{r}{r_0}\right)}$
4. $v\propto \ln \left(\frac{r}{r_0}\right)$

Q. The figure shows a non-conducting ring with positive and negative charge non uniformly distributed on it such that the total charge is zero. Which of the following statements is true?

1. The potential at all the points on the axis will be zero.
2. The electric field at all the points on the axis will be zero.
3. The direction of the electric field at all points on the axis will be along the axis
4. If the ring is placed inside a uniform external electric field, then net torque and force acting on the ring would be zero.

Q. The linear charge density on a dielectric ring of radius R varies with $\theta$ as $\lambda ={\lambda }_0\cos \frac{\theta }{2},$ where ${\lambda }_0$ is a constant. Find the potential at the centre of the ring. [in volt]

Q.
The ratio of the magnitude of electric field due to +q and – 2q at a point where net potential is zero on positive x-axis is z : 1, then z will be

Q. A charge of $10\mu C$ is distributed uniformly over the circumference of a ring of radius $3\text{m}$ placed on the $x-y$ plane with the axis at origin. Find electric potential at point $P(0,0,4\text{m})$

1. $18\times {10}^3\text{V}$
2. $18\times {10}^4\text{V}$
3. $1.8\times {10}^2\text{V}$
4. $1.8\times {10}^6\text{V}$

Q. What mass of a point charge kept above finite line charge at axial line can be balanced by the electric field produced by finite line charge.

1. $0.6\times {10}^{-7}\text{kg}$
2. $1.6\times {10}^{-7}\text{kg}$
3. $2.6\times {10}^{-7}\text{kg}$
4. $3.6\times {10}^{-7}\text{kg}$

Q. A non-conducting ring of radius $0.5\text{m}$ have charge $1.11\times {10}^{-10}\text{C}$ non-uniformly distributed over the circumference of the ring, which produces an electric field $E$ around itself. If $l=0$ is the center of the ring, then the value of $\underset{l=\infty }{\overset{l=0}{\int }}-\overrightarrow{E}\cdot \overrightarrow{dl}$ is

1. $-1\text{V}$
2. $2\text{V}$
3. $-2\text{V}$
4. Zero

Q. Four charges each of magnitude $q$ are placed at four corners of a square of side $a$. The work done in carrying a charge $-q$ from its centre to infinity will be

1. zero
2. $\frac{2\sqrt{2}{q}^2}{\pi {ϵ}_{0}a}$
   
3. $\frac{\sqrt{2}{q}^2}{\pi {ϵ}_{0}a}$
4. $\frac{q^2}{2\pi {ϵ}_{0}a}$
   

Q. Find the work done by external agent in moving a $1C$ charge from point P to Q in presence of an infinitely long uniformly charged line of linear charge density $\lambda$ as shown.

1. $\frac{\lambda }{2\pi {\epsilon }_0}\ln 1.5$
2. $\frac{\lambda }{2\pi {\epsilon }_0}$
3. $\frac{\lambda }{2\pi {\epsilon }_0}\ln 5$
4. $\frac{7\lambda }{2\pi {\epsilon }_0}$

Q. The following diagram represents a semi circular wire of linear charge density $\lambda ={\lambda }_0\sin \theta$, where ${\lambda }_0$ is a positive constant. The electric potential at $\text{O}$ is $(\text{take}k=\frac{1}{4\pi {ϵ}_0})$

1. $k{\lambda }_0\sin \theta$
2. $\frac{k{\lambda }_0\cos \theta }{2}$
3. Zero
4. $k{\lambda }_0\cos \theta$

Q. Three charges $2q$, $-q$ and $-q$ are located at the vertices of an equilateral triangle. At the center of the triangle,

1. the field is zero but potential is non-zero
2. the field is non-zero but potential is zero
3. both field and potential are zero
4. both field and potential are non-zero

Q. Ten charges are placed on the circumference of a circle of radius $R$ with constant angular separation between successive charges. Alternate charges $1,3,5,7,9$ have charge $(+q)$ each, while $2,4,6,8,10$ have charge $(-q)$ each. The potential $V$ and the electric field $E$ at the centre of the circle are respectively :
(Take $V=0$ at infinity)

1. $V=\frac{10q}{4\pi {\epsilon }_{0}R};E=0$
2. $V=0;E=\frac{10q}{4\pi {\epsilon }_0{R}^2}$
3. $V=0;E=0$
4. $V=\frac{10q}{4\pi {\epsilon }_{0}R};E=\frac{10q}{4\pi {\epsilon }_0{R}^2}$

Q. Two charges $q_1$ and $q_2$ are placed $30\text{cm}$ apart as shown in the figure. A third charge $q_3$ is moved along the arc of a circle of radius $40\text{cm}$ from $C$ to $D$. The change in potential energy of the system is $\frac{x{q}_3}{4\pi {ϵ}_0}$ where $x$ is

1. $8q_2$
   
2. $6q_2$
3. $8q_1$
4. $6q_1$

Q. The electric potential at a point $(x,y,z)$ is given by $V=-x^{2}y-x{z}^3+4$. The electric field $\overrightarrow{E}$ at that point is

1. $\overrightarrow{E}=\hat{i}(2xy-z^3)+x{y}^2\hat{j}+3z^{2}x\hat{k}$
2. $\overrightarrow{E}=\hat{i}(2xy+z^3)+x^2\hat{j}+3x{z}^2\hat{k}$
3. $\overrightarrow{E}=\hat{i}2xy+\hat{j}(x^2+y^2)+\hat{k}3(xz-y^2)$
4. $\overrightarrow{E}=\hat{i}{z}^2+\hat{j}xyz+\hat{k}\left(z^2\right)$

Q. Find out the magnitude of electric field intensity, $E$ at point $(2,0,0)$ due to a dipole moment, $\overrightarrow{p}=\hat{i}+\sqrt{3}\hat{j}$ kept at origin as shown in the figure and also find the potential, $V$ at that point.

