8.022 (E&M) – Lecture 14
Topics:
„
Electromagnetic Inductance
„
Faraday’s and Lentz’s laws
Last time
„
Parallel between Electric and Magnetic Fields
„
Toward Maxwell’s equations:
⎧ ∇ i E = 4πρ
⎪
⎨
⎪∇ × E = 0
⎩
⇔
∇iB = 0
⇔
∇×B =
„
Vector Potential: E = −∇ φ
„
Biot-Savart:
G. Sciolla – MIT
B =
I
c
∫ dl
wire
4π
J
c
⇔
×
B ≡ ∇×A
rˆ
r2
8.022 – Lecture 14
2
1
Moving rod in uniform B
„
Let’s move a conducting rod in a uniform B
„
„
„
Charges move with velocity v//x axis
B//y axis
What happens?
EF1
++
++
E2
v
= q × B = qE1
c
B
B
vv
---
FLorentz
1) Lorentz force:
2) Electric field E1 causes separation of charges on the wire
3) Separation of charges creates an opposite electric field E2 that
exactly compensates E1 and equilibrium is established:
v
E2 = − × B
c
G. Sciolla – MIT
8.022 – Lecture 14
3
Moving a loop in uniform B
„
Now move a rectangular loop of wire in B
„
„
Same velocity
Same B
E2
+
+
+
+
„
v
What happens?
„
„
„
E1
-
Lorentz force Æ E1
E1 Æ separation of charges on the wire
Separation of charges creates opposite electric field E2= -E1:
G. Sciolla – MIT
8.022 – Lecture 14
4
2
What if B is non uniform?
„
Now move the rectangular loop of wire in non uniform B
„
„
„
Velocity v
B = B0 above - - B = 0 below - - -
+
+
E1
-
I
„
„
„
„
„
v
What happens?
Lorentz force Æ E1
E1 Æ separation of charges on the wire
Separation of charges creates charges to flow in the loop (no
opposing force in the bottom part!)
This phenomenon is called electromagnetic induction
G. Sciolla – MIT
8.022 – Lecture 14
5
Comments on induction
Please notice the following:
„
+
+
End of electrostatics!
∫
E idl ≠ 0 or ∇ × E ≠ 0
loop
„
FB
E1
-
I
v
The current flowing in top leg of the loop will feel a
force FB from B pointing up
„
Lentz’s law
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8.022 – Lecture 14
6
3
Induced emf
„
Consider a sliding conducting bar on rails closed on a
resistor R in a region of constant magnetic field B
--
B
v
L
++
„
R
I
x
Charge separation in the bar will induce current Æ e.m.f.
e .m .f . =
„
1
q
W (− → +) =
1
q∫
+
−
F ids =
Current flowing in the loop: I =
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1
c
∫ (v × B )ids
+
−
=
vBL
c
vBL
cR
8.022 – Lecture 14
7
Faraday’s law
„
„
„
vBL
BL dx
=
EMF in the loop: e .m .f . =
c
c dt
Magnetic flux in the rectangle is defined as: ΦB = Blx
Combine the two keeping in mind that given the direction
of v, flux decreases with time:
Æ Faraday’s law: e .m .f . = −
„
1 ∂ΦB
c ∂t
The minus sign is important: Lentz’s law
„
It indicates that the direction of the current is such to oppose the
changes in flux of B: ~”electromagnetic inertia”
G. Sciolla – MIT
8.022 – Lecture 14
8
4
Thoughts on Lentz’s law
--
B
v
L
++
I
e .m .f . = −
1 ∂ΦB
c ∂t
x
Lentz’s law:
The current generated in wire opposes changes in flux of B
„
„
v is LÆR:
„ Flux of B decreases over time Æ e.m.f. is created with direction
that compensates this change: counterclockwise
v is RÆL:
„ Flux of B increases over time Æ e.m.f. is created with direction
that compensates this change: clockwise
G. Sciolla – MIT
8.022 – Lecture 14
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Another way of looking at Lentz
--
B
v
L
I
e .m .f . = −
1 ∂ΦB
c ∂t
++
When current flows in magnetic field it feels a force
Lentz’s law: the force will be will try to slow down the bar
„ If I clockwise:
„
„
It creates a B pointing into the board Æ I x B points to the left
If I counterclockwise:
„
It creates a B pointing out of the board Æ I x B points to the right
G.
Sciolla
NB:
the– MIT
– sign
8.022
– Lecture
14
in Lentz’s law
is what
allows
conservation of energy10
5
General proof of Faraday’s law
„
Consider a loop of arbitrary shape moving with velocity v through a
static magnetic field B
Loop at time t
Loop at time t+∆t
„
„
B
dl
v+∆t
∫
At time t, the flux through the loop is: ΦB = B ida
S
How does it change when t Æ t+∆t?
