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 58c

Chapter 26, Problem 58c

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.

Verified step by step guidance

Step 1: Understand the context of the problem. The mass spectrometer uses electric and magnetic fields to separate ions based on their mass-to-charge ratio. The question asks whether the sign of the ion charge (positive or negative) affects the separation process.

Step 2: Recall the forces acting on a charged particle in electric and magnetic fields. The electric force is given by \( F_E = qE \), where \( q \) is the charge and \( E \) is the electric field. The magnetic force is given by \( F_B = qvB \), where \( v \) is the velocity of the particle and \( B \) is the magnetic field.

Step 3: Analyze the motion of the particle. In the mass spectrometer, the electric and magnetic forces are used to control the trajectory of the ions. The direction of these forces depends on the sign of the charge \( q \). For a positive charge, the forces act in one direction, and for a negative charge, they act in the opposite direction.

Step 4: Consider the separation mechanism. The separation of isotopes in the mass spectrometer depends on the mass-to-charge ratio \( m/q \). While the sign of \( q \) determines the direction of the forces, the magnitude of \( q \) (not its sign) is what affects the trajectory and separation. Thus, the sign of the charge does not affect the ability to separate isotopes based on \( m/q \).

Step 5: Conclude the explanation. The sign of the ion charge (positive or negative) does not matter for the separation process in the mass spectrometer, as the separation depends only on the magnitude of the charge and the mass-to-charge ratio \( m/q \). However, the direction of the forces and the trajectory will differ for positive and negative charges.

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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 charged particles where other charged particles experience a force. It is quantified in volts per meter (V/m) and influences the motion of charged ions in devices like mass spectrometers. The strength of the electric field determines how quickly ions will accelerate towards the plates, affecting their trajectory and separation based on mass.

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 curved paths. The radius of curvature is dependent on the mass-to-charge ratio of the ions, which is crucial for their identification.

Ion Charge

The charge of an ion, whether positive or negative, significantly affects its behavior in electric and magnetic fields. Positive ions (lost electrons) and negative ions (gained electrons) will experience forces in opposite directions when subjected to these fields. This difference is essential in mass spectrometry, as it influences the trajectory and detection of ions, impacting the results of the analysis.

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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

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 uniform conducting rod of length ℓ and mass m sits atop a fulcrum, which is placed a distance ℓ/4 from the rod’s left-hand end and is immersed in a uniform magnetic field of magnitude B directed into the page (Fig. 27–54). An object whose mass M is 7.0 times greater than the rod’s mass is hung from the rod’s left-hand end. What current (direction and magnitude) should flow through the rod in order for it to be “balanced” (i.e., be at rest horizontally) on the fulcrum? (Flexible connecting wires which exert negligible force on the rod are not shown.)

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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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