Electric Field Due to an Arc at the Centre

Q.

A rod of length L has a total charge Q distributed uniformly along its length. It is bent in th shape of a semicircle. Find the magnitude of the electric field at the centre of curvatere of he semicircle.

Q. Figure shows a rod $AB$, which is bent in a ${120}^{\circ }$ circular arc of radius $R$. A charge $(-Q)$ is uniformly distributed over rod $AB$. What is the electric field $\overrightarrow{E}$ at the centre of curvature $O$ ?

1. $\frac{3\sqrt{3}Q}{16{\pi }^2{\epsilon }_0{R}^2}\left(\hat{i}\right)$
2. $\frac{3\sqrt{3}Q}{8{\pi }^2{\epsilon }_0{R}^2}\left(\hat{i}\right)$
3. $\frac{3\sqrt{3}Q}{8\pi {\epsilon }_0{R}^2}\left(\hat{i}\right)$
4. $\frac{3\sqrt{3}Q}{8{\pi }^2{\epsilon }_0{R}^2}(-\hat{i})$

Q. The linear charge density of a uniform semicircular wire varies with $\theta$ as shown in the figure as $\lambda ={\lambda }_0\cos \theta$. Find the charge on the wire if $R$ is the radius of the ring.

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

Q. For the given charge distribution on the ring, the net electric field at the centre of non-conducting ring is
(Assume the part of ring in first and third quadrant is neutral, second quadrant is positively charged, fourth quadrant is negatively charged)

1. $180\sqrt{2}\text{N/C}$
2. $180\sqrt{3}\text{N/C}$
3. $180\sqrt{5}\text{N/C}$
4. $180\sqrt{7}\text{N/C}$

Q.

If a long hollow copper pipe carries a direct current, the magnetic field associated with the current will be

Q. A thin Non-conducting rod is bent into a semicircle of radius $r$. A charge $+Q$ is uniformly distributed along the upper half and a charge $-Q$ is uniformly distributed along the lower half, as shown in figure. Find the electric field $E$ at $\text{P}$.

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

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

1. $4.2\times {10}^7\text{N/C}$
2. $5.2\times {10}^7\text{N/C}$
3. $6.2\times {10}^7\text{N/C}$
4. $7.2\times {10}^7\text{N/C}$

Q. To get the formula of electric field due to a ring at the centre from the formula of electric field due to an arc at centre, replace $\theta$ with .

1. ${90}^{\circ }$
2. ${180}^{\circ }$
3. ${45}^{\circ }$
4. ${360}^{\circ }$

Q.

A magnetic wire of dipole moment $4\pi {\mathrm{Am}}^2$ is bent in the form of a semi-circle. What is the new magnetic moment is?

1. ${\mathrm{\pi Am}}^2$

2. $2{\mathrm{\pi Am}}^2$

3. $8A{m}^2$

4. $4{\mathrm{\pi Am}}^2$

Q. A thin glass rod is bent into a semicircle of radius r. A charge +Q is uniformly distributed along the upper half and a charge –Q is uniformly distributed along the lower half, as shown in the figure. The electric field E at, P (the centre of the semicircle, is)

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

Q. A charge $q$ is uniformly distributed on a spherical shell of radius $R$. The electric field at a distance $\frac{4R}{3}$ from the centre will be

1. $\frac{9q}{64\pi {\epsilon }_0{R}^2}$
2. $\frac{q}{4\pi {\epsilon }_0{R}^3}$
3. $\frac{3q}{16\pi {\epsilon }_0{R}^2}$
4. Zero

Q. If $O$ is the center of a ring of radius $r$, then find the potential at point $O$ due to half ring that has a linear charge density $\lambda$.

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

Q. The linear charge density on a ring which varies with angle $\theta$ can be represented as $\lambda =Kcos\frac{\theta }{2},where$k = 2 cm^{-1}$and$\theta$ is the angle subtended by the radius of the ring with the horizontal. The potential at the centre of the ring is

Q. Charge q is uniformly distributed over a thin half ring of radius R. The electric field at the centre of the ring is

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

Q. For the given semi-infinite rod of uniformly distributed line charge, angle $\left(\theta \right)$ between net electric field and component of net electric field perpendicular to the axis of line charge at the point $\text{P}$ is

1. ${30}^{\circ }$
2. ${45}^{\circ }$
3. ${60}^{\circ }$
4. ${15}^{\circ }$

Q. A large plane charged sheet having surface charge density $\sigma =+2\times {10}^{-6}{\text{C/m}}^2$ lies in the $\text{x-y}$ plane. Find the flux of the electric field through a circular area of radius $1\text{cm}$ lying completely in the region where $\text{x, y, z}$ all are positive and with its normal making an angle of ${60}^{\circ }$ with the $\text{z-}$axis.

