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Electromagnetism Physics 15b

Lecture #14 Induction

Purcell 7.1–7.5

What We Did Last Time Moving charge near current receives force proportional to its velocity   Identified as Lorentz force due to magnetic field   Electricity and magnetism are connected through relativity

Relativistic transformation of E and B fields

  Linear transformation   Consistent with the force transformation

′F = F+F⊥ γ

′E= E

′E⊥ = γ (E⊥ + β ×B⊥ )

′B= B

′B⊥ = γ (B⊥ − β ×E⊥ )

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Today’s Goals Introduce electromagnetic induction   Changing magnetic field produces

electromotive force   First step into non-static E&M

Discuss Faraday’s Law   First in integral form

  Magnetic flux emf   Then in differential form

  B field E field   Applications: AC power generator   Eddy current

Michael Faraday (1791–1867)

Moving Rod in B Move a conducting rod perpendicular to a uniform B field   Movable charge q in the rod receives Lorentz force

  Net charges appear at the ends   This in turn produces E field opposite

to v × B

Total force on charge q is

B

v

++ ++

– – – –

F

E

F =

qc

v ×Bq

Ftotal =

qc

v ×B + qE = 0 Equilibrium because current cannot keep flowing on this rod

E = −

1c

v ×B must appear in the rod

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Moving Loop in B Now, move a rectangular conducting loop   Top and bottom bars feel the same effect

What happens when the loop goes outside the B field?   No more Lorentz force in the bottom rod   Top rod is still moving the charges to

produce E field

Current I will flow around the loop

This is called electromagnetic induction   The current is induced by the loop’s motion   Change of B is crucial

B

v

+ + + + + +

– – – – – –

F q

E F q

E

v

+ + + + + +

– – – – – –

F q

E

q

I

Faraday’s Law Potential difference across the top bar is

  It’s as if we have an emf in it

The emf appear only while the loop is crossing the boundary of the B field   Consider the flux of B through the loop:

  We can express the emf as

B

v

+ + + + + +

– – – – – –

F q

E

q

I

w

Δφ = Ew =

vBwc

E =

vBwc

x

Φ ≡ B ⋅da

loop∫ = Bwx

dΦdt

= Bwdxdt

= −Bwv

E = −

1c

dΦdt

Faraday’s Law

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Lenz’s Law

What’s the minus sign in ?

  It comes out naturally if you define the flux and emf consistently (w/ right-hand rule)

Lenz’s Law helps you check the sign   The induced current creates additional

magnetic field that opposes changes of flux   Since Φ is decreasing in this example,

the induced B has the same direction as the external B   The induced current receives force from the magnetic field that

slows down the motion   The magnetic force on the top bar points upward

This is an example of “Nature opposes changes”

B

v

+ + + + + +

– – – – – –

F q

E

q

I

w

x

E = −

1c

dΦdt

Moving-Rod Circuit A circuit is made of a movable metal rod on two rails   Rod moves with v   Area of the circuit

  Flux is

  Faraday’s law This result in current I

R L

x

vB

A = xL q

F

qE

E = −

1c

dΦdt

=BvL

c

dAdt

= vL

dΦdt

= −BdAdt

= −BvL

I =

E

R=

vBLcR

The resistor dissipates IE =

E 2

R=

v 2B2L2

c2RWhere did this energy com from?

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Lorentz force on the induced current is

  Direction of F opposite to v   To keep the bar moving,

we must pull the bar with this much force to the right

How much work per unit time do we have to do?

