INDUCTANCE (Chapter 30) below


KEY TERMS: induced current, induced emf, Faraday's law of induction, Lenz's law, motional emf, induced electric field, Maxwell's equations.


Fig. 29.1 Induction of a current in a coil of wire

Whenever there is a change in the number of magnetic field lines passing through a loop of wire a voltage (or emf) is generated (or induced) in the loop of wire. This is how an electric generator works. The phenomenon is known as electromagnetic induction and is explained by
Faraday's law of induction
e = -- dF / dt

where F is the magnetic flux given by the closed integral of the dot product B 0 dA.







According to Faraday's law,
if there is no change (with time)
in the number of lines of B field,
or magnetic flux,
through a closed loop(s)
there will be no induced,
or generated,
voltage set up in the loop(s).

Fig. 29.2 Coil in a uniform constant magnetic field



Fig. 29.3 Magnetic flux (F)





According to Faraday's law, as the
strength of a magnetic field (B) passing through a loop of wire increases, there
will be an increase in the number of magnetic field lines, and therefore an induced voltage, or emf,
set up in the wire loop.

If the strength decreases there will also
be an induced emf set up in the loop but with the opposite polarity.

Lenz's law indicates the polarity
of the induced emf.

Fig. 29.5 Coil in an increasing magnetic field




Fig. 29.6 Changing magnetic flux induces current (recall I = V / R)




Fig. 29.8 Alternator (AC generator)

In an AC electric generator, or alternator, the flux through the loops of wire (wound on the armature) changes, and therefore according to Faraday's law there will be an emf induced in the loops of wire.
This induced emf causes a current to flow in the loops.


Fig. 29.10 DC generator

In a DC generator the direction of the induced current in the loops must be changed every one-half
turn of the generator shaft. This is done with split rings on the rotating armature contacting the
stationary "brushes" which are in turn connected to the wires coming out of the generator.



There is another way to see how a current is set up in the loop.
The electric force equation F = q v x B and the right hand rule indicate there is a force on the charges (electrons) in the purple bar that is moving with velocity v.


Fig. 29.11Slidewire generator
Because the magnetic flux (F) through the loop is changing (increasing) there is an emf induced in the loop in accordance with Faraday's law.




Fig. 29.12 Induced current (I) causes a force F = I L x B to be exerted on the bar.
This force is in the direction opposite to the velocity v.




Fig. 29.13 Lenz's law describes the tendency of nature to resist any
change in magnetic flux passing through a loop of wire.
Changes in flux can be canceled by inducing a magnetic field in the appropriate direction.




Fig. 29.14 Induced currents caused by changes in magnetic flux




Fig. 29.15 (a) Induced emf in rod, (b) induced current in wire loop




Fig. 29.17 Solenoid with changing current.
The galvanometer G will record the presence of an induced emf in the loop of wire
caused by the change in flux passing through the loop.




Fig. 29.19 Eddy currents




Fig. 29.20 Metal detector




Maxwell's Equations (section 29.7) will be covered later in Chapter 32 "Electromagnetic Waves."

Maxwell's four equations

1. Gauss's Law for electric fields
2. Gauss's Law formagnetic fields,
3. Faraday's Law, and
4. Ampere's Law as modified by Maxwell

predict the existence of electromagnetic waves capable of propagating through empty space at the speed of light, which is an electromagnetic wave.






Fig. 29.31 Lenz's law




Fig. 29.33 Lenz's law (Exercise 29.17)




Fig. 29.34 Lenz's law (Exercise 29.18)




Fig. 29.37 Lenz's law (Exercise 29.21)





Fig. 29.39 Motional emf (Exercise 29.26)





Fig. 29.57 Inclined linear generator (Problem 29.77)





INDUCTANCE (Chapter 30)


KEY TERMS: inductor, magnetic energy density, R-L circuit, time constant, L-C circuit, electrical oscillation, series R-L-C circuit.




Fig. 30.4 An inductor, simply a loop or loops of conducting wire.
A magnetic field is set up when current passes through the wire loop.
The B field can be calculated using Ampere's Law or the Law of Biot and Savart.





Fig. 30.6 Voltage across resistor (R) and inductor (L).
The voltage across the inductor is actually a back emf in accordance with Lenz's Law.
As the current being pushed through the inductor (coil) is increased, the flux (F) through the coil increases and this results in an induced current in the opposite direction to the original current.







Fig. 30.11 R-L circuit




Fig. 30.12 and 30.13: Current versus time in an R-L circuit





Fig. 30.14 Oscillation of current in an LC circuit




Oscillation of current in an LC circuit
From Physics, 4th Edition, by Halliday, Resnick, and Krane, page 830,
John Wiley & Sons, Inc. New York, 1992.




Fig. 30.15 Oscillating LC circuit




Fig. 30.17 Series R-L-C circuit





Go to AC Circuits page next

 © 2009 J. F. Becker