Ch 29: The Magnetic Field

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 29: The Magnetic Field

Problem 29

Chapter 29, Problem 29

To five significant figures, what are the cyclotron frequencies in a 3.0000 T magnetic field of the ions (a) O₂⁺, (b) N₂⁺, and (c) CO⁺? The atomic masses are shown in the table; the mass of the missing electron is less than 0.001 u and is not relevant at this level of precision. Although N₂⁺ and CO⁺ both have a nominal molecular mass of 28, they are easily distinguished by virtue of their slightly different cyclotron frequencies. Use the following constants: 1 u = 1.6605 x 10⁻²⁷ kg, e = 1.6022 x 10⁻¹⁹ C.

Verified step by step guidance

Step 1: Understand the cyclotron frequency formula. The cyclotron frequency (f) is given by the formula:

f = \frac{q}{m} B

, where q is the charge of the ion, m is the mass of the ion, and B is the magnetic field strength.

Step 2: Determine the charge of the ions. Since the ions are singly charged (O₂⁺, N₂⁺, CO⁺), the charge q is equal to the elementary charge,

e = 1.6022 \times 10^{- 19} C

Step 3: Calculate the mass of each ion in kilograms. Use the atomic masses provided in the table and the conversion factor

1 u = 1.6605 \times 10^{- 27} kg

. For O₂⁺, the mass is

2 \times 15.995 \times 1.6605 \times 10^{- 27}

. Similarly, calculate for N₂⁺ and CO⁺ using their respective atomic masses.

Step 4: Substitute the values into the cyclotron frequency formula. For each ion, substitute q =

1.6022 \times 10^{- 19}

, m (calculated in Step 3), and B =

3.0000 T

into the formula

f = \frac{q}{m} B

Step 5: Compare the cyclotron frequencies of N₂⁺ and CO⁺. Note that although both have a nominal molecular mass of 28 u, their slightly different atomic masses will result in distinct cyclotron frequencies, allowing them to be distinguished.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Was this helpful?

Key Concepts

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

Cyclotron Frequency

Cyclotron frequency is the frequency at which a charged particle orbits in a magnetic field. It is determined by the charge of the particle, the strength of the magnetic field, and the mass of the particle. The formula for cyclotron frequency (f) is f = (qB)/(2πm), where q is the charge, B is the magnetic field strength, and m is the mass of the particle. This concept is crucial for understanding the motion of ions in magnetic fields.

Atomic Mass Unit (u)

The atomic mass unit (u) is a standard unit of mass used to express atomic and molecular weights. One atomic mass unit is defined as one twelfth of the mass of a carbon-12 atom, approximately 1.6605 x 10⁻²⁷ kg. In this context, the atomic masses of ions like O₂⁺, N₂⁺, and CO⁺ are essential for calculating their cyclotron frequencies, as the mass directly influences the frequency of their motion in a magnetic field.

Magnetic Field Strength

Magnetic field strength, denoted as B, is a measure of the intensity of a magnetic field at a given point in space. It is typically measured in teslas (T). In this problem, a magnetic field strength of 3.0000 T is provided, which is necessary for calculating the cyclotron frequencies of the ions. The strength of the magnetic field affects the radius and frequency of the charged particles' circular motion.

Recommended video:

Guided course

05:30

Magnetic Fields and Magnetic Dipoles

Related Practice

Textbook Question

Radio astronomers detect electromagnetic radiation at 45 MHz from an interstellar gas cloud. They suspect this radiation is emitted by electrons spiraling in a magnetic field. What is the magnetic field strength inside the gas cloud?

views

Textbook Question

The value of the line integral of $\vec{B} \cdot \vec{d s}$ around the closed path in FIGURE EX29.21 is 1.38 x 10^-5 T m. What are the direction (into or out of the figure) and magnitude of I₃?

views

Textbook Question

The two 10-cm-long parallel wires in FIGURE EX29.33 are separated by 5.0 mm. For what value of the resistor R will the force between the two wires be 5.4 x 10⁻⁵ N?

views

Textbook Question

A proton moves in the magnetic field B = 0.50 î T with a speed of 1.0 x 10⁷ m/s in the directions shown in FIGURE EX29.27. For each, what is magnetic force F on the proton? Give your answers in component form.

views

Textbook Question

The microwaves in a microwave oven are produced in a special tube called a magnetron. The electrons orbit the magnetic field at 2.4 GHz, and as they do so they emit 2.4 GHz electromagnetic waves. What is the magnetic field strength?

views

Textbook Question

The Hall voltage across a conductor in a 55 mT magnetic field is 1.9 μV. When used with the same current in a different magnetic field, the voltage across the conductor is 2.8 μV. What is the strength of the second field?

views