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

1. $E=\frac{k}{4},V=\frac{k}{8}$
2. $E=\frac{\sqrt{7}k}{8},V=\frac{k}{4}$
3. $E=\frac{8k}{3},V=\frac{5k}{2}$
4. $E=\frac{k}{3},V=\frac{3k}{2}$

Q. Determine the electric field strength vector if the potential of this field depends on $x,y$ coordinates as $V=10axy$.

1. $10a(y\hat{i}+x\hat{j})$
2. $-10a(y\hat{i}+x\hat{j})$
3. $-a(y\hat{i}+x\hat{j})$
4. $-10a(x\hat{j}+y\hat{k})$

Q. A rod of uniform charge density $\lambda$ and length L is placed vertically as shown. A point charge $Q_0$ is placed at point O at a distance L from its one end. Potential energy of charge $Q_0$ is equal to

1. $\frac{\lambda Q_{0}ln2}{4\pi {ϵ}_0}$
2. $\frac{\lambda Q_0}{4\pi {ϵ}_0}$
3. $\frac{\lambda Q_0}{2\pi {ϵ}_0}$
4. $\frac{\lambda Q_{0}ln2}{2\pi {ϵ}_0}$

Q. In a region, the potential is represented by $V(x,y,z)=6x-8xy-8y+6yz$, where $V$ is in volts and $x,y,z$ are in meters. The electric force experienced by a charge of $2$
coulomb situated at point $(1,1,1)$ is

1. $6\sqrt{5}\text{N}$
2. $30\text{N}$
3. $24\text{N}$
4. $4\sqrt{35}\text{N}$

Q. A non-conducting ring of radius $0.5\text{m}$ have charge $1.11\times {10}^{-10}\text{C}$ non-uniformly distributed over the circumference of the ring, which produces an electric field $E$ around itself. If $l=0$ is the center of the ring, then the value of $\underset{l=\infty }{\overset{l=0}{\int }}-\overrightarrow{E}\cdot \overrightarrow{dl}$ is

1. $2\text{V}$
2. $-2\text{V}$
3. $-1\text{V}$
4. Zero

Q. Determine the electric field strength vector if the potential of this field depends on $x,y$ coordinates as $V=10axy$.

1. $-10a(y\hat{i}+x\hat{j})$
2. $10a(y\hat{i}+x\hat{j})$
3. $-10a(x\hat{j}+y\hat{k})$
4. $-a(y\hat{i}+x\hat{j})$

Q. The electric potential at any point $(x,y,z)$ is given as $V=\left(2y^3\right)\text{volt}$ . The electric field at point $(2,1,0)$ is
(($x,y,z)$ are in metre)

1. $6\hat{j}{\text{Vm}}^{-1}$
2. $-6\hat{j}{\text{Vm}}^{-1}$
3. $2\hat{i}{\text{Vm}}^{-1}$
4. $-2\hat{i}{\text{Vm}}^{-1}$

Q. The electric potential $V$ at any point (x, y, z) all in meters in space is given by $V=4x^2$ volt. The electric field at the point (1, 0, 2) in volt/meter is:

1. 8 along negative X-axis
2. 8 along positive X-axis
3. 16 along negative X-axis
4. 16 along positive X-axis

Q. Determine the electric field strength vector if the potential of this field depends on $x,y$ coordinates as $V=10axy$.

1. $10a(y\hat{i}+x\hat{j})$
2. $-10a(y\hat{i}+x\hat{j})$
3. $-a(y\hat{i}+x\hat{j})$
4. $-10a(x\hat{j}+y\hat{k})$

Q. Two insulating plates, both uniformly charged in such a way that the potential difference between them is $V_2-V_1=20\text{V}$. The plates are separated by $d=0.1\text{m}$ and can be treated as infinitely large. An electron is released from rest on the inner surface of plate $1$. What is its speed when it hits the plate $2$?

$[e=1.6\times {10}^{-19}\text{C}$ and $m_e=9\times {10}^{-31}\text{kg}]$

1. $2.7\times {10}^6\text{m}/\text{s}$
2. $1.8\times {10}^6\text{m}/\text{s}$
3. $1.6\times {10}^6\text{m}/\text{s}$
4. $1.2\times {10}^6\text{m}/\text{s}$

Q. Find the work done by external agent in moving a $1C$ charge from point P to Q in presence of an infinitely long uniformly charged line of linear charge density $\lambda$ as shown.

1. $\frac{\lambda }{2\pi {\epsilon }_0}\ln 1.5$
2. $\frac{\lambda }{2\pi {\epsilon }_0}$
3. $\frac{\lambda }{2\pi {\epsilon }_0}\ln 5$
4. $\frac{7\lambda }{2\pi {\epsilon }_0}$

Q. An electron is projected at an angle $\theta$ with a uniform magnetic field. If the pitch of the helical path is equal to its radius, then the angle of projection is

1. $ta{n}^{-1}\pi$
2. $ta{n}^{-1}2\pi$
3. $co{t}^{-1}2\pi$
4. $co{t}^{-1}\pi$