A particle of charge \(+q\) and mass \(m\) moving under the influence of a uniform electric field \(E\hat i\) and a uniform magnetic field \(\mathrm B\hat k\) follows a trajectory from \(P\) to \(Q\) as shown in the figure. The velocities at \(P\) and \(Q\) are \(v\hat i\) and \(-2v\hat j\) respectively. Which of the following statement(s) is/are correct?

       

1.	\(E=\frac{3}{4} \frac{{mv}^2}{{qa}}\).
2.	Rate of work done by electric field at \(P\) is \(\frac{3}{4} \frac{{mv}^3}{a}\).
3.	Rate of work done by both fields at \(Q\) is zero.
4.	All of the above.

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

1. $\frac{\sqrt{2}{\mu }_{0}i}{3\mathrm{\pi a}}\odot$                                 

2. $\frac{\sqrt{2}{\mu }_{0}i}{3\mathrm{\pi a}}\otimes$

3. $\frac{\sqrt{2}{\mu }_{0}i}{\mathrm{\pi a}}\odot$                                   

4. $\frac{\sqrt{2}{\mu }_{0}i}{\mathrm{\pi a}}\otimes$   

For a positively charged particle moving in a  x-y­ plane initially along the x-axis, there is a sudden change in its path due to the presence of electric and/or magnetic fields beyond P. The curved path is shown in the x-y plane and is found to be non-circular. Which one of the following combinations is possible

1.  $\overrightarrow{\mathrm{E}}=0; \overrightarrow{\mathrm{B}}=\mathrm{b}\hat{\mathrm{i}}+\mathrm{c}\hat{\mathrm{k}}$                 

2.  $\overrightarrow{\mathrm{E}}=\mathrm{a}\hat{\mathrm{i}};\overrightarrow{ \mathrm{B}}=\mathrm{c}\hat{\mathrm{k}}+\mathrm{a}\hat{\mathrm{i}}$

3. $\overrightarrow{\mathrm{E}}=0;\overrightarrow{\mathrm{B}}=\mathrm{c}\hat{\mathrm{j}}+\mathrm{b}\hat{\mathrm{k}}$                   

4.  $\overrightarrow{\mathrm{E}}=\mathrm{a}\hat{\mathrm{i}};\overrightarrow{\mathrm{B}}=\mathrm{c}\hat{\mathrm{k}}+\mathrm{b}\hat{\mathrm{j}}$

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:

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  i

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

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

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. Net magnetic field at centre will be \(\left(\theta = \frac{\pi}{n}\right)\)

1. \(\frac{\left(\mu\right)_{o} i}{2 πa} tan \frac{\pi}{n}\)                                               
2. \(\frac{\left(\mu\right)_{0} n i}{2 πa} tan \frac{\pi}{n}\)

3.\(\frac{2}{\pi} \frac{n i}{a} \left(\mu\right)_{0} tan \frac{\pi}{n}\)                                           
4. \(\frac{n i}{2 a} \left(\mu\right)_{0} tan \frac{\pi}{n}\)

 The unit vectors \(\hat{i} ,   \hat{j}   ~\text{and} ~ \hat{k}\) are as shown below. What will be the magnetic field at \(O\) in the following figure?

1. \(\frac{\mu_{0}}{4 \pi} \frac{i}{a} 2 - \frac{\pi}{2} \hat{j}\)             
2. \(\frac{\mu_{0}}{4 \pi} \frac{i}{a}2 + \frac{\pi}{2} \hat{j}\)
3. \(\frac{\mu_{0}}{4 \pi} \frac{i}{a}2 + \frac{\pi}{2} \hat{i}\)             
4. \(\frac{\mu_{0}}{4 \pi} \frac{i}{a} 2 + \frac{\pi}{2} \hat{k}\)