R L

x

vB

I

F

P = Fv =

v 2B2L2

c2R= IE Exactly the power

dissipated in the resistor

Moving-Rod Circuit

F =

Ic

L ×B

F =

ILBc

=vB2L2

c2R

Proof of Faraday’s Law A loop of an arbitrary shape is moving with velocity v through a static magnetic field B   Flux through the loop at time t and t + Δt

Consider the volume enclosed by S, S′, and the “ribbon” R between them   Since div B = 0, the total flux must be zero

  Infinitesimal area da on the ribbon is

Φ(t) = B ⋅da

S∫ , Φ(t + Δt) = B ⋅da′S∫

0 = Φ(t) − Φ(t + Δt) + B ⋅da

R∫ da = vΔt( ) × dL

dL

vΔt

S

S′

B ⋅da

R∫ = Φ(t + Δt) − Φ(t)

B ⋅da

R∫ = B ⋅ (vΔt × dL)loop∫

dΦdt

= B ⋅ (v × dL)loop∫

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Proof of Faraday’s Law

  Use math identity:

Imagine a unit charge on, and moving with, the loop   The Lorentz force acting on it is v×B

The loop integral represents the work the Lorentz force would do if a unit charge were moved around the wire, i.e. the emf

a ⋅ (b × c) = b ⋅ (c × a) = c ⋅ (a ×b)

dΦdt

= dL ⋅ (B × v)loop∫ = − (v ×B) ⋅dL

loop∫

E = −

1c

dΦdt

dΦdt

= B ⋅ (v × dL)loop∫

dL

vΔt

S

S′

Relativity Faraday’s law says “flux changes emf happens”   It doesn’t say why the flux changes

What if B field itself changes while the loop is static?   Relativity: we should get the same result

  Same problem in another reference frame

Test this experimentally

  Current is generated in a loop of wire when   Magnet approaches   Current flows in a nearby wire

A N S A

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Ways to Change Flux Magnetic flux Φ depends on the B field, the size, shape and angle of the loop   Simple case: a flat loop of

area A in a uniform B field

Induction may occur because of   Changing B field   Changing area A of the loop   Changing angle θ between B and the loop

B

A θ

Φ = B ⋅dA∫ = BAcosθ

E = −

1c

dΦdt

= −1c

ddt

(BAcosθ)

AC Power Generator Alternate Current (AC) generators are very simple   For a loop area A rotating

with angular velocity ω

  If the loop has N turns

  For commercial 60 Hz power generator

Φ = BAcosθ = BAcosω t

E = −

1c

dΦB

dt=

BAωc

sinω t

E =

NBAωc

sinω t

ω = 2π f = 120π sec

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Eddy Currents Faraday’s Law works in conductor of any shape   Consider a simple plate   Increase B field Rotating current   Move the plate into a B field Ditto

Rotating current in a continuous body of conductor due to changing B field is called the eddy current   Direction is given by Lenz’s Law

Eddy currents always slow down the change   Used for braking systems of various machines

Differential Form of Faraday

Faraday’s law in integral form:   The emf can be expressed as

  RHS is

  Apply this to an arbitrary, but stationary, surface S

curl E is no longer zero — Leaving electrostatics

E = −

1c

dΦdt

E = E ⋅ds∫ = (∇ ×E) ⋅da

S∫

−

1c

dΦdt

= −1c

ddt

B ⋅daS∫

lhs − rhs = ∇ ×E +

1c∂B∂t

⎛⎝⎜

⎞⎠⎟⋅da

S∫ = 0 ∇ ×E = −

1c∂B∂t

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Maxwell’s Equations All the equations in differential form that we found so far:

Another step toward Maxwell’s equations One last term is missing — Where is it?   Hint #1: Symmetry   Hint #2: Look at the Lorentz transformation of fields

∇ ⋅E = 4πρ∇ ⋅B = 0

∇ ×E = −1c∂B∂t

∇ ×B =4πc

J

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪

Relates E and charge density ρ — Gauss

No magnetic monopoles

Change in B creates E — Faraday

Relates B and current density J — Ampere

Summary Induction: emf when the magnetic flux in a loop changes

Faraday’s Law

  Sign of the emf follows Lenz’s Law: the induced current opposes the change of the flux

Differential form:

  curl E no longer zero!   Just one last step before completing

Maxwell’s equations

E = −

1c

dΦdt

∇ ×E = −

1c∂B∂t

B

v

+ + + + + +

– – – – – –

FL q

E

q

I

w

E =

vwBc

x