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Chapter 32. Maxwell’s Equations. The electric field spreads into space proportional to the amount of static charge and how closely you space the static charges. Magnetic field lines are closed loops and always return to the source creating them. - PowerPoint PPT Presentation
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1
Chapter 32
2
Maxwell’s Equations
dt
EdJBor
AdEdt
disdB
dt
BdEorAdB
dt
dsdE
BorAdB
Eorq
AdE
enclosed
enclosed
enclosedenclosed
00
00
00
00
The electric field spreads into space proportional to the amount of static charge and how closely you space the static charges
Magnetic field lines are closed loops and always return to the source creating them
An electric field, resembling a magnetic field in shape, can be created by a time-varying magnetic field.
There are two ways to produce a magnetic field: 1) by a current and 2) by a time-varying electric field.
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Maxwell’s Equations for Vacuum
In vacuum, there is no charge so renc=0
Since no charge, no currents enclosed so J=0
Note the symmetry of the equations i.e. they look practically the same!! dt
EdB
dt
BdE
B
E
00
0
0
4
Without proof
AAA
2)(
5
Separating E from B
2
2
002
002
00
2
)(
0
)(
)(
dt
EdE
dt
Ed
dt
dE
dt
EdBandEBUT
Bdt
dEE
Bdt
d
dt
BdE
dt
BdE
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The Wave Equation
This equation is called a wave equation.
In order to simplify the math, let’s just work with 1-dimension i.e. in the x-direction
2
2
002
2
2
2
002
dt
Edxd
Ed
dt
EdE
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A solution is a sine function
)sin()sin(
)sin()cos()sin(
)sin()cos()sin(
)sin(),(
'
02
0002
02
002
2
2
2
02
002
2
2
2
0
2
2
002
2
tkxEtkxEk
tkxEtkxtd
dEtkxE
td
dtd
Ed
tkxEktkxxd
dkEtkxE
xd
dxd
Ed
tkxEtxE
nSoldt
Edxd
Ed
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Using the wavelength and wave number
ff
k
k
f
k
tkxEtkxEk
22
2
2
)sin()sin(2
002
02
0002
fl is the speed of the wave, which we will call c
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An important revelation
smc
cfk
k
/1031
)(1
8
00
222
2
00
200
2
10
However, I could have solved for B
2
2
002
002
002
0000
00
)(
0
)(
)(
dt
BdB
dt
Bd
dt
dB
dt
BdEandBBUT
Edt
dBB
Edt
d
dt
EdB
dt
EdB
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Very similar functions
)sin(),(
)sin(),(
0
0
tkxBtxB
tkxEtxE
So the solution for E and B are mathematically similar
Now, let’s assign a direction for E in the y-direction
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Using the curl
0000
00
00
0
0
)cos(ˆ)cos(ˆ
)cos(ˆ)cos()(ˆ
ˆ
)cos(ˆˆ
00
ˆˆˆ
)sin(ˆˆ),(
cBEBk
E
tkxBztkxkEz
tkxBztkxBzdt
Bd
zBdt
BdE
tkxkEzdx
dEzE
Edz
d
dy
d
dx
dzyx
E
tkxEyytxEE
y
y
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Implications
)sin(ˆ),(
)sin(ˆ),(
0
0
tkxBztxB
tkxEytxE
The wave is called transverse; both E and B are perpendicular to the propagation of the wave. The direction of propagation is in the direction of E x B.
E=cB The wave travels in vacuum with a definite and unchanging
speed What is the wave propagating through?
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“Common Sense” on Waves
Ocean waves propagate in water Sound waves propagate in air Mechanical waves propagate through material
where they are transmitting Ergo, the 19th Century physicists thought that
EM waves propagate through the “ether”. Ether surrounds us and we move through it
without any drag.
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Michelson-Morley Experiment
In 1887, Michelson and Morley invented an experiment to measure the speed of light in the direction of Earth’s motion and in the direction against Earth’s motion
If there is ether, then there should be a slight difference in the speed of light.
Michelson-Morley found NO evidence of any difference of the speed of light.
Why?
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The New Physics
Actually, if they trusted their equations, they would have seen that there is no need to have a medium
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A completely different direction
Einstein thought about the results of the MM experiment.
He assumed that there was no mistake and the c is always constant
The Postulates of Special Relativity1. The laws of physics are the same in every inertial frame of
reference2. The speed of light in vacuum is the same in all inertial
frames of reference and is independent of the motion of the source
This is the beginning of the new physics of the 20th century
From here, we can get E=mc2 and from there, quantum mechanics
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Waves in Matter
Recall =k0
=m
c=1/sqrt(0)Let v=speed of light in a material
v=1/sqrt() < cv=c/sqrt(k*m)
Index of refraction, nn=c/v=sqrt(k*m)
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Radiation Pattern
The Poynting Vector, S,describes the energy flow per unit area and per unit time through a cross-sectional area perpendicular to propagation direction
S=(E x B)/0
The “intensity” of the EM wave in vacuum is defined as I=Sav=(E0B0)/20= ½ 0cE0
2
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EM Spectrum
