A current loop consists of two identical semicircular parts each of radius R, one lying in the x-y plane, and the other in the x-z plane. If the current in the loop is i. The resultant magnetic field due to the two semicircular parts at their common centre is:

1. $\frac{{\mu }_{\mathit{0}}i}{\mathit{2}\sqrt{\mathit{2}}R}$                                        2. $\frac{{\mu }_{\mathit{0}}i}{\mathit{2}R}$

3.  $\frac{{\mu }_{\mathit{0}}i}{\mathit{4}R}$                                            4. $\frac{{\mu }_{\mathit{0}}i}{\sqrt{\mathit{2}}R}$

A closely wound solenoid of 2000 turns and area of cross-section $1.5\times {10}^{-4} {\mathrm{m}}^2$ carries a current of $2.0 \mathrm{A}.$ It is suspended through its centre and perpendicular to its length, allowing it to turn in a horizontal plane in a uniform magnetic field $5\times {10}^{-2} \mathrm{T}$ making an angle of $30°$ with the axis of the solenoid. The torque on the solenoid will be

(1) $3\times {10}^{-3} \mathrm{N}‐\mathrm{m}$                                 

(2) $1.5\times {10}^{-3} \mathrm{N}‐\mathrm{m}$

(3) $1.5\times {10}^{-2} \mathrm{N}‐\mathrm{m}$                               

(4) $3\times {10}^{-2} \mathrm{N}‐\mathrm{m}$

A particle having a mass of ${10}^{-2} \mathrm{kg}$carries a charge of $5\times {10}^{-8} \mathrm{C}.$ The particle is given an initial horizontal velocity of ${10}^5 {\mathrm{ms}}^{-1}$ in the presence of electric field $\overrightarrow{\mathbf{E}}$ and magnetic field $\overrightarrow{\mathbf{B}}$. To keep the particle moving in a horizontal direction, it is necessary that

(1) $\overrightarrow{\mathbf{B}}$ should be perpendicular to the direction of velocity and $\overrightarrow{\mathbf{E}}$ should be along the direction of velocity.

(2) Both $\overrightarrow{\mathbf{B}}$ and $\overrightarrow{\mathbf{E}}$ should be along the direction of velocity.

(3) Both $\overrightarrow{\mathbf{B}}$ and $\overrightarrow{\mathbf{E}}$ are mutually perpendicular and perpendicular to the direction of velocity.

(4) $\overrightarrow{\mathbf{B}}$ should be along the direction of velocity and $\overrightarrow{\mathbf{E}}$ should be perpendicular to the direction of velocity.

Which one of the following pairs of statements are possible?

1.   (1) and (3)                                       2.    (3) and (4)

3.   (2) and (3)                                       4.   (2) and (4)

Under the influence of a uniform magnetic field, a charged particle moves with constant speed v in a circle of radius R. The time period of rotation of the particle -

(1) depends on v and not on R

(2) depends on R and not on v

(3) is independent of both v and R

(4) depends on both v and R

The magnetic force acting on a charged particle of charge $-2\mu C$ in a magnetic field of 2T acting in y-direction, when the particle velocity is $\left(2\hat{i}+3\hat{j}\right)\times {10}^6 m{s}^{-1}$ is.

(a) 8 N in -z-direction

(b) 4 N in z-direction

(c) 8 N in y-direction

(d) 8 N in z-direction

A closed-loop PQRS carrying a current is placed in a uniform magnetic field. If the magnetic forces on segments PS, SR and RO are $F_1, F_2 and F_3$ respectively and are in the plane of the paper and along with the directions shown the force on the segment QP is 

(a)$F_3+F_1-F_2$

(b)$\sqrt{{\left(F_3-F_1\right)}^2+F_2^2}$

(c)$\sqrt{{\left(F_3-F_1\right)}^2-F_2^2}$

(d)$F_3-F_1+F_2$

A particle mass m, charge Q, and kinetic energy T enter a transverse uniform magnetic field of induction $\overrightarrow{B}$. After 3sec the kinetic energy of the particle will be : 

1. 3T

2. 2T

3. T

4. 4T

 A narrow electron beam passes undeviated through an electric field E = $3\times {10}^4 \mathrm{volt}/\mathrm{m}$ and an overlapping magnetic field $\mathrm{B}=2\times {10}^{-3} \mathrm{Weber}/{\mathrm{m}}^2$. If electric field and magnetic field are mutually perpendicular. The speed of the electrons is 
1. 60 m/s                          2. $10.3\times {10}^7 \mathrm{m}/\mathrm{s}$
3. $1.5\times {10}^7 \mathrm{m}/\mathrm{s}$                4. $0.67\times {10}^{-7 }\mathrm{m}/\mathrm{s}$

A beam of electrons is moving with constant velocity in a region having electric and magnetic fields of strength $20 {\mathrm{Vm}}^{-1}$ and 0.5 T at right angles to the direction of motion of the electrons. What is the velocity of the electrons

1. 20 ${\mathrm{ms}}^{-1}$                  2. 40 ${\mathrm{ms}}^{-1}$ 
3. 8 ${\mathrm{ms}}^{-1}$                    4. 5.5 ${\mathrm{ms}}^{-1}$