Chapter 29Electromagnetic Induction and Faraday’s Law

HW#9: Chapter 28: Pb.18, Pb. 31, Pb.40Chapter 29:Pb.3, Pb 30, Pb. 48 Due Wednesday 22

28-4 Ampère’s LawConceptual Example 28-7: Coaxial cable.

A coaxial cable is a single wire surrounded by a cylindrical metallic braid. The two conductors are separated by an insulator. The central wire carries current to the other end of the cable, and the outer braid carries the return current and is usually considered ground. Describe the magnetic field (a) in the space between the conductors, and (b) outside the cable.

28-4 Ampère’s LawExample 28-8: A nice use for Ampère’s law.

Use Ampère’s law to show that in any region of space where there are no currents the magnetic field cannot be both unidirectional and non-uniform as shown in the figure.

28-4 Ampère’s LawSolving problems using Ampère’s law:

• Ampère’s law is only useful for solving problems when there is a great deal of symmetry. Identify the symmetry.

• Choose an integration path that reflects the symmetry (typically, the path is along lines where the field is constant and perpendicular to the field where it is changing).

• Use the symmetry to determine the direction of the field.

• Determine the enclosed current.

28-5 Magnetic Field of a Solenoid and a Toroid

A solenoid is a coil of wire containing many loops. To find the field inside, we use Ampère’s law along the path indicated in the figure.

28-5 Magnetic Field of a Solenoid and a Toroid

The field is zero outside the solenoid, and the path integral is zero along the vertical lines, so the field is (n is the number of loops per unit length)

28-5 Magnetic Field of a Solenoid and a Toroid

Example 28-9: Field inside a solenoid.

A thin 10-cm-long solenoid used for fast electromechanical switching has a total of 400 turns of wire and carries a current of 2.0 A. Calculate the field inside near the center.

28-6 Biot-Savart Law

The Biot-Savart law gives the magnetic field due to an infinitesimal length of current; the total field can then be found by integrating over the total length of all currents:

𝑑𝐵=𝜇 0 𝐼𝑑𝑙sin 𝜃4𝜋𝑟 2

�⃗�=∫𝑑 �⃗�=𝜇0 𝐼4 𝜋∫ �⃗� ℓ×𝒓

𝑟2

28-6 Biot-Savart Law

Example 28-11: B due to current I in straight wire.

For the field near a long straight wire carrying a current I, show that the Biot-Savart law gives B = μ0I/2πr.

B))))))))))))))

Almost 200 years ago, Faraday looked for evidence that a change in a magnetic field would induce an electric current with this apparatus:

29-1 Induced EMF

He found no evidence when the current was steady, but did see a current induced when the switch was turned on or off.

29-1 Induced EMF

Therefore, a changing magnetic field induces an emf.

Faraday’s experiment used a magnetic field that was changing because the current producing it was changing; the previous graphic shows a magnetic field that is changing because the magnet is moving.

29-1 Induced EMF

The induced emf in a wire loop is proportional to the rate of change of magnetic flux through the loop.

Magnetic flux:

Unit of magnetic flux: weber, Wb:

1 Wb = 1 T·m2.

29-2 Faraday’s Law of Induction; Lenz’s Law

This drawing shows the variables in the flux equation:

29-2 Faraday’s Law of Induction; Lenz’s Law

∅𝐵= �⃗� ∙ �⃗�

The magnetic flux is analogous to the electric flux – it is proportional to the total number of magnetic field lines passing through the loop.

29-2 Faraday’s Law of Induction; Lenz’s Law

29-2 Faraday’s Law of Induction; Lenz’s LawConceptual Example 29-1: Determining flux.

A square loop of wire encloses area A1. A uniform magnetic field B perpendicular to the loop extends over the area A2. What is the magnetic flux through the loop A1?

B))))))))))))))

Faraday’s law of induction: the emf induced in a circuit is equal to the rate of change of magnetic flux through the circuit:

29-2 Faraday’s Law of Induction; Lenz’s Law

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