Figure shows a square loop ABCD with edge length a. The resistance of the wire ABC is r and that of ADC is 2r. The value of magnetic field at the centre of the loop assuming uniform wire is

                   

(a) $\frac{\sqrt{2}{\mu }_{0}i}{3\mathrm{\pi a}}\odot$                                  (b) $\frac{\sqrt{2}{\mu }_{0}i}{3\mathrm{\pi a}}\otimes$

(c) $\frac{\sqrt{2}{\mu }_{0}i}{\mathrm{\pi a}}\odot$                                   (d) $\frac{\sqrt{2}{\mu }_{0}i}{\mathrm{\pi a}}\otimes$   

When a positively charged particle moves in an x-y plane, its path abruptly changes due to the presence of electric and/or magnetic fields beyond P. The curved path is depicted in the x-y plane and is discovered to be noncircular. Which of the following combinations is true?
          
1. \(\overrightarrow{\mathrm{E}}=0 ; \overrightarrow{\mathrm{B}}=\mathrm{b} \hat{\mathrm{i}}+\mathrm{c} \widehat{\mathrm{k}}\) 
2. \(\overrightarrow{\mathrm{E}}=\mathrm{ai} ; \overrightarrow{\mathrm{B}}=\mathrm{c} \widehat{\mathrm{k}}+\mathrm{a} \hat{\mathrm{i}}\)
3. \(\vec{E}=0 ; \vec{B}=c \hat{j}+b \widehat{k}\)
4. \(\overrightarrow{\mathrm{E}}=\mathrm{ai} ; \overrightarrow{\mathrm{B}}=c \widehat{\mathrm{k}}+\mathrm{b} \hat{\mathrm{j}}\)

Figure shows the cross-sectional view of the partially hollow cylindrical conductor with inner radius 'R' and outer radius '2R' carrying uniformly distributed current along it's axis. The magnetic induction at point 'P' at a distance $\frac{3R}{2}$ from the axis of the cylinder will be:

1. Zero                               

2. $\frac{5{\mu }_{0}i}{72\mathrm{\pi R}}$

3. $\frac{7{\mu }_{0}i}{18\mathrm{\pi R}}$                             

4.  $\frac{5{\mu }_{o}i}{36\mathrm{\pi R}}$

A long wire AB is placed on a table. Another wire PQ of mass 1.0 g and length 50 cm is set to slide on two rails PS and QR. A current of 50A is passed through the wires. At what distance above AB, will the wire PQ be in equilibrium

(1) 25 mm                           

(2) 50 mm

(3) 70 mm                           

(4) 100 mm

A particle with charge q, moving with a momentum p, enters a uniform magnetic field normally. The magnetic field has magnitude B and is confined to a region of width d, where $d<\frac{p}{Bq}$. The particle is deflected by an angle $\theta$ in crossing the field, then :

(a) $\sin \theta =\frac{Bqd}{p}$                              (b) $\sin \theta =\frac{p}{Bqd}$

(c) $\sin \theta =\frac{Bp}{qd}$                                (d) $\sin \theta =\frac{pd}{Bq}$

The same current i = 2A is flowing in a wireframe as shown in the figure. The frame is a combination of two equilateral triangles ACD and CDE of side 1m. It is placed in uniform magnetic field B = 4T acting perpendicular to the plane of the frame. The magnitude of the magnetic force acting on the frame is:

1. 24 N                                         2. Zero

3. 16 N                                         4. 8 N

In the given figure, net magnetic field at O will be 

1. $\frac{2{\mu }_{0}i}{3\mathrm{\pi a}}\sqrt{4-{\pi }^2}$                                 
2. $\frac{{\mu }_{0}i}{3\mathrm{\pi a}}\sqrt{4+{\pi }^2}$

3. $\frac{2{\mu }_{0}i}{3\mathrm{\pi a}}\sqrt{4+{\pi }^2}$                               
4. $\frac{2{\mu }_{0}i}{3\mathrm{\pi a}}\sqrt{\left(4-{\mathrm{\pi }}^2\right)}$

In the following figure, a wire bent in the form of a regular polygon of n sides is inscribed in a circle of radius a. The net magnetic field at centre will be $\left(\theta =\frac{\pi }{n}\right)$

(a) $\frac{{\mu }_{o}i}{2\mathrm{\pi a}}\tan \frac{\mathrm{\pi }}{n}$                                                (b) $\frac{{\mu }_{0}ni}{2\mathrm{\pi a}}\tan \frac{\mathrm{\pi }}{n}$

(c)$\frac{2}{\mathrm{\pi }}\frac{ni}{a}{\mu }_0\tan \frac{\mathrm{\pi }}{n}$                                             (d) $\frac{ni}{2a}{\mu }_0\tan \frac{\mathrm{\pi }}{n}$

An elastic circular wire of length L carries a current I. It is placed in a uniform magnetic field $\overrightarrow{B}$ (out of paper) such that its plane is perpendicular to the direction of $\overrightarrow{B}$ . The wire will experience

(1) No force                           

(2) A stretching force

(3) A compressive force           

(4) A torque

Wires 1 and 2 carrying currents $i_1$ and $i_2$ respectively are inclined at an angle $\theta$ to each other. What is the force on a small element dl of wire 2 at a distance of r from wire 1 (as shown in figure) due to the magnetic field of wire 1

(a) $\frac{{\mu }_0}{2\mathrm{\pi r}}{i}_1{i}_{2}dl \tan \theta$                              (b) $\frac{{\mu }_0}{2\mathrm{\pi r}}{i}_1{i}_{2}dl \sin \theta$

(c) $\frac{{\mu }_0}{2\mathrm{\pi r}}{i}_1{i}_{2}dl \cos \theta$                               (d) $\frac{{\mu }_0}{4\mathrm{\pi r}}{i}_1{i}_{2}dl \sin \theta$