Faraday Law (of Induction), (Maxwell-)Faraday Law &

Lenz LawLecture 22

Announcements

• Reading for Monday: 28-6 to 28-8

• Exam Thursday next week (3/5).

• Today’s material is last new material on MT3.

Faraday Law (of Induction)

• Changes in the Magnetic Flux through a loop induces an EMF:

• Transformer EMF: Changes in B generate E

• Motional EMF: Changes in M generate FB

E = � d

dt�B

�B ⌘Z

Md2A n̂ · ~B

Motional EMF: Generator

E = 2vLB cos ✓Result:

Using magnetic flux…• Magnetic Flux:

• EMF:

• Generator:

�B ⌘Z

Md2A n̂ · ~B

E = � d

dt�B

E = �ABd

dtcos�(t) = � d

dtAn̂ · ~B

Faraday Law of Induction

• Changes in the Magnetic Flux through a loop induces an EMF:

• Transformer EMF: Changes in B generate E

• Motional EMF: Changes in M generate FB

E = � d

dt�B

�B ⌘Z

Md2A n̂ · ~B

But there are two mechanisms for changing the flux!

I

~v

�~vI

(A)Loop moves Motional EMF

(B) B changes (wire moves) Transformer EMF

�(A)B (t) = �(B)

B (t)

But there are two mechanisms for changing the flux!

• Faraday Law:

• EMF’s are equal!

• How could this be?

• v = 0, Must be E — Not conservative!

I

~v

�~vI

�(A)B (t) = �(B)

B (t)

E = � d

dt�B E(A) = E(B)

E ⌘I

@M~d` ·

⇣~E + ~v ⇥ ~B

⌘

Demo: Run experiment to test hypothesis!

I

~v

�~vI

(A) Loop moves (B) B changes (wire moves)

E(A) = E(B)?

(Maxwell) Faraday Law:• Faraday Law of Induction for a stationary loop

• (Maxwell-)Faraday Law (Maxwell Equations)

• New law of physics! Describes E and B field, not just loops!

E =

I

@M~d` ·

⇣~E + ~v ⇥ ~B

⌘= � d

dt

Z

Md2A n̂ · ~B

I

@M~d` · ~E = �

Z

Md2A n̂ · @

~B

@t

�~vI

Electrodynamics & the Maxwell Equations

• Gauss Law (E):

• Gauss Law (B):

• Ampère Law:

• Faraday Law:

I

Md2A n̂ · ~E = Qinside/✏0

I

Md2A n̂ · ~B = 0

I

@M~d` · ~B = µ0I +µ0✏0

d

dt

Z

Md2A n̂ · ~E

I

@M~d` · ~E = � d

dt

Z

Md2A n̂ · ~B

M = const

M = const

Which way does the current flow? (Lenz’s Law)

• Lenz’s Law = “How to get the signs right!”

• Lenz’s Law = Currents are generated to oppose the change that created it.

• Lenz’s Law = Newton’s Third Law + Energy conservation

Lenz’s Law: Example 1

Lenz’s Law: Example 1

Lenz’s Law: Example 1

I

Lenz’s Law: Example 1

I

I

F

Lenz’s Law: Example 2

Lenz’s Law: Example 2

Lenz’s Law: Example 2

Lenz’s Law: Example 2

Lenz’s Law: Example 2

Eddy Current Demos

~FMagnet

~FPlate

IPlate

Lenz Law