(L09) A mass spectrometer is used to separate uranium isotopes 235U and 238U. A beam of mixed 235U and 238U ions travelling at 6500 m/s enters in a uniform magnetic field. Then the isotopes separate from each other by travelling along different semicircles within the magnetic field. Both the isotope ions carry a +4.8E-19 C charge, and the isotope masses are 3.901E-25 kg for 235U and 3.951E-25 kg for 238U. If the semicircle radius of the 235U ions is 13.1 m, what is the separation of the isotopes (the difference between semicircle radii of the two isotopes in meter)?
(L09) A mass spectrometer is used to separate uranium isotopes 235U and 238U. A beam of mixed 235U and 238U ions travelling at 6500 m/s enters in a uniform magnetic field. Then the isotopes separate from each other by travelling along different semicircles within the magnetic field. Both the isotope ions carry a +4.8E-19 C charge, and the isotope masses are 3.901E-25 kg for 235U and 3.951E-25 kg for 238U. If the semicircle radius of the 235U ions is 13.1 m, what is the separation of the isotopes (the difference between semicircle radii of the two isotopes in meter)?
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Author:Raymond A. Serway, Chris Vuille
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Chapter1: Units, Trigonometry. And Vectors
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![### Uranium Isotope Separation using a Mass Spectrometer
**Overview:**
A mass spectrometer is a device used to separate isotopes of uranium (^235U and ^238U). The device works by sending a beam of mixed ^235U and ^238U ions, each travelling at a speed of 6500 m/s, through a uniform magnetic field. Due to differences in their masses, the ions travel in different semicircular paths within the magnetic field.
**Charge and Mass of Ions:**
- Both ^235U and ^238U ions carry a charge of +4.8E-19 C.
- The mass of ^235U is 3.901E-25 kg.
- The mass of ^238U is 3.951E-25 kg.
**Experimental Details:**
If the semicircular radius for ^235U ions is known to be 13.1 m, the task is to determine the separation of the isotopes (calculated as the difference between the radii of the semicircles traced by each isotope in the magnetic field).
### Explanation of the Physical Principle:
1. **Lorentz Force**: When a charged particle enters a magnetic field, it experiences a force perpendicular to both its velocity and the magnetic field, causing it to move in a curved path.
2. **Radius of Curvature**: The radius of the path taken by a charged particle in a magnetic field depends on its mass (m), velocity (v), charge (q), and the magnetic field strength (B). This relationship is given by the formula:
\[
r = \frac{mv}{qB}
\]
3. **Calculation**: Given the velocities, charges, and the radius for ^235U, we can derive the radius for the ^238U ions and then find the separation.
### Problem Solving:
Given:
- \( r_{235} \) = 13.1 m (radius for ^235U)
- \( m_{235} \) = 3.901E-25 kg
- \( m_{238} \) = 3.951E-25 kg
- \( v \) = 6500 m/s
- \( q \) = 4.8E-19 C
Using the relationship:
\[ r = \frac{mv}{qB} \]
For ^235U:
\[ r_{235} = \frac{m](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6401e82b-1884-43b6-8725-0773860dca12%2F834d666b-0489-4a0d-947f-0257bd6387b4%2Fhtep9dr_processed.png&w=3840&q=75)
Transcribed Image Text:### Uranium Isotope Separation using a Mass Spectrometer
**Overview:**
A mass spectrometer is a device used to separate isotopes of uranium (^235U and ^238U). The device works by sending a beam of mixed ^235U and ^238U ions, each travelling at a speed of 6500 m/s, through a uniform magnetic field. Due to differences in their masses, the ions travel in different semicircular paths within the magnetic field.
**Charge and Mass of Ions:**
- Both ^235U and ^238U ions carry a charge of +4.8E-19 C.
- The mass of ^235U is 3.901E-25 kg.
- The mass of ^238U is 3.951E-25 kg.
**Experimental Details:**
If the semicircular radius for ^235U ions is known to be 13.1 m, the task is to determine the separation of the isotopes (calculated as the difference between the radii of the semicircles traced by each isotope in the magnetic field).
### Explanation of the Physical Principle:
1. **Lorentz Force**: When a charged particle enters a magnetic field, it experiences a force perpendicular to both its velocity and the magnetic field, causing it to move in a curved path.
2. **Radius of Curvature**: The radius of the path taken by a charged particle in a magnetic field depends on its mass (m), velocity (v), charge (q), and the magnetic field strength (B). This relationship is given by the formula:
\[
r = \frac{mv}{qB}
\]
3. **Calculation**: Given the velocities, charges, and the radius for ^235U, we can derive the radius for the ^238U ions and then find the separation.
### Problem Solving:
Given:
- \( r_{235} \) = 13.1 m (radius for ^235U)
- \( m_{235} \) = 3.901E-25 kg
- \( m_{238} \) = 3.951E-25 kg
- \( v \) = 6500 m/s
- \( q \) = 4.8E-19 C
Using the relationship:
\[ r = \frac{mv}{qB} \]
For ^235U:
\[ r_{235} = \frac{m
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