Electric Field Due to a Ring Along the Axis

Q. Find the electric field at point $P$ as shown in the figure on the perpendicular bisector of a uniformly charged thin wire of length $L$ carrying a charge $Q$. The distance of the point $P$ from the centre of the rod is $a=\frac{\sqrt{3}}{2}L$.

1. $\frac{Q}{4\pi {ϵ}_0{L}^2}$
2. $\frac{Q}{3\pi {ϵ}_0{L}^2}$
3. $\frac{\sqrt{3}Q}{4\pi {ϵ}_0{L}^2}$
4. $\frac{Q}{2\sqrt{3}\pi {ϵ}_0{L}^2}$

Q. What will be the magnitude of electric field at point $O$ as shown in figure? Each side of the figure is $l$ and perpendicular to each other ?

1. $\frac{1}{4\pi {ϵ}_0}\frac{q}{l^2}$
2. $\frac{1}{4\pi {ϵ}_0}\frac{q}{\left(2l^2\right)}(2\sqrt{2}-1)$
3. $\frac{q}{4\pi {ϵ}_0{\left(2l\right)}^2}$
4. $\frac{1}{4\pi {ϵ}_0}\frac{2q}{2l^2}\left(\sqrt{2}\right)$

Q. Four charges $+q,+q,-q\text{and}-q$ are placed respectively at the corners $A,B,C\text{and}D$ of a square of side $a$ arranged in the given order. $E\text{and}F$ are midpoints of side $BC$ and $CD$ respectively. $O$ is the center of the square.

$\left(i\right)$ The electric field at $O$ is

​​​​​​​$[$1 Mark$]$

1. $\frac{q}{\sqrt{2}\pi {\epsilon }_0{a}^2}$
2. $\frac{q}{\sqrt{3}\pi {\epsilon }_0{a}^2}$
3. $\frac{\sqrt{3}q}{\pi {\epsilon }_0{a}^2}$
4. $\frac{\sqrt{2}q}{\pi {\epsilon }_0{a}^2}$

Q. Two charges $4q$ and $q$ are placed $30\text{cm}$ apart. At what point, the value of electric field will be zero?

1. $10\text{cm}$ away from $q$ and between the charges.
2. $20\text{cm}$ away from $q$ and between the charges.
3. $15\text{cm}$ away from $q$ and between the charges.
4. $25\text{cm}$ away from $q$ and between the charges.

Q. A cube of side $l$ has point charges $(+q)$ at each of its vertices except at the origin where the charge is $(-q)$ as shown in the figure. The electric field at the centre of the cube is

1. $\frac{q}{3\sqrt{3}\pi {ϵ}_0{l}^2}(\hat{x}+\hat{y}+\hat{z})$
2. $\frac{2q}{3\sqrt{3}\pi {ϵ}_0{l}^2}(\hat{x}+\hat{y}+\hat{z})$
3. $\frac{-q}{3\sqrt{3}\pi {ϵ}_0{l}^2}(\hat{x}+\hat{y}+\hat{z})$
4. $\frac{-2q}{3\sqrt{3}\pi {ϵ}_0{l}^2}(\hat{x}+\hat{y}+\hat{z})$

Q. A ring of radius $R$ is charged uniformly with a charge $+Q$ . The electric field at any point on its axis at a distance $r$ from the circumference of the ring will be

1. $\frac{KQ}{r}$
2. $\frac{KQr}{R^3}$
3. $\frac{KQ}{r^2}$
4. $\frac{KQ}{r^3}(r^2-R^2{)}^{\frac{1}{2}}$

Q. Four positive charges $(2\sqrt{2}-1)Q$ are arranged at the four corners of a square. Another charge $q$ is placed at the centre of the square. Resulting field acting on each corner will be zero if $q$ is

1. $-\frac{7Q}{4}$
2. $-\frac{4Q}{7}$
3. $-Q$
4. $-(\sqrt{2}+1)Q$

Q.

Electric field vs. distance along the axis is shown for a uniformly charged circular ring

Which of the following represents the complete graph?