1. $16.3{\text{Nm}}^2\text{/C}$
2. $17.3{\text{Nm}}^2\text{/C}$
3. $8.15{\text{Nm}}^2\text{/C}$
4. $34.6{\text{Nm}}^2\text{/C}$

Q. For the given uniformly charged ring sector symmetrically placed about $Y$ axis, net electric filed at the point $\text{P}$ along $X$-direction is
$[l$ is the length of the arc, $\lambda$ is linear charge density, $R$ is radius of sector$]$

1. $\frac{2k\lambda }{R}\sin \left(\frac{l}{2R}\right)$
2. $\frac{2k\lambda }{R}\sin \left(\frac{l}{R}\right)$
3. $\frac{2k\lambda }{R}$
4. Zero

Q. For the given charge distribution, the net electric field at the centre of the non-conducting ring is

1. $180\text{N/C}$
2. $220\text{N/C}$
3. $320\text{N/C}$
4. $360\text{N/C}$

Q. For the given uniformly charged ring of linear charge density $+10\text{nC/m}$, the electric field in $y-$ direction at point $\text{P}$ is

1. $100\text{N/C}$
2. $200\text{N/C}$
3. Zero
4. $150\text{N/C}$

Q. Two mutually perpendicular infinite wires carry positive charge densities ${\lambda }_1$ and ${\lambda }_2$. The electric lines of force makes angle $\alpha$ with second wire then $\frac{{\lambda }_1}{{\lambda }_2}$ is

1. ${\tan }^2\alpha$
2. ${\cot }^2\alpha$
3. ${\sin }^2\alpha$
4. ${\cos }^2\alpha$

Q. An insulating rod of length $l$ carries a charge $q$ uniformly distributed on it. The rod is pivoted at one of its ends, and it is rotated at a frequency $f$ about a fixed perpendicular axis in the horizontal plane. The magnetic moment of the rod is

1. $\frac{\pi qf{l}^2}{12}$
   
2. $\frac{\pi qf{l}^2}{2}$
   
3. $\frac{\pi qf{l}^2}{6}$
   
4. $\frac{\pi qf{l}^2}{3}$
   

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

1. $4.2\times {10}^7\text{N/C}$
2. $5.2\times {10}^7\text{N/C}$
3. $6.2\times {10}^7\text{N/C}$
4. $7.2\times {10}^7\text{N/C}$

Q.

A circular arc has a radius of 0.1m. The angle it subtends at the Centreis $\frac{\pi }{6}$ . If the charge it carries is 10C. what is the linear charge density of the wire?

1. 192.3 c/m

2. 215.2 c/m

3. 50 c/m

4. 500 c/m

Q. A large charged conducting sheet is placed in a uniform electric field, perpendicularly to the electric field lines. After placing the sheet into the field, the electric field on the left side of the sheet is $E_1=5\times {10}^5\text{V/m}$ and on the right it is $E_2=3\times {10}^5\text{V/m}$. The sheet experiences a net electric force of $0.08\text{N}$. Find the area of one face of the sheet.$($ Assume the external field to remain constant after introducing the large sheet$)$

$[k=\frac{1}{4\pi {\epsilon }_0}=9\times {10}^9\text{N}{\text{m}}^2{\text{C}}^{-2}]$

1. $3.6\pi \times {10}^{-2}{\text{m}}^2$
2. $0.9\pi \times {10}^{-2}{\text{m}}^2$
3. $1.8\pi \times {10}^{-2}{\text{m}}^2$
4. None

Q. Find the magnitude of the electric field $\left(|\overrightarrow{E_{ind}}|\right)$ at the center of the sphere due to the induced charges on the sphere.

1. $\frac{Kq}{2}$
2. $\frac{Kq}{8}$
3. $\frac{Kq}{5}$
4. $\frac{Kq}{4}$

Q. Charge Q is distributed uniformly on length $l$ of a wire. It is bent in the form of a semicircular ring. Find the electric field at the centre of the ring.

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

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

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

Q. For the given charge distribution on the ring, the net electric field at the centre of non-conducting ring is
(Assume the part of ring in first and third quadrant is neutral, second quadrant is positively charged, fourth quadrant is negatively charged)

1. $180\sqrt{2}\text{N/C}$
2. $180\sqrt{3}\text{N/C}$
3. $180\sqrt{5}\text{N/C}$
4. $180\sqrt{7}\text{N/C}$

Q. A long straight conductor, carrying a current $i$, is bent to form an almost complete circular loop of radius $r$ as shown. The magnetic field at the centre of the loop :

1. has magnitude $\frac{{\mu }_{0}i}{r}(1-\frac{1}{\pi })$
2. has magnitude $\frac{{\mu }_{0}i}{r}(1+\frac{1}{\pi })$
3. has magnitude $\frac{{\mu }_{0}i}{2r}(1-\frac{1}{\pi })$
4. has magnitude $\frac{{\mu }_{0}i}{2r}(1+\frac{1}{\pi })$

Q. A thin glass rod is bent into a semicircular shape of radius R. A charge $+Q$ is uniformly distributed along the upper half and a charge $-Q$ is distributed uniformly along the lower half as shown. The electric field at the centre P is:

1. $\frac{\sqrt{2}Q}{4\pi {\epsilon }_0{R}^2}$
2. $\frac{Q}{\sqrt{2}\pi {\epsilon }_0{R}^2}$
3. $\frac{Q}{2\pi {\epsilon }_0{R}^2}$
4. $\frac{Q}{{\pi }^2{\epsilon }_0{R}^2}$