Chapter 29

Electromagnetic Induction

Induced current

You mean you can generate electricity this way??!

For my next magic act…

Note: No moving parts

Summary

Faraday’s Law of InductionAn emf is induced when the number of magnetic field lines that pass through the loop changes

Magnetic Flux

ΦB =

rB ⋅d

rA∫

If rB is uniform and parallel to

rA

ΦB =BA

Similar to electric flux

Unit: Weber

1Wb =1Tm2

If rB is uniform: ΦB =

rB⋅

rA=BAcosθ

Magnetic Flux

Faraday’s Law (restated)Emf is induced whenever ΦB changes

The minus sign will be explained later

€

ξ =−dΦ

dt

What if you have a coil?ξ =−N

dΦ1

dt= −

dΦN

dt (Coil of N turns)

where

Φ1 : flux of one turn

ΦN = NΦ1 : flux of N turns

EMF induced in a solenoidA=1m2, N=2000 turnsAn external magnetic field of B = 1mT is removed suddenly in 1s. What is the emf generated?

Solution

What are Φi and Φ f for one turn?

(initial and final flux)

Φi = Bi A = (10−3T )(1m2 ) = 10−3Wb

Φ f = B f A = (0T )(1m2 ) = 0Wb

A=1m2, N=2000 turnsAn external magnetic field of B = 1mT is removed suddenly in 1s. What is the emf generated?

ξ =−NdΦB

dt≈ −N

ΔΦB

Δt

⇒ ξ ≈ −NΦ f − Φ i

Δt= −(2000)

(0 −10−3)Wb

1s⇒ ξ ≈ 2V

Lenz’s LawAn induced current has a direction such that the B field due to the current opposes the change in the magnetic flux

Lenz’ Law – Example 1

When the magnet is moved toward the stationary loop, a current is induced as shown in aThis induced current produces its own magnetic field that is directed as shown in b to counteract the increasing external flux

The Logic

Bext:

Bext: increasing

BI: (to oppose the increase)

I: counterclockwise (view from left)

Lenz’ Law – Example 2

When the magnet is moved away the stationary loop, a current is induced as shown in cThis induced current produces its own magnetic field that is directed as shown in d to counteract the decreasing external flux

The Logic

Bext:

Bext: decreasing

BI: (to slow down the decrease)

I: clockwise (view from left)

Summary

Direction of currentWhat is the direction of current in B when the switch S is closed?

I

Do it yourself!

Which way do the currents flow?

What is the current?

Resistance: R

ξ =−dBA

dt= −B

dA

dt

but dA

dt= −Lv

⇒ ξ = BLv

⇒ I =ξ

R=

BLv

R

What is the force?

Resistance: R

rF =I

rL ×

rB

⇒ F =ILB=(BLvR

)LB

⇒ F =B2L2v

R(Pulling you back!!!)

Displacement CurrentThere is something wrong with Ampere’s Law

€

rB ⋅d

r s = μ 0Iencl∫ (Ampere's Law)

Depending on the surface, Iencl could be either zero or non-zero. Inside the capacitor there is no conduction current.

€

rB ⋅d

r s ∫ = μ 0Iencl (plane) = μ 0Iencl (bulge)

Iencl (plane) =dq

dt,

but there is no charge in the empty space,

Iencl (bulge) = 0.

Contradiction!

Displacement CurrentWe need to account for the E field in Ampere’s Law.

€

Two types of currents :

Iencl = IC + ID

IC =dq

dt (conduction current)

ID = ε 0

dΦ E

dt (displacement current)

where Φ E =r E ⋅d

r A (electric flux)∫

€

rB ⋅d

r s = μ 0Iencl∫ (Ampere's Law)

€

⇒ r

B ⋅dr s = μ 0(IC + ID )∫ (Ampere Maxwell Law)

Does it work?

€

Apply the generalized Ampere's Law to the bulging surface :

IC (bulge) = 0 on that surface, but ID is non - zero.

ID (bulge) = ε 0

dΦ E

dt= ε 0

d(EA)

dt= ε 0

d

dt(σ

ε 0

A) =dq

dt

⇒ ID (bulge) = IC (plane)

€

IC (plane) =dq

dtID (plane) = 0

⎧ ⎨ ⎪

⎩ ⎪

IC (bulge) = 0

ID (bulge) =dq

dt

⎧ ⎨ ⎪

⎩ ⎪

⇒ Iencl (plane) = Iencl (bulge)

⇒r B ⋅d

r s ∫ = μ 0Iencl (plane) = μ 0Iencl (bulge)

Displacement current density

€

JD =ID

A

ExampleWhat is the B field at point a given IC?

€

Iencl = ID = ε 0

dΦ E

dt= ε 0

d(Eπr2)

dt

E =σ

ε 0

=q

πR2ε 0

⇒ ID = ε 0

d

dt(

r2

R2

q

ε 0

) =r2

R2

dq

dt=

r2

R2 IC

€

rB ⋅d

r s = μ 0Iencl∫

⇒ B(2πr) = μ 0

r2

R2 IC

⇒ B =μ 0r

2πR2 IC

Ampere-Maxwell law

rB⋅d

rs—∫ =μ0 I + μ0ε0

dΦE

dtAssume the capacitor has radius r.

At distance r around the wire:

Bw (2πr) =μ0 I ⇒ Bw =μ0 I2πr

The E field inside the capacitor:

E =σε0

=q

Aε0

⇒ ΦE =EA=qε0

At distance r around the capacitor:

Bc(2πr) =μ0ε0

dΦE

dt=μ0

dqdt

=μ0 I

⇒ Bc =μ0 I2πr

=Bw

Isolated rod vs closed circuit

€

Einstein observed :r F = q

r v ×

r B = q

r E v

where r E v =

r v ×

r B .

The B field in our stationary frame

looks like an E field in the frame of

the moving charge.

Eddy Currents

Eddy currents want to stop whatever you are doing!

Which one falls faster?

Movie

Faraday’s Law (modern form)

ξ is really just rE ⋅d

rs—∫

Therefore, we have:

rE ⋅d

rs—∫ =−

dΦB

dt

rE : Induced electric field

Magnetic materials

Diamagnetism

Paramagnetism

Ferromagnetism

Diamagnetism No net magnetic dipole for each atom when B=0.

When magnetic field is switched on, an induced magnetic dipole points in the opposite direction to B due to Lenz’s Law, this causes the object to be repelled.

Copper, lead, NaCl, water, superconductor

Paramagnetism• Each atom already has a permanent dipole moment.• This dipole will align with external B field. • Forces points from weak field to strong (attraction).

Oxygen, aluminum, chromium, sodium

MovieLiquid Oxygen

Ferromagnetism• Each atom has a net magnetic dipole.• Atoms arrange themselves into domains.• External fields can affect the alignment of the

domains.• Heat can destroy the domains.• Magnets are made this way.

Insert Picture

B Field

Iron, Permalloy

Details

Picture

Applications of Faraday’s Law

Power plants

Flashlight with no battery

Toothbrush?

Transformers (a.c. versus d.c.)

The wonders of magnetic field

View from afar

Big magnetic field