Phy 213: General Physics III

Chapter 30: Induction & Inductance

Lecture Notes

Electromagnetic Induction• We have observed that force is exerted on a charge by

either and E field or a B field (when charge is moving):

• Consequences of the Lorentz Force:– A B field can exert a force on an electric current (moving charge)– A changing B-field (such as a moving magnet) will exert a magnetic

force on a static charge, producing an electric current → this is called electromagnetic induction

• Faraday’s contribution to this observation:– For a closed loop, a current is induced when:1. The B-field through the loop changes2. The area (A) of the loop changes3. The orientation of B and A changes

on a charge {together this is the Lorentz Force}F = qE + qv B

q

v

N

S

B

F

q

v

N

S

B

F

• A current is induced ONLY when any or all of the above are changing

• The magnitude of the induced current depends on the rate of change of 1-3

Moving charge

Moving magnet

Magnetic Flux• Faraday referred to changes in B field, area and orientation

as changes in magnetic flux inside the closed loop• The formal definition of magnetic flux (B(analogous to

electric flux)

When B is uniform over A, this becomes:

• Magnetic flux is a measure of the # of B field lines within a closed area (or in this case a loop or coil of wire)

• Changes in B, A and/or change the magnetic fluxFaraday’s Law: changing magnetic flux induces

electromotive force (& thus current) in a closed wire loop

B = B dA

A

B

B = BA cos

Faraday’s Law• When no voltage source is present, current will flow around a

closed loop or coil when an electric field is present parallel to the current flow.

• Charge flows due to the presence of electromotive force, or emf () on charge carriers in the coil. The emf is given by:

• An E-field is induced along a coil when the magnetic flux changes, producing an emf (). The induced emf is related to:– The number of loops (N) in the coil– The rate at which the magnetic flux is changing inside the loop(s), or

Note: magnetic flux changes when either the magnetic field (B), the area (A) or the orientation (cos ) of the loop changes:

d d = E d = -N = -N BA cos

dt dtB

d dB=A cos

dt dtB

d dA=B cos

dt dtB

d cosd

=BAdt dt

B

coil = E d = iR

ds

E i

Changing Magnetic Field

dB-NA cos

dt

A magnet moves toward a loop of wire (N=10 & A is 0.02 m2). During the movement, B changes from is 0.0 T to 1.5 T in 3 s (Rloop is 2 ).

1) What is the induced in the loop?2) What is the induced current in the loop?

Changing Area

A loop of wire (N=10) contracts from 0.03 m2 to 0.01 m2 in 0.5 s, where B is 0.5 T and is 0o (Rloop is 1 ).

dA-NB cos

dt

1) What is the induced in the loop?

2) What is the induced current in the loop?

Changing Orientation

A loop of wire (N=10) rotates from 0o to 90o in 1.5 s, B is 0.5 T and A is 0.02 m2 (Rloop is 2 ).

1) What is the average angular frequency, ?2) What is the induced in the loop?3) What is the induced current in the loop?

( )

( )

d cos-NAB

dt

d cos ωt-NAB

r

dt

o

Lenz’s Law• When the magnetic flux changes within a loop of wire, the

induced current resists the changing flux• The direction of the induced current always produces a

magnetic field that resists the change in magnetic flux (blue arrows)

• Review the previous examples and determine the direction of the current

B

Magnetic flux, B

B

Increasing B

i

B

Increasing B

i

Operating a light bulb with motional EMFConsider a rectangular loop placed

within a magnetic field, with a moveable rail (Rloop= 2 ).

B = 0.5 Tv = 10 m/sL = 1.0 m

Questions:1) What is the area of the loop?2) How does the area vary with v?3) What is the induced in the loop?4) What is the induced current in the loop?5) What is the direction of the current?

Force & Magnetic Induction

What about the force applied by the hand to keep the rail moving?• The moving rail induces an electric current and also produces

power to drive the current:

P = .i = (5 V)(2.5 A) = 12.5 W• The power (rate of work performed) comes from the effort of the

hand to push the rail– Since v is constant, the magnetic field exerts a resistive force on the rail:

The force of the hand can be determined from the power:

Net hand B hand BF = F + F = 0 or F = F

hand hand

PP = F v F =

v

hand Bms

12.5 WF = = 12.5 N =F

10

BF

handF

Generators & Alternating Current• Generators are devices that utilize electromagnetic

induction to produce electricity• Generators convert mechanical energy into

electrical energy– Mechanical energy is utilized to either:

• Rotate a magnet inside a wire coil• Rotate a wire coil inside a magnetic field

– In both cases, the magnetic flux inside the coil changes producing an induced voltage

– As the magnet or coil rotates, it produces an alternating current (AC) {due to the changing orientation of the coil and the magnetic field}

• Motors and Generators are equivalent devices– A generator is a motor running in reverse:

Maxwell’s EquationsTaken in combination, the electromagnetic equations are

referred to as Maxwell’s Equations:

1. Gauss’ Law (E)

2. Gauss’ Law (B)

3. Ampere’s Law

4. Faraday’s Law

enc

o o

ρdVqE dA = =

B dA = 0

o enc o o o

dq dB d = i = = E dA

dt dt

o o

EB d = dA

t

Bd dE d = - =- B dA

dt dt

B

E d = - dAt

Significance of Maxwell’s Equations1. A time changing E field induces a B field.

2. A time changing B field induces an E field.

3. Together, 1 & 2 explain all electromagnetic behavior (in a classical sense) AND suggest that both E & B propagate as traveling waves, directed perpendicular to each other AND the propagation of the waves, where:

and

The product, oo, has special significance:

or

22

o o 2

EE =

t

22

o o 2

EE =

t

22

o o 2

BB =

t

8 mwave s

o o

1v = = 2.99x10 = c

o o o o

o o 2wave

1 =

v