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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
All textbooksGiancoli Douglas 5th editionCh. 27 - MagnetismProblem 58b
Chapter 26, Problem 58b

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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Step 1: Understand the problem. The mass spectrometer separates ions based on their mass-to-charge ratio (m/q). The ions are subjected to both electric and magnetic fields, and their trajectories are influenced by these fields. The goal is to calculate the separation of the marks formed by ions of different masses on the detector for both singly and doubly charged ions.
Step 2: Use the velocity selector principle. The velocity selector ensures that only ions with a specific velocity pass through. The condition for this is that the electric force equals the magnetic force: F_e = F_m, or qE = qvB. From this, solve for the velocity v: v = E / B, where E is the electric field strength and B is the magnetic field strength.
Step 3: Determine the radius of the circular path in the magnetic field. Once the ions exit the velocity selector, they enter a region with only a magnetic field. The magnetic force provides the centripetal force, so qvB = mv² / r. Rearrange to solve for the radius r: r = mv / (qB). Substitute the velocity v from Step 2 into this equation to express r in terms of known quantities: r = m(E / B) / (qB) = mE / (qB²).
Step 4: Calculate the separation of the marks on the detector. The separation between the marks for two isotopes is determined by the difference in their radii. For isotopes with masses m₁ and m₂, the separation Δr is given by Δr = r₂ - r₁ = (m₂E / (qB²)) - (m₁E / (qB²)). Factor out common terms: Δr = E / (qB²) * (m₂ - m₁).
Step 5: Repeat the calculation for doubly charged ions. For doubly charged ions, the charge q is doubled. This affects the radius formula: r = mE / (2qB²). Use the same approach as in Step 4 to calculate the separation Δr for doubly charged ions, substituting q = 2e (where e is the elementary charge).

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Key Concepts

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

Electric Field

An electric field is a region around a charged particle where other charged particles experience a force. It is quantified in volts per meter (V/m) and influences the motion of charged particles, such as ions in a mass spectrometer. The strength of the electric field affects the acceleration of ions, which is crucial for determining their trajectory and the distance they travel before hitting a detector.
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Magnetic Field

A magnetic field is a vector field that exerts a force on moving charged particles, described by the magnetic flux density (measured in teslas, T). In a mass spectrometer, the magnetic field interacts with the velocity of ions, causing them to follow a curved path. The radius of this curvature depends on the charge and mass of the ions, which is essential for distinguishing between different isotopes based on their trajectories.
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Mass Spectrometry

Mass spectrometry is an analytical technique used to measure the mass-to-charge ratio of ions. In this process, ions are accelerated by electric fields and then deflected by magnetic fields, allowing for the separation of isotopes based on their mass. The distance between marks on a detector corresponds to the different trajectories of ions with varying masses, providing insights into the composition of the sample being analyzed.
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Related Practice
Textbook Question

Near the equator, the Earth’s magnetic field points almost horizontally to the north and has magnitude B = 0.50 x 10⁻⁴ T. What should be the magnitude and direction for the velocity of an electron if its weight is to be exactly balanced by the magnetic force?

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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

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

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;

(c) the density of free electrons in the metal.

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Textbook Question

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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Textbook Question

A mass spectrometer is monitoring air pollutants. It is difficult, however, to separate molecules of nearly equal mass such as CO (28.0106 u) and N₂ (28.0134 u). How large a radius of curvature must a spectrometer have (Fig. 27–34) if these two molecules are to be separated on the detector by 0.50 mm?

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