A uniform magnetic field B exists in the region between x =0 and x =(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 x-axis enters region 2 from region 1 at point P1(y=-R). Which of the following option(s) is/are correct?

Question

A uniform magnetic field B exists in the region between x =0 and x =(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 x-axis enters region 2 from region 1 at point P1(y=-R). Which of the following option(s) is/are correct? (A) For ðµ > (2/3) (p/QR), the particle will re-enter region 1 (B) For ðµ = (8/13) (p/QR), the particle will enter region 3 through the point ð2 on x-axis (C) 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 𝑦-axis is 𝑝/√2 (D) 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

Tardigrade Admin · Accepted Answer

R '=( p / qB ) ( R'- R )2+ R 2= R ' ⇒ R '=(13/8) R

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

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

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 𝑦-axis is $p / 2$

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

$R^{'} = \frac{p}{qB}$
$(R^{'} - R)^{2} + R^{2} = R^{'}$
$\Rightarrow R^{'} = \frac{13}{8} R$

For B>32​QRp​, the particle will re-enter region 1

For B=138​QRp​, the particle will enter region 3 through the point P2​ on x-axis

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 𝑦-axis is p/2​

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

R′=qBp​ (R′−R)2+R2=R′ ⇒R′=813​R

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

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

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 𝑦-axis is $p / 2$

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

$R^{'} = \frac{p}{qB}$
$(R^{'} - R)^{2} + R^{2} = R^{'}$
$\Rightarrow R^{'} = \frac{13}{8} R$