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
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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.
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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)∫
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rB ⋅d
r s = μ 0Iencl∫ (Ampere's Law)
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⇒ 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?
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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
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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