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| ### Statement |
| ### Statement |
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| $10.1.8.$ [Insert the problem statement] |
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| ### Solution |
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| 10.1.8. The figure shows a simple mass spectrometer in which the magnetic field induction is 0.1 T. Ions are formed in the ionizer A and are accelerated by a voltage of 10 kV. After turning in the magnetic field, the ions hit a photographic plate and cause it to blacken. At what distance from the slit will the bands of $^1H^+$, $^2H^+$, $^3H^+$, and $^4He^+$ ions be located on the photographic plate? What should be the slit width for the bands of $^{16}O^+$ and $^{15}N^+$ ions to separate? |
| 10.1.8. The figure shows a simple mass spectrometer in which the magnetic field induction is 0.1 T. Ions are formed in the ionizer A and are accelerated by a voltage of 10 kV. After turning in the magnetic field, the ions hit a photographic plate and cause it to blacken. At what distance from the slit will the bands of $^1H^+$, $^2H^+$, $^3H^+$, and $^4He^+$ ions be located on the photographic plate? What should be the slit width for the bands of $^{16}O^+$ and $^{15}N^+$ ions to separate? |
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| ### Solution |
| ### Solution |
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| Let us use the solution to problem 10.1.1. There, the radius of the circle along which a charged particle moves in a magnetic field was found: |
| Let us use the solution to problem 10.1.1. There, the radius of the circle along which a charged particle moves in a magnetic field was found: |
| $$R = \sqrt{\frac{2mU}{qB^2}}$$ | | $$R = \sqrt{\frac{2mU}{qB^2}}$$ |
| where $m$ is the mass of the particle, $q$ is the charge of the particle, $U$ is the accelerating voltage, and $B$ is the magnetic field induction. | | where $m$ is the mass of the particle, $q$ is the charge of the particle, $U$ is the accelerating voltage, and $B$ is the magnetic field induction. |
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| The trajectory of the particle is a semicircle, so the distance from the slit to the point of impact on the photographic plate is: | | The trajectory of the particle is a semicircle, so the distance from the slit to the point of impact on the photographic plate is: |
| $$x = 2R = 2 \cdot \sqrt{\frac{2mU}{qB^2}} = \sqrt{\frac{8mU}{qB^2}}$$ | | $$x = 2R = 2 \cdot \sqrt{\frac{2mU}{qB^2}} = \sqrt{\frac{8mU}{qB^2}}$$ |
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| In our problem, all particles are ions, which means their charge is the same and is equal to the electron charge with a plus sign. | | In our problem, all particles are ions, which means their charge is the same and is equal to the electron charge with a plus sign. |
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| For $^1H^+$, the mass is $m = 1.67 \cdot 10^{-27}$ kg. Substituting the numerical data, we obtain: $x_1 = 0.29$ m. | | For $^1H^+$, the mass is $m = 1.67 \cdot 10^{-27}$ kg. Substituting the numerical data, we obtain: $x_1 = 0.29$ m. |
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| For $^2H^+$, the mass is $m = 3.34 \cdot 10^{-27}$ kg. Substituting the numerical data, we obtain: $x_2 = 0.41$ m. | | For $^2H^+$, the mass is $m = 3.34 \cdot 10^{-27}$ kg. Substituting the numerical data, we obtain: $x_2 = 0.41$ m. |
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| For $^3H^+$, the mass is $m = 5.01 \cdot 10^{-27}$ kg. Substituting the numerical data, we obtain: $x_3 = 0.5$ m. | | For $^3H^+$, the mass is $m = 5.01 \cdot 10^{-27}$ kg. Substituting the numerical data, we obtain: $x_3 = 0.5$ m. |
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| For $^4He^+$, the mass is $m = 6.68 \cdot 10^{-27}$ kg. Substituting the numerical data, we obtain: $x_4 = 0.58$ m. | | For $^4He^+$, the mass is $m = 6.68 \cdot 10^{-27}$ kg. Substituting the numerical data, we obtain: $x_4 = 0.58$ m. |
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| For the bands of $^{16}O^+$ and $^{15}N^+$ ions to separate, the distance between them, $x_O - x_N$, must be greater than the slit width: | | For the bands of $^{16}O^+$ and $^{15}N^+$ ions to separate, the distance between them, $x_O - x_N$, must be greater than the slit width: |
| $$\Delta l < x_O - x_N$$ | | $$\Delta l < x_O - x_N$$ |
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| For $^{15}N^+$, the mass is $m = 2.505 \cdot 10^{-26}$ kg; for $^{16}O^+$, the mass is $m = 2.672 \cdot 10^{-26}$ kg. Substituting the numerical values, we obtain: | | For $^{15}N^+$, the mass is $m = 2.505 \cdot 10^{-26}$ kg; for $^{16}O^+$, the mass is $m = 2.672 \cdot 10^{-26}$ kg. Substituting the numerical values, we obtain: |
| $$\Delta l < 3.7 \text{ cm}$$ | | $$\Delta l < 3.7 \text{ cm}$$ |
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| #### Answer | | #### Answer |
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| $$x_1 = 0.29 \text{ m}$$ | | $$x_1 = 0.29 \text{ m}$$ |
| $$x_2 = 0.41 \text{ m}$$ | | $$x_2 = 0.41 \text{ m}$$ |
