Ch 29: Electromagnetic Induction

Young & Freedman Calc - University Physics 15th Edition

Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook

Young & Freedman Calc 15th Edition

Ch 29: Electromagnetic Induction

Problem 1a

Chapter 29, Problem 1a

A single loop of wire with an area of 0.0900 m² is in a uniform magnetic field that has an initial value of 3.80 T, is perpendicular to the plane of the loop, and is decreasing at a constant rate of 0.190 T/s. What emf is induced in this loop?

Verified step by step guidance

Start by recalling Faraday's Law of Induction, which states that the induced electromotive force (emf) in a loop is equal to the negative rate of change of magnetic flux through the loop. The formula is:

ε = - d Φ d t

, where

Φ

is the magnetic flux.

Calculate the initial magnetic flux through the loop using the formula:

Φ = B A

, where

B

is the magnetic field strength and

A

is the area of the loop. Substitute the given values:

Φ = 3.80 T × 0.0900 m

².

Determine the rate of change of the magnetic flux. Since the magnetic field is decreasing at a constant rate, the change in magnetic flux can be expressed as:

d Φ d t = A d B d t

. Substitute the area and the rate of change of the magnetic field:

d Φ d t = 0.0900 m

² × (-0.190 T/s).

Apply Faraday's Law to find the induced emf. Substitute the rate of change of magnetic flux into the formula:

ε = - d Φ d t

. This will give you the magnitude of the induced emf.

Remember that the negative sign in Faraday's Law indicates the direction of the induced emf according to Lenz's Law, which states that the induced emf will generate a current that opposes the change in magnetic flux. However, for the magnitude, you can consider the absolute value.

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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 equal to the negative rate of change of magnetic flux through the loop. This principle is crucial for understanding how varying magnetic fields can generate electric currents.

Magnetic Flux

Magnetic flux refers to the total magnetic field passing through a given area, such as a loop of wire. 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. In this problem, the magnetic field is perpendicular to the loop, simplifying the calculation of flux.

Rate of Change of Magnetic Field

The rate of change of the magnetic field is a key factor in determining the induced emf. In this scenario, the magnetic field is decreasing at a constant rate, which directly affects the rate of change of magnetic flux. Understanding this rate is essential for applying Faraday's Law to calculate the induced emf in the loop.

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Textbook Question

In a physics laboratory experiment, a coil with 200 turns enclosing an area of 12 cm² is rotated in 0.040 s from a position where its plane is perpendicular to the earth's magnetic field to a position where its plane is parallel to the field. The earth's magnetic field at the lab location is 6.0 × 10^-5 T. What is the total magnetic flux through the coil before it is rotated? After it is rotated?

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