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 $x$-axis enters region $2$ from region $1$ at point $P_{1}(y=-R)$. Which of the following option(s) is/are correct?

• A. For $𝐵 > \frac{2}{3} \frac{p}{QR},$ the particle will re-enter region $1$
• B. For $𝐵 = \frac{8}{13} \frac{p}{QR},$ the particle will enter region $3$ through the point $𝑃_2$ 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 $𝑃_1$ and the farthest point from 𝑦-axis is $𝑝/\sqrt{2}$
• D. 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

Answer 1

Answer: (A), (B)

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