∆ΦB = ΦB (t + ∆t ) − ΦB (t ) = Φ ribbon =
„
∫
B ida
ribbon
On the ribbon:
da = (v ∆t ) × dl
G. Sciolla – MIT
8.022 – Lecture 14
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Proof of Faraday’s law(2)
„
This means that:
∆ΦB =
∫
ribbon
„
B ida =
∫
B i(v ∆t ) × dl =
ribbon
Using the identity a i(b × c ) = (a × b )ic
∆ΦB =
∫
∆t B i(v × dl ) = ∆t
loop
For ∆t → 0:
∫
∆t B i(v × dl )
ribbon
∫
we obtain:
B × v idl = −∆tloop
loop
∫
v × B idl
ribbon
∂ΦB
⎛v
⎞
= −c ∫ ⎜ × B ⎟idl
∂t
⎠
loop ⎝ c
Since v/c x B is the magnetic force for unit charge
Æ its line integral on the loop is the work necessary to move a unit
charge around the wire: e.m.f!
1 ∂ΦB
Æ e .m .f . = −
c ∂t
G. Sciolla – MIT
8.022 – Lecture 14
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6
Work from B???
„
„
Faraday’s law:
e .m .f . = −
1 ∂ΦB
c ∂t
This means that v / c × B integrated over the loop is the work that
we have to do to move a unit charge around the loop
„
But last time we proved that B cannot do work
„
Are these 2 statements inconsistent???
„
„
No, the work done to move the charges is not done by B
It’s done by whoever is moving the loop in B
G. Sciolla – MIT
8.022 – Lecture 14
13
Verification of Faradya’s law
„
„
1 ∂Φ
B
Faraday’s law: e .m .f . = −
c ∂t
What does it mean?
„
„
„
E.m.f. Is produced when the flux of B changes over time
Æ area of the loop cannot be null!
Demo H1:
- move loop in B
- current flows in wire
- if we used instead a
wire with 0 area: no I
B
vs.
A
G. Sciolla – MIT
8.022 – Lecture 14
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7
“Relativity”
„
What if loop is static and B changes?
„ Relativity tells me that we should get the same result
„
„
Does this make sense?
„
„
„
Same problem from another reference frame
Charges do not move in the other reference frame
What causes the force? The induced electric field
Since e.m.f. is the work necessary to move a unit charge
around the loop:
e .m .f . =
„
∫C E idl
Demo H3: magnet bar moving in the loop
G. Sciolla – MIT
8.022 – Lecture 14
15
Application of Lentz’s law
„
H5: disk falling in a magnetic field B
„ Create B with e electromagnets (solenoid on Fe core)
B
„
What happen if we drop a disk of conductor?
„
„
With and without B
What if we drop a full disk
What if we drop a disk with a cut?
„
G. Sciolla – MIT
8.022 – Lecture 14
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8
Explanation
„
Falling Loop:
„ B perpendicular to loop is limited in space Æ flux
changes during fall Æ induced I
Æ loop will levitate
(Eddie currents)
„
Falling Disk
„
„
Iback
Ifront
Will it slow down?
Falling open ring
„
Will it levitate?
G. Sciolla – MIT
8.022 – Lecture 14
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More demos on Faraday’s law
„
„
H8: current generated by a solenoid
„ Where to put the loop of wire to have current?
„ Remember: B of solenoid is 0 outside
„ Switch I on and off
F
I
B
H22 Levitating rings
G. Sciolla – MIT
A
A
8.022 – Lecture 14
18
9
More demos on Faraday’s law
„
H15a: current generated by a solenoid
„ Spinning disk of conductor
„ Magnet sitting on top separated by a plastic sheet
„ When disk starts spinning, magnet levitates
„ Why?
N
+++
---
v
G. Sciolla – MIT
8.022 – Lecture 14
19
Faraday’s law in differential form
„
Faraday’s law in integral form: e .m .f . = −
„
Right term (apply Stokes): e .m .f . =
„
Left term:
„
Since this is valid for any surface:
„
„
1 ∂ΦB
c ∂t
∫C E idl =S∫ ∇ × E ida
1 ∂ΦB
1 ∂
=−
B ida
c ∂t
c ∂t S∫
⎛
1 ∂B ⎞
Æ ∫ ⎜⎜ ∇ × E + c ∂t ⎟⎟ida = 0
S ⎝
⎠
−
∇×E = −
1 ∂B
c ∂t
curl E is not longer zero: bye bye electrostatics!
Explicit link between E and B, as in relativity!
G. Sciolla – MIT
8.022 – Lecture 14
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10
Another step toward Maxwell’s
equations…
„
All the equations in differential form that we found so far:
⎧ ∇ i E = 4πρ
⎪
⎪∇ i B = 0
⎪
⎨∇ × E = − 1 ∂B
⎪
c ∂t
⎪
4π
⎪∇ × B =
J
c
⎩
„
Å Relates E and charge density (ρ) - Gauss
Å Magnetic field lines are closed
Å Change in B creates E - Faraday
Å Relates B and its sources (J) - Ampere
Another step toward Maxwell’s equations: one last
missing ingredient… Can you guess what?
„
„
Symmetry will guide you… Hint:
Or vector calculus… Hint: take the divergence of Faraday’s law…
G. Sciolla – MIT
8.022 – Lecture 14
21
Summary and outlook
„
Today:
„
„
Faraday’s (and Lentz’s) law:
e .m .f . = −
1 ∂ΦB
„
Integral form:
„
Differential form: ∇ × E = −
c ∂t
1 ∂B
c ∂t
Next time:
„
Mutual and self inductance
G. Sciolla – MIT
8.022 – Lecture 14
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11