Ch. 27 - Magnetism

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 27 - Magnetism

Problem 56

Chapter 26, Problem 56

In a mass spectrometer, germanium atoms have radii of curvature equal to 21.0, 21.6, 21.9, 22.2, and 22.8 cm. The largest radius corresponds to an atomic mass of 76 u. What are the atomic masses of the other isotopes?

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Understand the relationship between the radius of curvature in a mass spectrometer and the mass of the isotope. The radius of curvature \( r \) is proportional to the mass \( m \) of the isotope, given by the equation \( r = \frac{mv}{qB} \), where \( v \) is the velocity of the particle, \( q \) is the charge, and \( B \) is the magnetic field. Since \( v \), \( q \), and \( B \) are constant for all isotopes in this problem, the ratio of radii corresponds directly to the ratio of masses.

Set up a proportionality equation to relate the radii and masses of the isotopes. For example, \( \frac{r_1}{r_2} = \frac{m_1}{m_2} \), where \( r_1 \) and \( r_2 \) are the radii of curvature, and \( m_1 \) and \( m_2 \) are the corresponding masses of the isotopes.

Use the given data: the largest radius \( r_5 = 22.8 \ \text{cm} \) corresponds to an atomic mass of \( m_5 = 76 \ \text{u} \). For each of the other radii \( r_1 = 21.0 \ \text{cm} \), \( r_2 = 21.6 \ \text{cm} \), \( r_3 = 21.9 \ \text{cm} \), and \( r_4 = 22.2 \ \text{cm} \), calculate the corresponding masses using the proportionality \( m = m_5 \cdot \frac{r}{r_5} \).

Substitute the values of \( r \) and \( r_5 \) into the proportionality equation for each isotope. For example, for the first isotope, \( m_1 = 76 \cdot \frac{21.0}{22.8} \). Repeat this calculation for \( r_2, r_3, \) and \( r_4 \).

Express the results for the atomic masses of the isotopes in atomic mass units (u). These values will correspond to the calculated masses for \( r_1, r_2, r_3, \) and \( r_4 \), based on the proportionality relationship.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Mass Spectrometry

Mass spectrometry is an analytical technique used to measure the mass-to-charge ratio of ions. In this process, atoms or molecules are ionized and accelerated through a magnetic or electric field, causing them to follow curved paths. The radius of curvature is influenced by the mass and charge of the ions, allowing for the determination of their mass based on their trajectory.

Isotopes

Isotopes are variants of a chemical element that have the same number of protons but different numbers of neutrons, resulting in different atomic masses. For example, germanium has several isotopes, each with a unique mass. The mass spectrometer can separate these isotopes based on their mass-to-charge ratios, enabling the identification of their respective atomic masses.

Radius of Curvature

The radius of curvature in a mass spectrometer is a measure of the path that an ion takes as it moves through a magnetic or electric field. It is directly related to the mass and charge of the ion; heavier ions or those with lower charge will have larger radii. By analyzing the radii of curvature of different isotopes, one can infer their atomic masses relative to a known isotope.

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Textbook Question

Suppose the electric field between the electric plates in the mass spectrometer of Fig. 27–34 is 2.84 x 10⁴ V/m and the magnetic fields are B = B'= 0.58 T. The source contains carbon isotopes of mass numbers 12, 13, and 14 from a long-dead piece of a tree. (To estimate atomic masses, multiply by 1.67 x 10⁻²⁷ kg.) Does it matter if the ion charge is positive (lost electrons) or negative (gained electrons)? Explain.

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Textbook Question

\(\What\) is the value of q/m for a particle that moves in a circle of radius 8.0 mm in a 0.46-T magnetic field if a crossed 320-V/m electric field will make the path straight?

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Textbook Question

One form of mass spectrometer accelerates ions by a voltage V before they enter a magnetic field B. The ions are assumed to start from rest. Show that the mass of an ion is m = qB²R²/2V, where R is the radius of the ions’ path in the magnetic field and q is their charge.

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Textbook Question

Suppose the electric field between the electric plates in the mass spectrometer of Fig. 27–34 is 2.84 x 10⁴ V/m and the magnetic fields are B = B'= 0.58 T. The source contains carbon isotopes of mass numbers 12, 13, and 14 from a long-dead piece of a tree. (To estimate atomic masses, multiply by 1.67 x 10⁻²⁷ kg .) How far apart are the marks formed by the singly charged ions of each type on a detector or photographic film? What if the ions were doubly charged?

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Textbook Question

A long copper strip is 3.0 cm wide and thick. When it carries a steady 42-A current in a 0.80-T magnetic field it produces a 6.5-μV Hall emf. Determine:

(a) the Hall field in the conductor;

(b) the drift speed of the conduction electrons;

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Textbook Question

A circular coil 18.0 cm in diameter and containing twelve loops lies flat on the ground. The Earth’s magnetic field at this location has magnitude 5.50 x 10⁻⁵ T and points into the Earth at an angle of 54.0° below a line pointing due north. If a 7.10-A clockwise current passes through the coil, (a) determine the torque on the coil; (b) which edge of the coil rises up : north, east, south, or west?

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