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A proton, a deuteron and an $\alpha$ particle are moving with same momentum in a uniform magnetic field. The ratio of magnetic forces acting on them is.......... and their speed is.................. in the ratio.
$1: 2: 4$ and $2: 1: 1$
$2: 1: 1$ and $4: 2: 1$
$4: 2: 1$ and $2: 1: 1$
$1: 2: 4$ and $1: 1: 2$
Solution
$F=q(\vec{v} \times \vec{B})=\frac{q}{m}(\vec{P} \times \vec{B})$
$\Rightarrow F \propto \frac{ q }{ m }$
thus $F _{1}: F _{2}: F _{3}=\frac{ q _{1}}{ m _{1}}: \frac{ q _{2}}{ m _{2}}: \frac{ q _{3}}{ m _{3}}$
$=\frac{ e }{ m _{ p }}: \frac{ e }{2 m _{ p }}: \frac{2 e }{4 m _{ p }}$
$=\frac{1}{1}: \frac{1}{2}: \frac{2}{4}$
$=2: 1: 1$
Now for speed calculation
$P = constant \Rightarrow v \propto \frac{1}{ m }$
thus $v _{1}: v _{2}: v _{3}=\frac{1}{ m _{ p }}: \frac{1}{2 m _{ p }}: \frac{1}{4 m _{ p }}$
$=\frac{1}{1}: \frac{1}{2}: \frac{1}{4}$
$=4: 2: 1$
Similar Questions
A charged particle (electron or proton) is introduced at the origin $(x=0, y=0, z=0)$ with a given initial velocity $\overrightarrow{\mathrm{v}}$. A uniform electric field $\overrightarrow{\mathrm{E}}$ and magnetic field $\vec{B}$ are given in columns $1,2$ and $3$ , respectively. The quantities $E_0, B_0$ are positive in magnitude.
|
column $I$ |
column $II$ | column $III$ |
| $(I)$ Electron with $\overrightarrow{\mathrm{v}}=2 \frac{\mathrm{E}_0}{\mathrm{~B}_0} \hat{\mathrm{x}}$ | $(i)$ $\overrightarrow{\mathrm{E}}=\mathrm{E}_0^2 \hat{\mathrm{Z}}$ | $(P)$ $\overrightarrow{\mathrm{B}}=-\mathrm{B}_0 \hat{\mathrm{x}}$ |
| $(II)$ Electron with $\overrightarrow{\mathrm{v}}=\frac{\mathrm{E}_0}{\mathrm{~B}_0} \hat{\mathrm{y}}$ | $(ii)$ $\overrightarrow{\mathrm{E}}=-\mathrm{E}_0 \hat{\mathrm{y}}$ | $(Q)$ $\overrightarrow{\mathrm{B}}=\mathrm{B}_0 \hat{\mathrm{x}}$ |
| $(III)$ Proton with $\overrightarrow{\mathrm{v}}=0$ | $(iii)$ $\overrightarrow{\mathrm{E}}=-\mathrm{E}_0 \hat{\mathrm{x}}$ | $(R)$ $\overrightarrow{\mathrm{B}}=\mathrm{B}_0 \hat{\mathrm{y}}$ |
| $(IV)$ Proton with $\overrightarrow{\mathrm{v}}=2 \frac{\mathrm{E}_0}{\mathrm{~B}_0} \hat{\mathrm{x}}$ | $(iv)$ $\overrightarrow{\mathrm{E}}=\mathrm{E}_0 \hat{\mathrm{x}}$ | $(S)$ $\overrightarrow{\mathrm{B}}=\mathrm{B}_0 \hat{\mathrm{z}}$ |
($1$) In which case will the particle move in a straight line with constant velocity?
$[A] (II) (iii) (S)$ $[B] (IV) (i) (S)$ $[C] (III) (ii) (R)$ $[D] (III) (iii) (P)$
($2$) In which case will the particle describe a helical path with axis along the positive $z$ direction?
$[A] (II) (ii) (R)$ $[B] (IV) (ii) (R)$ $[C] (IV) (i) (S)$ $[D] (III) (iii)(P)$
($3$) In which case would be particle move in a straight line along the negative direction of y-axis (i.e., more along $-\hat{y}$ )?
$[A] (IV) (ii) (S)$ $[B] (III) (ii) (P)$ $[C]$ (II) (iii) $(Q)$ $[D] (III) (ii) (R)$
