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A positive charge 'q' of mass 'm' is, moving along the +x axis. We wish to apply a uniform magnetic field B for time $∆$t so that the charge reverses its direction crossing the y-axis at a distance 'd'. Then
1. $\mathrm{B} = \frac{2\mathrm{mv}}{\mathrm{qd}} \mathrm{and} ∆\mathrm{t} = \frac{\mathrm{\pi d}}{2\mathrm{v}}$
2. $\mathrm{B} = \frac{\mathrm{mv}}{\mathrm{qd}} \mathrm{and} ∆\mathrm{t} = \frac{\mathrm{\pi d}}{\mathrm{v}}$
3. $\mathrm{B} = \frac{2\mathrm{mv}}{\mathrm{qd}} \mathrm{and} ∆\mathrm{t} = \frac{\mathrm{\pi d}}{\mathrm{v}}$
4. $\mathrm{B} = \frac{\mathrm{mv}}{2\mathrm{qd}} \mathrm{and} ∆\mathrm{t} = \frac{\mathrm{\pi d}}{2\mathrm{v}}$

Two long current carrying thin wires, both with current I, are held by insulating threads of length L and are in equilibrium as shown in figure, with threads making an angle $\mathrm{\theta }$ with the verticle. If wires have mass $\mathrm{\lambda }$ per unit length then the value of I is : (g = gravitational acceleration)
1. $\sin \theta \sqrt{\frac{\pi \lambda gL}{{\mu }_0 \cos \theta }}$
2. $2\mathrm{sin\theta }\sqrt{\frac{\mathrm{\pi \lambda gL}}{{\mathrm{\mu }}_0 \mathrm{cos\theta }}}$
3. $2\sqrt{\frac{\mathrm{\pi gL}}{{\mathrm{\mu }}_0}\mathrm{tan\theta }}$
4. $\sqrt{\frac{\mathrm{\pi \lambda gL}}{{\mathrm{\mu }}_0}\mathrm{tan\theta }}$

A rectangular loop of sides 10 cm and 5 cm carrying a current I of 12 A, is placed in different orientation as shown in figure below.
(A) 
(B)
(C) 
(D) 
If there is a uniform magnetic field of 0.3 T in the positive Z-direction in which orientations the loop will be (i) stable equilbrium and (ii) unstable equilibrium:
1. A and D respectively
2. B and D respectively
3. B and C respectively
4. A and B respectively

A proton (mass m) accelerated by a potential difference V flies through a uniform transverse magnetic field (B). The field occupies a region of space of width 'd'. If '$\alpha$' be the angle of deviation of proton from initial direction of motion (Fig), the value of sin $\mathrm{\alpha }$ will be:

1. $\frac{\mathrm{B}}{2}\sqrt{\frac{qd}{mV}}$
2. $\frac{\mathrm{B}}{\mathrm{d}}\sqrt{\frac{q}{2mV}}$
3. $\mathrm{Bd}\sqrt{\frac{\mathrm{q}}{2\mathrm{mV}}}$
4. $\mathrm{qV}\sqrt{\frac{\mathrm{Bd}}{2\mathrm{m}}}$
 

Two long straight parallel wires, carrying (adjustable) currents ${\mathrm{I}}_1$ and ${\mathrm{I}}_2$, are kept at a distance d apart. If the force 'F' between two wires is taken as 'positive' when the wires repel each other and 'negative' when the wires attract each other, the graph showing the dependence of F on the product ${\mathrm{I}}_1{\mathrm{I}}_2$, would be:
1. 
2. 
3. 
4. 

A wire carrying current I is tied between points P and Q and is in the shape of a circular arch of radius R due to uniform magnetic field B (perpendicular to the plane of the paper, shown by xxx) in the vicinity of the wire (fig). If the wire subtends an angle $2{\mathrm{\theta }}_0$ at the centre of the circle (of which it forms an arch) then the tension in the wire is :

1. IBR
2. $\frac{\mathrm{IBR}}{\sin {\mathrm{\theta }}_0}$
3. $\frac{\mathrm{IBR}}{2 \sin {\mathrm{\theta }}_0}$
4. $\frac{\mathrm{IBR} {\mathrm{\theta }}_0}{\sin {\mathrm{\theta }}_0}$

In a certain region, static electric and magnetic field exist. The magnetic field is given by $\overrightarrow{\mathrm{B}}=B_0\left(\hat{i}+2\hat{j}-4\hat{k}\right)$. If a test charge moving with a velocity $\overrightarrow{\mathrm{v}}=v_0\left(3\hat{i}-\hat{j}+2\hat{k}\right)$ experiences no force in that region, then the electric field in the region, in SI units is :
 
1. $\overrightarrow{E}=-{\mathrm{v}}_0{B}_0\left(\mathit{3}\hat{i}\mathit{-}2\hat{j}-4\hat{k}\right)$
2. $\overrightarrow{E}=-{\mathrm{v}}_0{B}_0\left(\hat{i}\mathit{+}\hat{j}+7\hat{k}\right)$
3. $\overrightarrow{E}=v_0{B}_0\left(14\hat{j}+7\hat{k}\right)$
4. $\overrightarrow{E}=-v_0{B}_0\left(14\hat{j}+7\hat{k}\right)$

