CHAPTER 20CHAPTER 20Induced Voltages and InductanceInduced Voltages and Inductance

An electric current produces a magnetic field.

B = o I2r

Scientists in the 19th century saw that electricity (a relatively new discovery) and magnetism (a relatively old discovery) were related.

They hypothesized:

“If electricity (current) produces a magnetic field, then a magnetic field should be able to produce electricity.”

As with many initial hypotheses, this one was close to correct…but not exactly.

Michael Faraday (as in Farad) 1791-1867Michael Faraday (as in Farad) 1791-1867Faraday discovered the correct relationship between electricity and magnetism. The relationship came as a surprise to him and was discovered somewhat accidentally.

- The iron in the illustration is not a magnet.- With the switch open, it contains no magnetic field.-When the switch is closed, the primary coil will have current flowing and a B-field is established.

Use one of the 2 RHR’s to determine the direction of the B-field lines established in the iron.

B

- Notice the B-field extends all the way around the inside of the iron.- A current in the secondary coil wire develops only momentarily when the switch is closed (indicated by the Galvanometer) and then returns to zero.- A current in the secondary coil also develops (in the opposite direction) momentarily when the switch is opened.

Faraday: It is the change in a magnetic field that can produce current, not the magnetic field itself.

The change in B-field strength produces an emf (voltage) in the wire that then causes current.

Application:Application:

Current produced by a battery producing current elsewhere seems at first like an interesting novelty, but of little practical value.

However, a magnet alone moving through a coil of wire produces the same effect (current) for the same reason.

This is essentially how electricity has been produced for most of the past 150 years and how it is still produced today.

Demo: magnets, coil, galvanometer

Faraday’s Law of Magnetic InductionFaraday’s Law of Magnetic Induction = emf = voltageN = number of turns of wire in the loop = change in flux (Webers)t = change in time

= - Nt

Lenz’s Law: “the no free lunch law”Lenz’s Law: “the no free lunch law”The direction of the induced emf (voltage) is such

that it produces a current whose magnetic field opposes the change in magnetic flux through the

loop. That is, the induced current tends to maintain the original flux through the loop.

Example #1Example #1A Conducting Bar Moving Perpendicular to a Uniform A Conducting Bar Moving Perpendicular to a Uniform Magnetic FieldMagnetic Field

Size of loop increases as the bar is pulled right at a velocity v.

MagnitudeMagnitude

= t

=B At

= B Lxt

= B L v

Direction of CurrentDirection of Current

Downward B-field lines being added

Current creates a B-field with upward pointing B-field lines

Current goes up on right side of loop and down through

resistor on left. (CCW)

Lenz Law ExamplesLenz Law ExamplesExample #2Example #2Square Loop of Wire Pulled at Constant Velocity into a Square Loop of Wire Pulled at Constant Velocity into a Magnetic FieldMagnetic Field

v = xt

A = Area exposed to B-field = x L = B x L = B x L t t = = B v L

t

What will be the direction of current? Counter Clockwise

B’-field lines pointed up canceling out the increase in B-field lines pointing down.

NOTE: Current stops once loop is entirely in field.

Field Strength = B (down)

v

L

Example #3Example #3Stationary Loop I a Spacially Uniform Magnetic Field Stationary Loop I a Spacially Uniform Magnetic Field Whose Magnitude is Changing at a Constant RateWhose Magnitude is Changing at a Constant Rate

o = Bo A = B A = (B – Bo) A = B A

= = A

t ( )B

tFind Direction of CurrentFind Direction of Current

Decreasing BDecreasing B Increasing BIncreasing BCurrent produced B ‘ tends to restore B-field lines lost.

I = Clockwise

Current produced B ‘ tends to oppose B-field lines gained.

I = Counter Clockwise

t = t

B

A

t = 0

Bo

A

Example #4Example #4Loop of Wire Rotates at a Constant Rate about an Axis Loop of Wire Rotates at a Constant Rate about an Axis Perpendicular to a Uniform Magnetic FieldPerpendicular to a Uniform Magnetic Field

= 2 B l v = 2 B l v sin2 in formula because 2 wires of length l (left and right side of loop)

Remember:Remember:If loop is horizontal

and moving up (or down) the velocity component of

movement to the B-field is at a maximum.

Isin = 1

v

Rotation

Right half of loop

Magnetic Flux (Magnetic Flux ():): A measure of the B-field lines that actually pass through a loop of given area (A)

Flux ExampleFlux Example (Perpendicular Situation)

B = 20 T = 20 Wbm2

Wb (Weber) can be thought of as magnetic field lines.

A = .25m2

If B-field lines are perpendicular to the area of the loop, then the magnetic flux through the loop is:

= B A

= 20 x .25m2 Wbm2

= 5 Wb

A

B

Flux ExampleFlux Example (Angled Situation)

The B-field only “sees” an area of A cos• If = 90 Area exposed to B-field view is 0.

(cos(90)=0)• If = 0 Area exposed to B-field view is A.

(cos(0)=1) = B A

= B A cos

= (20 )(.25m2)(cos30)Wbm2

= 4.3 Wb

Notice: As loop rotated in a stationary B-field, the changed

from 5.0Wb to 4.3Wb. This will induce emf and current in the wire

(as long as the loops is rotating and the flux through it is

changing.)

Area remains equal to .25 meters2, but is at an angle of (30) with the horizontal while B (20T) is perpendicular () to the horizontal.

B A

Caution: = B A cos only whencertain angles are referenced

= B A sin Think!Think!

Faraday’s Law of Magnetic InductionFaraday’s Law of Magnetic Induction = emf = voltageN = number of turns of wire in the loop = change in flux (Webers)t = change in time

= - Nt

Lenz’s Law: “the no free lunch law”Lenz’s Law: “the no free lunch law”The direction of the induced emf (voltage) is such

that it produces a current whose magnetic field opposes the change in magnetic flux through the

loop. That is, the induced current tends to maintain the original flux through the loop.

As loop rotates CCW, gets smaller (less B-field lines intersect the opening).The current sets up additional B-field lines going down through the loop.Find direction of current.Find direction of Force on the current carrying wire in a magnetic field.

I

B

A

FB