A positive, singly ionized atom of mass number is accelerated from rest by the voltage 192 V. Thereafter, it enters a rectangular region of width with magnetic field as shown in the figure. The ion finally hits a detector at the distance below its sta

Question

A positive, singly ionized atom of mass number $A_{M}$ is accelerated from rest by the voltage 192 V. Thereafter, it enters a rectangular region of width $w$ with magnetic field ${\vec{B}}_{0} = 0.1 \hat{k} Tesla,$ as shown in the figure. The ion finally hits a detector at the distance $x$ below its starting trajectory. [Given: Mass of neutron/proton $= (\frac{5}{3}) \times 10^{- 27} kg,$ charge of the electron $= 1.6 \times 10^{- 19}$ C] [2024]

Which of the following option(s) is(are) correct

Which of the following option(s) is(are) correct?

Smart Achievers · Accepted Answer

(1, 2)

From figure $x = 2 R$

$\Rightarrow x = 2 \frac{P}{q B} \Rightarrow x = \frac{2 \sqrt{2 m q V}}{q B} \Rightarrow x = \frac{2}{B} \sqrt{\frac{2 m V}{q}}$

For $H^{+}$ $\to m = \frac{5}{3} \times 10^{- 27} kg$ and $V = 192$ given

$∴$ $x = \frac{2}{0.1} \sqrt{\frac{2 \times \frac{5}{3} \times 10^{- 27} \times 192}{1.6 \times 10^{- 19}}} = 4 cm$

so option (1) is correct.

For $A_{m} = 144$ ,

$x = \frac{2}{0.1} \sqrt{\frac{2 \times 144 \times \frac{5}{3} \times 10^{- 27} \times 192}{1.6 \times 10^{- 19}}} = 48 cm$

so option (2) is correct.

For $A_{m} = 1$ ,

$x = 4 cm$ and for $A_{m} = 196$

$x = 56 cm$

so $x_{0} = 4 cm \times x_{1} = 56 cm$

$∴$ $x_{1} - x_{0} = 52 cm$

so option (3) is incorrect.

For $A_{M} = 196$ ,

$Minimum width W_{\min} = \frac{P}{q B} = \frac{\sqrt{2 m q V}}{q B} = \frac{1}{B} \sqrt{\frac{2 m V}{q}}$

$∴$ $W_{\min} = R = \frac{1}{0.1} \sqrt{\frac{2 \times 196 \times \frac{5}{3} \times 10^{- 27} \times 192}{1.6 \times 10^{- 19}}} = 28 cm$

so option (4) is incorrect.

Solution

1 The value of $x$ for $H^{+}$ ion is 4 cm.

2 The value of $x$ for an ion with $A_{M} = 144$ is 48 cm.

3 For detecting ions with $1 \leq A_{M} \leq 196$ , the minimum height $(x_{1} - x_{0})$ of the detector is 55 cm.

4 The minimum width $w$ of the region of the magnetic field for detecting ions with $A_{M} = 196$ is 56 cm.

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Solution

1 The value of x for H+ ion is 4 cm.

2 The value of x for an ion with AM=144 is 48 cm.

3 For detecting ions with 1≤AM≤196, the minimum height (x1-x0) of the detector is 55 cm.

4 The minimum width w of the region of the magnetic field for detecting ions with AM=196 is 56 cm.

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1 The value of $x$ for $H^{+}$ ion is 4 cm.

2 The value of $x$ for an ion with $A_{M} = 144$ is 48 cm.

3 For detecting ions with $1 \leq A_{M} \leq 196$ , the minimum height $(x_{1} - x_{0})$ of the detector is 55 cm.

4 The minimum width $w$ of the region of the magnetic field for detecting ions with $A_{M} = 196$ is 56 cm.