Skip to main content
Ch. 29 - Electromagnetic Induction and Faraday's Law
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. 29 - Electromagnetic Induction and Faraday's LawProblem 55
Chapter 28, Problem 55

(III) In a circular region, there is a uniform magnetic field B→\(\overrightarrow{B}\) pointing into the page (Fig. 29–56). An xy coordinate system has its origin at the circular region’s center. A free positive point charge +Q = 1.0 μC is initially at rest at a position x = +10 cm on the x axis. If the magnitude of the magnetic field is now decreased at a rate of -0.10 T/s, what force (magnitude and direction) will act on +Q?


Verified step by step guidance
1
Understand the problem: A positive charge +Q is placed in a region where the magnetic field is decreasing at a constant rate. This changing magnetic field induces an electric field according to Faraday's Law of Electromagnetic Induction. The force on the charge is due to this induced electric field.
Apply Faraday's Law of Induction: The induced electric field is related to the rate of change of the magnetic flux. The magnetic flux Φ is given by Φ = B × A, where B is the magnetic field and A is the area of the circular region. Since the magnetic field is uniform and decreasing, the rate of change of flux is dΦ/dt = A × dB/dt.
Determine the induced electric field: The magnitude of the induced electric field E at a distance r from the center of the circular region is given by |E| = (1 / (2πr)) × |dΦ/dt|. Substituting dΦ/dt = A × dB/dt, we get |E| = (1 / (2πr)) × (πR² × dB/dt), where R is the radius of the circular region and r is the distance of the charge from the center.
Calculate the force on the charge: The force on the charge is given by F = Q × E, where Q is the charge and E is the induced electric field. Substituting the expression for E, we get F = Q × (1 / (2πr)) × (πR² × dB/dt). Simplify this expression to find the magnitude of the force.
Determine the direction of the force: The induced electric field forms circular loops around the center of the circular region (as per Lenz's Law). At the position of the charge (x = +10 cm), the electric field will be tangential to the circle and directed counterclockwise (since the magnetic field is decreasing into the page). The force on the positive charge will be in the same direction as the electric field at that point.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
15m
Was this helpful?

Key Concepts

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

Magnetic Field and Force on a Charge

A magnetic field exerts a force on a moving charge, described by the Lorentz force law. The force is given by F = q(v × B), where F is the force, q is the charge, v is the velocity of the charge, and B is the magnetic field. In this scenario, since the charge is initially at rest, the force will be influenced by changes in the magnetic field rather than its motion.
Recommended video:
Guided course
10:47
Magnetic Field Produced by Moving Charges

Faraday's Law of Electromagnetic Induction

Faraday's Law states that a change in magnetic flux through a circuit induces an electromotive force (EMF) in the circuit. In this case, as the magnetic field decreases at a rate of -0.10 T/s, it creates a changing magnetic flux, which induces a current and subsequently a force on the charge due to the induced electric field.
Recommended video:
Guided course
09:26
Faraday's Law

Direction of Induced Force

The direction of the induced force on a charge can be determined using the right-hand rule and Lenz's Law. Lenz's Law states that the induced current will flow in a direction that opposes the change in magnetic flux. For a positive charge in a decreasing magnetic field, the induced electric field will act in a direction that creates a force on the charge, which can be determined by analyzing the orientation of the magnetic field and the charge's position.
Recommended video:
09:49
Direction of Current in Loop Equations
Related Practice
Textbook Question

If 75 MW of power at 45 kV (rms) arrives at a town from a generator via transmission lines of total resistance 3.0 Ω, calculate (a) the emf at the generator end of the lines, and (b) the fraction of the power generated that is wasted in the lines.

1
views
Textbook Question

The primary windings of a transformer which has an 85% efficiency are connected to 110-V ac. The secondary windings are connected across a 2.4-Ω, 75-W lightbulb.

(a) Calculate the current through the primary windings of the transformer.

(b) Calculate the ratio of the number of primary windings of the transformer to the number of secondary windings of the transformer.

1
views
Textbook Question

(II) For the electric power transmission system shown in Fig. 29–26, what is the ratio Ns/Np for (a) the step-up transformer, (b) the step-down transformer next to the home?

1
views
Textbook Question

A model-train transformer plugs into 120-V ac and draws 0.35 A while supplying 6.5 A to the train.

(a) What voltage is present across the tracks?

(b) Is the transformer step-up or step-down?

2
views
Textbook Question

A circular loop of area 12 m² encloses a magnetic field perpendicular to the plane of the loop; its magnitude is B(t) = (8.0 T/s)t. The loop is connected to a 7.5-Ω resistor and a 6.5-pF capacitor in series. When fully charged, how much charge is stored on the capacitor?

1
views
Textbook Question

A high-intensity desk lamp is rated at 35 W but requires only 12 V. It contains a transformer that converts 120-V household voltage.

(a) Is the transformer step-up or step-down?

(b) What is the current in the secondary coil when the lamp is on?

1
views