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Ch 29: Electromagnetic Induction
Young & Freedman Calc - University Physics 14th Edition
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook
All textbooksYoung & Freedman Calc 14th EditionCh 29: Electromagnetic InductionProblem 40d
Chapter 29, Problem 40d

The magnetic field B at all points within the colored circle shown in Fig. E29.15 has an initial magnitude of 0.750 T. (The circle could represent approximately the space inside a long, thin solenoid.) The magnetic field is directed into the plane of the diagram and is decreasing at the rate of -0.0350 T/s. What is the emf between points a and b on the ring?
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Verified step by step guidance
1
Identify that the problem involves electromagnetic induction, specifically Faraday's law of induction, which relates the electromotive force (emf) to the rate of change of magnetic flux through a loop.
Recognize that the magnetic field is decreasing at a rate of -0.0350 T/s, and this change in magnetic field will induce an emf in the loop according to Faraday's law.
Use Faraday's law of induction, which states that the induced emf (ε) is equal to the negative rate of change of magnetic flux through the loop: ε = -dΦ/dt, where Φ is the magnetic flux.
Calculate the magnetic flux (Φ) through the loop initially using the formula Φ = B * A, where B is the magnetic field and A is the area of the loop. Since the magnetic field is uniform and perpendicular to the plane of the loop, A = πr², where r is the radius of the loop.
Substitute the rate of change of the magnetic field and the area of the loop into the expression for the induced emf: ε = -A * dB/dt. This will give you the magnitude of the emf induced between points a and b on the ring.

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Key Concepts

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

Faraday's Law of Electromagnetic Induction

Faraday's Law states that a change in magnetic flux through a loop induces an electromotive force (emf) in the loop. The induced emf is proportional to the rate of change of the magnetic flux. In this scenario, the decreasing magnetic field within the circle causes a change in flux, leading to an induced emf between points a and b.
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Magnetic Flux

Magnetic flux is the measure of the quantity of magnetism, considering the strength and extent of a magnetic field. It is calculated as the product of the magnetic field strength, the area it penetrates, and the cosine of the angle between the field and the normal to the surface. Here, the flux changes due to the decreasing magnetic field, affecting the induced emf.
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Lenz's Law

Lenz's Law states that the direction of the induced emf and current will oppose the change in magnetic flux that produced them. This principle ensures the conservation of energy and explains the direction of the induced current in the ring, which will be such that it opposes the decrease in the magnetic field within the circle.
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Related Practice
Textbook Question

The magnetic field B at all points within the colored circle shown in Fig. E29.15 has an initial magnitude of 0.750 T. (The circle could represent approximately the space inside a long, thin solenoid.) The magnetic field is directed into the plane of the diagram and is decreasing at the rate of -0.0350 T/s. What is the current in the ring if its resistance is 4.00 Ω?

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Textbook Question

A long, straight solenoid with a cross-sectional area of 8.00 cm2 is wound with 90 turns of wire per centimeter, and the windings carry a current of 0.350 A. A second winding of 12 turns encircles the solenoid at its center. The current in the solenoid is turned off such that the magnetic field of the solenoid becomes zero in 0.0400 s. What is the average induced emf in the second winding?

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Textbook Question

A long, thin solenoid has 400 turns per meter and radius 1.10 cm. The current in the solenoid is increasing at a uniform rate di/dt. The induced electric field at a point near the center of the solenoid and 3.50 cm from its axis is 8.00 × 10-6 V/m. Calculate di/dt.

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Textbook Question

A parallel-plate, air-filled capacitor is being charged as in Fig. 29.23. The circular plates have radius 4.00 cm, and at a particular instant the conduction current in the wires is 0.520 A. (a) What is the displacement current density jD in the air space between the plates?

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Textbook Question

The magnetic field B at all points within the colored circle shown in Fig. E29.15 has an initial magnitude of 0.750 T. (The circle could represent approximately the space inside a long, thin solenoid.) The magnetic field is directed into the plane of the diagram and is decreasing at the rate of -0.0350 T/s. If the ring is cut at some point and the ends are separated slightly, what will be the emf between the ends?

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1795
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Textbook Question

The magnetic field B at all points within the colored circle shown in Fig. E29.15 has an initial magnitude of 0.750 T. (The circle could represent approximately the space inside a long, thin solenoid.) The magnetic field is directed into the plane of the diagram and is decreasing at the rate of -0.0350 T/s. What is the shape of the field lines of the induced electric field shown in Fig. E29.15 , within the colored circle?

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