chapter 29 electromagnetic induction. induced current you mean you can generate electricity this...
TRANSCRIPT
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Chapter 29
Electromagnetic Induction
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Induced current
You mean you can generate electricity this way??!
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For my next magic act…
Note: No moving parts
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Summary
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Faraday’s Law of InductionAn emf is induced when the number of magnetic field lines that pass through the loop changes
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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
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Magnetic Flux
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Faraday’s Law (restated)Emf is induced whenever ΦB changes
The minus sign will be explained later
€
ξ =−dΦ
dt
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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
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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?
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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
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Lenz’s LawAn induced current has a direction such that the B field due to the current opposes the change in the magnetic flux
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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
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The Logic
Bext:
Bext: increasing
BI: (to oppose the increase)
I: counterclockwise (view from left)
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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
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The Logic
Bext:
Bext: decreasing
BI: (to slow down the decrease)
I: clockwise (view from left)
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Summary
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Direction of currentWhat is the direction of current in B when the switch S is closed?
I
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Do it yourself!
Which way do the currents flow?
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What is the current?
Resistance: R
ξ =−dBA
dt= −B
dA
dt
but dA
dt= −Lv
⇒ ξ = BLv
⇒ I =ξ
R=
BLv
R
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What is the force?
Resistance: R
rF =I
rL ×
rB
⇒ F =ILB=(BLvR
)LB
⇒ F =B2L2v
R(Pulling you back!!!)
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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!
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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)
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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)
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Displacement current density
€
JD =ID
A
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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
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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
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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.
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Eddy Currents
Eddy currents want to stop whatever you are doing!
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Which one falls faster?
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Movie
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Faraday’s Law (modern form)
ξ is really just rE ⋅d
rs—∫
Therefore, we have:
rE ⋅d
rs—∫ =−
dΦB
dt
rE : Induced electric field
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Magnetic materials
Diamagnetism
Paramagnetism
Ferromagnetism
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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
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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
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MovieLiquid Oxygen
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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
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Details
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Picture
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Applications of Faraday’s Law
Power plants
Flashlight with no battery
Toothbrush?
Transformers (a.c. versus d.c.)
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The wonders of magnetic field
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View from afar
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Big magnetic field
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