1. 

2. 

3. 

4. 

Q. Two point charges (+ Q) and (-2 Q) are fixed on the X-axis at positions a and 2a from origin respectively. At what positions on the axis, the resultant electric field is zero

1. Only $x=\sqrt{2}a$
2. Only $x=-\sqrt{2}a$
3. $x=\frac{3a}{2}$ only
4. Both $x=\pm \sqrt{2}a$

Q. Two charges $q_1$ and $q_2$ are kept on $x-$ axis and electric field at different points on $x-$ axis is plotted against $x$. Choose correct statement about nature and magnitude of $q_1$ and $q_2$.

1. $q_1+ve,q_2-ve;\left|q_1\right|>\left|q_2\right|$
2. $q_1+ve,q_2-ve;\left|q_1\right|<\left|q_2\right|$
3. $q_1-ve,q_2+ve;\left|q_1\right|>\left|q_2\right|$
4. $q_1-ve,q_2+ve;\left|q_1\right|<\left|q_2\right|$

Q. Three charged particles $A,B$ and $C$ with charges $-4q,2q$ and $-2q$ are present on the circumference of a circle of radius $d$. The charged particle $A,C$ and centre $O$ of the circle formed an equilateral triangle as shown in figure. Electric field at $O$ along $x-$ direction is:

1. $\frac{\sqrt{3}q}{\pi {ϵ}_0{d}^2}$
2. $\frac{2\sqrt{3}q}{\pi {ϵ}_0{d}^2}$
3. $\frac{\sqrt{3}q}{4\pi {ϵ}_0{d}^2}$
4. $\frac{3\sqrt{3}q}{4\pi {ϵ}_0{d}^2}$

Q.

Consider a uniformly charged ring of radius R. Find the point on the axis where the electric field is maximum.

Q. A charge ${10}^{-9}\text{C}$ is located at the origin of a coordinate system and another charge $Q$ at $(2,0,0)$. If the $x-$ component of the electric field at $(3,1,1)$ is zero, calculate the value of $Q$.

1. $4.3\times {10}^{-10}\text{C}$
2. $2.3\times {10}^{-10}\text{C}$
3. $-4.3\times {10}^{-10}\text{C}$
4. $-2.3\times {10}^{-10}\text{C}$

Q. For the two fixed infinite line charges, the net electric field is zero at a point $\text{P}$ between them. The distance of the point $\text{P}$ from the left infinite line charge is

1. $0.25\text{m}$
2. $1.25\text{m}$
3. $2.25\text{m}$
4. $3.25\text{m}$

Q. Five point charges $(+q$ each$)$ are placed at the five vertices of a regular hexagon of side $2a$. What is the magnitude of the net electric field at the centre of the hexagon?

1. $\frac{1}{4\pi {ϵ}_0}\frac{q}{a^2}$
2. $\frac{q}{16\pi {ϵ}_0{a}^2}$
3. $\frac{\sqrt{2}q}{4\pi {ϵ}_0{a}^2}$
4. $\frac{5q}{16\pi {ϵ}_0{q}^2}$

Q.

The distance of point P on the axis from the Centre of a uniformly charged circular ring where the electric field is maximum, is $2^{-n}R$(given radius of the ring is R). The value of n in decimal form is___

Q. A charge $q$ is placed in the middle of a line joining the two equal and like point charges $Q$. This system will remain in equilibrum for which the value of $q$ is-

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

Q. An infinite number of identical charges, each $+q\text{coulomb}$, are placed along $\text{x-}$axis at $x=1\text{m},3\text{m},9\text{m},...$ Calculate the electric field at the point $x=0$ due to these charges.

1. $\frac{1}{4\pi {\epsilon }_0}\frac{9q}{8}{\text{NC}}^{-1}$
2. $\frac{1}{4\pi {\epsilon }_0}\frac{9q}{10}{\text{NC}}^{-1}$
3. $\frac{1}{4\pi {\epsilon }_0}\frac{9q}{7}{\text{NC}}^{-1}$
4. $\frac{1}{4\pi {\epsilon }_0}\frac{9q}{5}{\text{NC}}^{-1}$

Q. Two concentric rings, one of radius ‘a’ and the other of radius ‘b’, have the charges +q, and $-{\left(\frac{2}{5}\right)}^{\frac{-3}{2}}$ respectively, as shown in the figure. Find the ratio $\frac{b}{a}$ if a charge particle placed on the axis at z = a is in equilibrium.

