Chapter 31: Induction and Inductance

31.1. Two Symmetric Situations

31.2. Two Experiments

31.3. Faraday's Law of Induction

31.4. Lenz's Law

31.5. Induction and Energy Transfers

31.6. Induced Electric Fields

31.7. Inductors and Inductance

31.8. Self-Induction

31.9. RL Circuits

31.10. Energy Stored in a Magnetic Field

31.11. Energy Density of a Magnetic Field

31.12. Mutual Inductance

31.1. Two Symmetric Situations

•If a current loop is placed in a magnetic field, the forces produced by the magnetic field act as a torque to turn the current loop

•In the 1820's, Michael Faraday speculated that the reverse might be true

•Faraday reasoned that since moving charges produce magnetic fields, changing magnetic fields might produce moving charges

•Faraday was right

•When a magnetic field is changed, an electric field is produced

•If there is a conducting loop in the changing magnetic field, a current is induced in it

31.2. Two Experiments

•If a bar magnet is brought close to a conducting loop, the induced current can be measured on an ammeter

•The induced current is proportional to the rate of change of the magnetic field

•If the magnet stops moving, so does the induced current

•Another way to make a changing magnetic field is to turn on the current through a primary conducting loop

•The primary loop then produces a changing magnetic field which induces a current in the secondary loop

•A changing magnetic field induces a voltage

•But Faraday was not certain that the voltage induced by a changing magnetic field is the same as the voltage from a battery

•So Faraday gave induced voltages the name "Electromotive Force" (EMF)

•It later turned out that the EMF is the same as ordinary voltage, but the name stuck

•So EMF and voltage can be used interchangeably

31.3. Faraday's Law of Induction

•It was soon realized that an EMF is induced

(1) whenever the strength of a magnetic field is increased inside a loop

OR

(2) whenever the area of the magnetic field inside the loop is increases

•This brings up the idea of magnetic flux

•Remember that the electric flux is defined as the E field strength times the area

•The magnetic field flux is also defined as the magnetic field strength times the area of the magnetic field

•Since the B field strength often changes with position, an integral must be used to define the magnetic flux

•The minus sign indicates that the EMF is induced in such a direction as to oppose the change in magnetic field

31.4. Lenz's Law

•Faraday's Law is easy to use when only the magnitude is needed

•But when the direction of the induced EMF and induced current is needed, a corollary to Faraday's Law is used

•Lenz's law says that the direction of the induced EMF is such as to create a magnetic field that would oppose the change in the magnetic field that creates the EMF

•Lenz's Law must be practiced before it is fully understood

Rules for Using Lenz's Law

Lenz's Law: The polarity of an induced EMF is such as to create a current whose magnetic flux will oppose a change in the magnetic flux through a loop.

1. Find the direction of the existing B-field.

2. Determine if the existing B-field is increasing or decreasing.

3. If the existing field is increasing, the induced B-field is in the opposite direction. If the existing field is decreasing, the induced B-field is in the same direction.

4. Find the direction of the induced current needed to produce the induced B-field.

5. If required, the direction of the induced EMF can be found by noting that the induced current in #4 (above) flows from positive to negative.

Examples of Using Lenz's Law

and the Right Hand Rule

31.5. Induction and Energy Transfers

•A good example of induced EMF and current is the withdrawal of a conducting loop from a magnetic field

•So work must be done to pull the loop from the field with a force F at a speed v

•The rate of doing work is:

P = Fv

•The induced voltage can be found by calculating the rate of change of the flux

•If the resistance of the conducting loop is R, the current is:

•The removal of a conducting loop from a magnetic field is a simple example of the phenomenon of eddy currents

•Whenever a conducting sheet slices through a magnetic field, it is slowed up by the same kind of retarding force found for the conducting loop

•That is, current loops, called eddy currents, are formed and the motion of a conductor through a magnetic field is retarded

•Some of the gravitational potential energy of a conducting sheet falling through a magnetic field is converted to heat energy dissipated by eddy currents in the conductor

•Magnetic induction heating stove tops are one example

31.6. Induced Electric Fields

•To get a clearer idea of what happens in a conducting loop placed in a changing magnetic field, a copper disc is substituted for the loop

•Now there is a more complete picture of the induced electric fields and the current they induce

•The electric field lines are found to be complete circles that are centered on the axis of the disc

•The induced currents follow the concentric lines of electric field

•The strength of the electric fields is dependent on the area enclosed by them

•A good way to study this effect is to rewrite Faraday's Law to concentrate on the induced electric fields

•If the induced EMF is due to the summation of electric fields over a complete line, then faraday's Law can be rewritten as:

•This brings up a problem with Kirchoff's voltage sum law however

•In looking at circuits, use was made of the fact that the electric field is conservative and thus the potential difference between two points is independent of the path taken between them.

•So a clarification is needed

•Electric fields created by static charges are conservative

•Electric fields created by magnetic induction are not conservative

•The reason for this difference is that electric fields created by static charges drop off with distance from the source

•But electric fields created by magnetic induction do not drop off with distance along a field line

31.7. Inductors and Inductance

•Coils and solenoids are often used as components in electrical circuits

•In this application they are called inductors

•Inductors store energy in magnetic fields just as capacitors store energy in electric fields

•But where there is a delay in the the voltage due to the time required to charge the capacitor, the reverse is true for an inductor

•The voltage across an inductor comes from Faraday's Law

where B is the magnetic field strength, A is the area of the coil, and N is the number of turns in the coil

•Assuming that A and N are constants:

•The individual constants are usually lumped into a single constant called the inductance:

•In general, Faraday's Law for an inductor is written as:

where L is defined as:

31.8. Self-Induction

•It is possible for inductors in circuits to link magnetically to other inductors

•But most inductors are used for their self-inductance, the voltage they create in response to a change in current through them

•Care must be taken to keep inductors well separated to avoid the flux lines of one inductor from penetrating (and thus linking to) another inductor

•Actual inductance is part of the circuit model of a coil

•But coils also have the resistance due to the wires in their windings

•The smaller the winding resistance the closer to a pure inductance a coil is

31.9. RL Circuits

•As inductors store energy in their magnetic fields, they behave somewhat like capacitors in actual circuits

•But inductors respond to changes in the current through them

•A simple series RL circuit has the Kirchhoff Law equation:

31.10. Energy Stored in a Magnetic Field

•The energy stored in an inductor can be found by integrating the power into the inductor over time

31.11. Energy Density of a Magnetic Field

•For an inductor the energy density is

•So the energy density of the magnetic field inside a solenoid is

•The energy density u is independent of the shape of the inductor

•This is similar to the energy density for a capacitor

31.12. Mutual Inductance

•When two coils are placed nearby, some of the magnetic flux from each can penetrate the other

•This means that a change in current through one will affect the magnetic flux and thus the current through the other

•This situation can alter the inductance of each inductor

•It is called mutual inductance

•The mutual inductance of coil #2 due to a current in #1 is:

•Which has the same form as the self-inductance

•Rearranging the mutual inductance equation:

*And since the M values are due to the fraction of the flux from each coil that penetrates the other coil