A uniform magnetic field exists in the region between and (region 2 in the figure) pointing normally into the plane of the paper. A particle with charge and momentum directed along the -axis enters region 2 from region 1 at point . Which of the foll

Question

A uniform magnetic field $B$ exists in the region between $x = 0$ and $x = \frac{3 R}{2}$ (region 2 in the figure) pointing normally into the plane of the paper. A particle with charge $+ Q$ and momentum $p$ directed along the $x$ -axis enters region 2 from region 1 at point $P_{1}$ $(y = - R)$ . Which of the following option(s) are correct [2017]

Smart Achievers · Accepted Answer

(1, 2)

(1) For the charge $x = 0$ to return region 1,

$x = \frac{3 R}{2}$ $\Rightarrow \frac{2 p}{3 R} = Q B$ $[Here, radius r = \frac{3}{2} R]$

$∴$ $B = \frac{2 p}{3 Q R}$

Therefore, for $B \geq \frac{2 p}{3 Q R},$ the particle will re-enter region 1.

(2) When $B = \frac{8 p}{13 Q R},$

$\frac{m v^{2}}{r} = Q v (\frac{8 p}{13 Q R})$ $∴$ $r = \frac{13 R}{8}$

Thus $' C'$ is the centre of the circular path of radius $\frac{13 R}{8}$

Also, $C P_{2} = \sqrt{C O^{2} + O P_{2}^{2}} = \sqrt{{(\frac{5 R}{8})}^{2} + {(\frac{3 R}{2})}^{2}}$

$∴$ $C P_{2} = \frac{13 R}{8}$

Thus the particle will enter region 3 through the point $P_{1}$ on the $X$ -axis.

(3) Change in momentum $= \sqrt{2} p$

(4) Further, $\frac{m v^{2}}{r} = q v B$ $∴$ $r = \frac{m v}{q B}$ $∴$ $r \propto m$

i.e., distance is directly proportional to mass.

Solution

1 For $B > \frac{2}{3} \frac{p}{Q R},$ the particle will re-enter region 1.

2 For $B = \frac{8}{13} \frac{p}{Q R},$ the particle will enter region 3 through the point $P_{2}$ on the $x$ -axis.

3 When the particle re-enters region 1 through the longest possible path in region 2, the magnitude of the change in its linear momentum between point $P_{1}$ and the farthest point from the $y$ -axis is $\frac{p}{\sqrt{2}} .$

4 For a fixed B, particles of same charge Q and same velocity $v$ , the distance between the point $P_{1}$ and the point of re-entry into region 1 is inversely proportional to the mass of the particle.

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Solution

1 For B>23pQR, the particle will re-enter region 1.

2 For B=813pQR, the particle will enter region 3 through the point P2 on the x-axis.

3 When the particle re-enters region 1 through the longest possible path in region 2, the magnitude of the change in its linear momentum between point P1 and the farthest point from the y-axis is p2.

4 For a fixed B, particles of same charge Q and same velocity v, the distance between the point P1 and the point of re-entry into region 1 is inversely proportional to the mass of the particle.

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1 For $B > \frac{2}{3} \frac{p}{Q R},$ the particle will re-enter region 1.

2 For $B = \frac{8}{13} \frac{p}{Q R},$ the particle will enter region 3 through the point $P_{2}$ on the $x$ -axis.

3 When the particle re-enters region 1 through the longest possible path in region 2, the magnitude of the change in its linear momentum between point $P_{1}$ and the farthest point from the $y$ -axis is $\frac{p}{\sqrt{2}} .$

4 For a fixed B, particles of same charge Q and same velocity $v$ , the distance between the point $P_{1}$ and the point of re-entry into region 1 is inversely proportional to the mass of the particle.