Lecture 11
Faraday Disk Dynamo
! + !v × B)
! =0
F! = q(E
Induced Currents,
Eddy Currents and
Maxwell’s Equations
dE = vB dr = ωrB dr
! R
1
ωrB dr = BωR2
|E| =
2
0
1
2
Eddy Currents
Faraday Disc Machine at ANU
2MA supply!!
We have learnt that changing magnetic fields can induce
electric fields in conductors - is this useful?
yes - e.g. hard-drive heads, metal detectors, electrical
power generators, credit-card readers, alternators in cars
etc etc
Can it be fun - also YES!
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4
Can Crusher
Driving too fast? - Arago’s Disk
20µF Capacitance charged to 10kV
discharged through a coil ?
A really big dB/dt
induces a big current in
the can
This current feels the
B-field and the can
thus experiences
inward force
How much energy?
Icoil
Coil
+
+
Can
+
Ican
+
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!
Maxwell’s Equations (Nearly)
! + !v × B)
!
Lorentz Force Law: F! = q(E
{
!
!
!
! · dA
! = qencl
E
0 Gauss’s Law
"0
! · dA
!=0
B
Think
w/o charge,
currents
Gauss’s Law for Mag.
! · d!l = E = − dΦB
E
dt
! · d!l = µ0 Ipenetrating
B
0
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Return to Ampère’s Law and Capacitors
Everything we know so far...
!
+
Faraday’s Law
Ampère’s Law
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! · d!l = µ0 Ipenetrating
B
Maxwell:
Let’s create
a fictitious
current
related to this
strengthening
electric field
so that Ampere’s
Law works
i.e. assume
that the current
exists even in
empty space
8
Displacement Current
!
Displacement Current: Part II
So let us write Ampère’s Law as
follows and then it works in
all situations:
! · d!l = µ0 Ipenetrating
B
C = !0 A/d V = E d
Capacitor Stuff Q = C V
Q = !0 A/d(Ed) = !0 ΦE
dΦE
dQ
= !0
Now, as current flows into the capacitor, the charge builds up: ic =
dt
dt
or
!
So our displacement current (which we require to equal to the input current):
!
! · d!l = µ0 (Ireal + Icrazy )
B
! · d!l = µ0 (Ireal + "0 dΦE )
B
dt
But, is it real ? ...
dΦE
id = !0
dt
If there is a real current in the space
then there should be an associated
magnetic field - is there?
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10
Magnetic field of capacitor
Finally, the pinnacle of EM...
We have solved this one before essentially what is the magnetic
field inside a conducting zone?
Maxwell’s Equations
{
(see Lect 8. - slide 12)
zero on axis, increasing radially to
end of conducting zone, then
falling off as 1/r
!
!
!
! · dA
! = qencl Gauss’s Law
E
"0
! · dA
!=0
B
Gauss’s Law for Mag.
! · d!l = E = − dΦB
E
Faraday’s Law
dt
!
! · d!l = µ0 (Ireal + "0 dΦE ) Ampère’s Law
B
dt
! + !v × B)
!
Lorentz Force Law: F! = q(E
If you measure it this is exactly what you see
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Hertz
James Clerk
Maxwell
1865 - unified E & M
with light
wait till
Thursday
for
final installment
1888 most
exciting Experiments
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Maxwell’s Last 2 equations...
no currents, no charges
Maxwell’s last two equations...
!
a
changing E-fields make
B-fields
! · d!l = µ0 "0 dΦE
B
dt
x=vt
E
?
!
!
changing B-fields make
E-fields
! · d!l = − dΦB
E
dt
changing E-fields make
B-fields
! · d!l = µ0 "0 dΦE
B
dt
B d = µ0 !0
dΦB
Ea = −
dt
dΦE
dt
= −µ0 !0 (v d) E
So, B = −µ0 !0 v E
= −B a v
!
changing B-fields make
E-fields
! · d!l = − dΦB
E
dt
x=vt
d
B
E
But also, E = −Bv
E = −Bv
So,
15
v=√
1
!0 µ0
?B
E
3.00 x 108 ms-1
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Ampère’s Law Practice
In this lecture and the next
!
! · d!l = µ0 I
B
Eddy Currents: We have covered Sect. 29.6 - this is not
examinable - read for interest
Displacement Current: Sect. 29.7 - not examinable
Maxwell’s Eqns and Speed of Light: Sect. 32-32.2. Details are not
examinable just the overall knowledge that light and E&M are
related. Recognition of M.E.s and their meaning is required
The last lecture will be on the speed of light and the momentum
of electromagnetic radiation and some important EM constants
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