A Helmholtz coil has a pair of loops, each with N turns and radius R. They are placed coaxially at distance R and the sane current I flows through the loops in the same direction. The magnitude of magnetic field P, midway between centres A and C, is given by

1. $\frac{8\mathrm{N} {\mathrm{\mu }}_0\mathrm{I}}{\sqrt{5}\mathrm{R}}$
2. $\frac{8\mathrm{N} {\mathrm{\mu }}_0\mathrm{I}}{5^{3/2}\mathrm{R}}$
3. $\frac{4\mathrm{N} {\mathrm{\mu }}_0\mathrm{I}}{5^{1/2}\mathrm{R}}$
4. $\frac{4\mathrm{N} {\mathrm{\mu }}_0\mathrm{I}}{5^{3/2}\mathrm{R}}$

A current of 1 A is flowing on the sides of an equilateral triangle of side $4.5\times {10}^{-2} \mathrm{m}$. The magnetic field at the centre of the triangle will be :
1. $2\times {10}^{-5} \mathrm{Wb}/{\mathrm{m}}^2$
2. zero
3. $8\times {10}^{-5} \mathrm{Wb}/{\mathrm{m}}^2$
4. $4\times {10}^{-5} \mathrm{Wb}/{\mathrm{m}}^2$

A charge q is spread uniformly over an insulated loop of radiur r. If it is rotated with an angular velocity $\mathrm{\omega }$ with respect to normal axis then the magnetic moment of the loop is :
1. ${\mathrm{q\omega r}}^2$
2. $\frac{4}{3}{\mathrm{q\omega r}}^2$
3. $\frac{3}{2}{\mathrm{q\omega r}}^2$
4. $\frac{1}{2}{\mathrm{q\omega r}}^2$

A loosely wound helix made of stiff wire is mounted vertically with a lower end just touching a dish of mercury. When a current from the battery is started in the coil through the mercury
1. the wire oscillates
2. the wire continues making contact
3. the wire breaks contact just when the current is passed
4. the mercury will expand by heating due to passage of current

Two particles X and Y having equal charges, after being accelerated through the same potential difference, enter region of uniform magnetic field and describe circular paths of radius ${\mathrm{R}}_1 \mathrm{and} {\mathrm{R}}_2$, respectively. The ratio of mass of X to that of Y is 
1. ${\left(\frac{{\mathrm{R}}_1}{{\mathrm{R}}_2}\right)}^{1/2}$
2. $\frac{{\mathrm{R}}_2}{{\mathrm{R}}_1}$
3. ${\left(\frac{{\mathrm{R}}_1}{{\mathrm{R}}_2}\right)}^2$
4. $\frac{{\mathrm{R}}_1}{{\mathrm{R}}_2}$

Two particles, each of mass m and charge q, are atttached to the two ends of a light rigid rod of length 2R. The rod is rotated at constant angular speed about a perpendicular axis passing through its center. The ratio of the magnitudes of the magnetic moment of the system and its angular momentum about the center of the rod is 
1. $\frac{\mathrm{q}}{2\mathrm{m}}$
2. $\frac{\mathrm{q}}{\mathrm{m}}$
3. $\frac{2\mathrm{q}}{\mathrm{m}}$
4. $\frac{\mathrm{q}}{\mathrm{\pi }}m$

Two long parallel wires are at a distance 2d. They carry steady equal currents flowing out of the plane of the paper, as shown. The variation of the magnetic field B along the line XX' is given by 
1. 
2. 
3. 
4. 

A long straight wire along the z-axis carries a current I in the negative z direction. The magnetic vector field $\overrightarrow{\mathrm{B}}$ at a point having coordinates (x,y) in the z = 0 plane is 
1. $\frac{{\mathrm{\mu }}_0\mathrm{I}\left(\mathrm{y}\hat{\mathrm{i}}-\mathrm{x}\hat{\mathrm{j}}\right)}{2\mathrm{\pi }\left({\mathrm{x}}^2+{\mathrm{y}}^2\right)}$
2. $\frac{{\mathrm{\mu }}_0\mathrm{I}\left(\mathrm{x}\hat{\mathrm{i}}+\mathrm{y}\hat{\mathrm{j}}\right)}{2\mathrm{\pi }\left({\mathrm{x}}^2+{\mathrm{y}}^2\right)}$
3. $\frac{{\mathrm{\mu }}_0\mathrm{I}\left(\mathrm{x}\hat{j}-\mathrm{y}\hat{i}\right)}{2\mathrm{\pi }\left({\mathrm{x}}^2+{\mathrm{y}}^2\right)}$
4. $\frac{{\mathrm{\mu }}_0\mathrm{I}\left(\mathrm{x}\hat{\mathrm{i}}-\mathrm{y}\hat{\mathrm{j}}\right)}{2\mathrm{\pi }\left({\mathrm{x}}^2+{\mathrm{y}}^2\right)}$