## Physics II: Chapter 9

ReferenceBeginning Physics II

Chapter 9: INDUCED EMF

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## KEY WORD LIST

Generator, Motional EMF, Induced EMF, Magnetic Flux, Faraday’s Law, Lenz’s Law, Alternating Current, Direct Current, Induced Electric Fields

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## GLOSSARY

For details on the following concepts, please consult Chapter 9.

GENERATOR
When a wire moves through a magnetic field an EMF is generated in the wire, which has the ability to move charges through the wire. This means that it is possible to build an apparatus that makes use of magnetic effects to produce EMFs that drive electrical circuits connected to the apparatus. The apparatus is called a generator, and, like a battery, it pumps positive charges within the apparatus toward the high-potential end of the apparatus, so that, in an external circuit, the charges produce a current flowing from the high- to the low-voltage terminals. As in a battery, the voltage produced by the generator on open circuit is its EMF.

MOTIONAL EMF
An EMF produced in wires moving through a magnetic field is called motional EMF. For a wire of length L, with uniform electric field E, the potential difference is Vba = EL, and using the relation result E = vB, we have for our moving wire:

INDUCED EMF
An EMF produced in stationary wires that are situated in a changing magnetic field is called induced EMF. Its characteristics are given by Faraday’s law.

MAGNETIC FLUX
It is a concept similar to electric flux. As in the case of electric flux one can visualize the magnetic flux by drawing magnetic field lines, with the number of field lines passing through a unit area perpendicular to the lines proportional to B at that location.

In the following diagram there is a small planar area A represented by a vector A that has a magnitude equal to the area, and a direction perpendicular to the plane of the area. We use the notion of circulating current and the right and rule to determine the positive direction of A. For a magnetic field B that passes through the area in the positive direction at an angle θ to A, we define the magnetic flux as

The total magnetic flux through any area is just proportional to the total number of field lines through that area. The unit for magnetic flux is T – m2, which is given the name Weber (Wb).

This law says that whenever there is a change in flux within a circuit there will be an EMF induced in the circuit. This EMF depends on the time rate of change of the flux through the circuit,

where ∆φ is the change in magnetic flux through the circuit in a short time interval, ∆t. The minus sign is necessary to assure that the correct direction is given for the EMF. The requirement of the minus sign is called Lenz’s law.

LENZ’S LAW
This law states that the EMF produced by a changing flux is always in a direction to produce a current whose own flux is in the opposite direction to the initial change in flux.

ALTERNATING CURRENT
One can generate an EMF by rotating a coil with an angular velocity ω in a magnetic field B. Then the angle θ = ωt. The flux (φ) through a single turn of the coil is given by BAcos θ, or BAcos ωt.

By differentiating, we get the EMF to be,

The EMF varies as sin ωt. The EMF produced in this manner will change its direction and then change back again at an angular frequency ω, or at a frequency f = ω/2π and period T = 2π/ω. This is what we call an alternating voltage which produces an alternating current (AC). Thus, by rotating a coil in a magnetic field, we can easily generate an AC voltage. The magnitude of the voltage can be increased by constructing the coil out of many turns, N, of wire, in which case the voltage becomes,

DIRECT CURRENT
It is also possible to construct generators to produce DC (direct current) voltage. To accomplish this, we reverse the connection to the outside wires every time the direction of the EMF in the coil reverses direction. The resultant EMF in the outside circuit will then take the form shown below.

Secondly, we use several coils (armature), in which each coil will produce a voltage which reaches its maximum at a different time, and the total voltage will vary very little with time.

INDUCED ELECTRIC FIELDS
Any changing magnetic field actually produces a new type of electric field in the vicinity of the circuit that pushes the charges and creates the EMF in the stationary circuit. This new electric field is fundamentally different from the “electrostatic” field produced by point charges. . Thus, Faraday’s law has profound implications for our concept of the electric field.

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