Q.

P is a point on the axis of a ring with Centre O and radius 0.1m. The ring is uniformly charged with linear charge density 0.5 c/m. Given OP = 0.1 m, the electric field at P is approximately ten to the power

___in SI units.

Q. ABC is a right-angle triangle with right angle $A$ and $AB=AC=30\text{cm}$ if the charge on $B$ and $C$ is $4\times {10}^{-3}\mu \text{C}$ then the electric field at $A$ will be

1. $400\text{N/C}$ parallel to BC
2. $800\text{N/C}$ parallel to BC
3. $400\sqrt{2}\text{N/C}$ perpendicular to BC
4. $800\text{N/C}$ perpendicular to BC
   

Q. The magnitude of electric field $E$ in the annular region of a charged cylindrical capacitor

1. varies as $\frac{1}{r}$, where $r$ is the distance from the axis.
2. varies as $\frac{1}{r^2}$, where $r$ is the distance from the axis.
3. is the same throughout.
4. is higher near the outer cylinder than near to it.

Q.
Uniformly charged ring is shown is figure, $E$ due to ring is maximum at

1. Centre
2. $\infty$
3. $R$ distance from centre
4. $\frac{R}{\sqrt{2}}$ distance from centre

Q. Charges $-q$ and $+q$ located at $\text{A}$ and $\text{B}$ respectively and $\text{OP}$ is perpendicular to line $\text{AB}$. A charge $Q$ is placed at $\text{P}$, where $\text{OP}=y$ $(y>>2a)$. The charge $Q$ experiences an electrostatics force $F$. If $Q$ is now moved along the equatorial line to $P^{\prime }$ such that $O{P}^{\prime }=\frac{y}{3}$, the force experienced by charge $Q$ will be close to
$(\text{Assume}\frac{y}{2}>>2a)$

1. $F/3$
2. $F/27$
3. $9F$
4. $27F$

Q. Consider two point charges, $\frac{q}{3}$ and $-\frac{2q}{3}$ at point $\text{B}$ and $\text{C}$ respectively as shown in the figure. Take $O$ to be the centre of the circle of radius $R$ and $\angle CAB={60}^{\circ }$. The magnitude of force between the charges is

1. $\frac{q^2}{54\pi {ϵ}_0{R}^2}$
2. $\frac{q^2}{27\pi {ϵ}_0{R}^2}$
3. $\frac{q^2}{25\pi {ϵ}_0{R}^2}$
4. $\frac{q^2}{60\pi {ϵ}_0{R}^2}$

Q. A conducting sphere of radius $R$ having charge $q$ is joined to another conducting sphere of radius $2R$ having charge $-2q$. The charge flowing between them will be

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

Q. For the given uniformly charged ring, which of the following curve of electric field $\left(E\right)$ versus distance from the centre of ring $\left(x\right)$ along $x-$ axis is correct?
[Consider only right side of ring]

1. 
2. 
3. 
4. 

Q. Which of the following graphs correctly represent the variation of electric field $E$ vs $r$ for a ring, where $r$ represents the distance from the centre of ring along its axis.

1. 
2. 
3. 
4. 

Q.

Positive charge Q is distributed uniformly over a ircular ring of radius R. A particle having having a mass ma and a negative charge q, is placed on its axis at a distanc ex from the centre. Find the force on the particle. Assuming x<< R, Find the time period of oscillation of the particle if it is released from there.

Q.

P is a point on the axis of a ring with Centre 0 and radius 0.1m. The ring is uniformly charged with linear charge density 0.5 c/m. Given OP = 0.1 m, the electric field at P is approximately ten to the power ____ in SI units.

___