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 48

Chapter 26, Problem 48

$\What$ 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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Understand the problem: The particle is moving in a circular path due to the magnetic field, and the electric field is applied to make the path straight. This means the electric force and magnetic force must balance each other. Use this condition to find the charge-to-mass ratio (q/m).

Write the force balance equation: The electric force $ F_E $ is given by $ F_E = qE $, and the magnetic force $ F_B $ is given by $ F_B = qvB $. For the path to be straight, $ F_E = F_B $, so $ qE = qvB $. Simplify to get $ v = \frac{E}{B} $, where $ v $ is the velocity of the particle.

Relate the circular motion to the magnetic force: In the absence of the electric field, the particle moves in a circle due to the magnetic force. The centripetal force is provided by the magnetic force, so $ F_B = \frac{mv^2}{r} $. Substituting $ F_B = qvB $, we get $ qvB = \frac{mv^2}{r} $. Simplify to find $ \frac{q}{m} = \frac{v}{Br} $.

Substitute $ v = \frac{E}{B} $ into $ \frac{q}{m} = \frac{v}{Br} $: Replace $ v $ with $ \frac{E}{B} $ to get $ \frac{q}{m} = \frac{\frac{E}{B}}{Br} $. Simplify this to $ \frac{q}{m} = \frac{E}{B^2r} $.

Substitute the given values into the formula: Use $ E = 320 \ \text{V/m} $, $ B = 0.46 \ \text{T} $, and $ r = 8.0 \ \text{mm} = 8.0 \times 10^{-3} \ \text{m} $ in the equation $ \frac{q}{m} = \frac{E}{B^2r} $ to calculate the charge-to-mass ratio.

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Key Concepts

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

Lorentz Force

The Lorentz force is the force experienced by a charged particle moving through electric and magnetic fields. It is given by the equation F = q(E + v × B), where F is the force, q is the charge, E is the electric field, v is the velocity of the particle, and B is the magnetic field. This concept is crucial for understanding how charged particles behave in electromagnetic fields.

Cyclotron Motion

Cyclotron motion refers to the circular path that a charged particle follows when it moves perpendicular to a uniform magnetic field. The radius of this path is determined by the particle's velocity, charge, and the strength of the magnetic field. This concept is essential for analyzing the motion of the particle in the given magnetic field and understanding how the electric field can alter this motion.

Charge-to-Mass Ratio (q/m)

The charge-to-mass ratio (q/m) is a measure of how much charge a particle has relative to its mass. It plays a significant role in determining the radius of the circular path of a charged particle in a magnetic field. By calculating q/m, one can predict the behavior of the particle under the influence of electric and magnetic fields, which is key to solving the problem presented.

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Related Practice

Textbook Question

How much work is required to rotate the current loop (Fig. 27–23) in a uniform magnetic field $\vec{B}$ from (a) θ = 0° ( $\vec{μ}$ ∣∣ $\vec{B}$ ) to θ = 180°, (b) θ = 90° to θ = -90°.

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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

For a particle of mass m and charge q moving in a circular path in a magnetic field B, (a) show that its kinetic energy is proportional to r², the square of the radius of curvature of its path. Show that its angular momentum is L=qBr² , around the center of the circle.

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

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 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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\(\What\) 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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Key Concepts

Lorentz Force

Cyclotron Motion

Charge-to-Mass Ratio (q/m)