Transcript Firth and Rakoski: Maxwell`s Theory

Maxwell’s Equations:
The Final Classical
Theory
Catherine Firth & Alexa Rakoski
James Clerk Maxwell
• Born June 13th 1831 in
Edinburgh, Scotland
• Born James Clerk but
took the name of
Maxwell when he
inherited an estate from
the Maxwell family
• Published first scientific
paper at the age of 14:
method to draw
mathematical curves
using twine, and the
properties of multifocal
curves
Maxwell’s Later Years
• Attended university first at the
University of Edinburgh, and later at
Cambridge University, where he
graduated with a degree in
mathematics
• In 1856 at the age of 25, he became
Chair of Natural Philosophy at
Marischal College, Aberdeen
• In 1860 moved to King’s College
London, where he worked until 1865
• It was during this time that ‘A
Dynamical Theory of the
Electromagnetic Field’ was
published
• In 1871 he became the director of the
newly-formed Cavendish Laboratory
at Cambridge
• Died November 5th 1879 at the age of
48
Before Maxwell
• Two empirical laws:
• Coulomb’s Law for electrostatics
𝑞
1 𝑞
𝑬 = 𝑘1 2 𝒓 =
𝒓
2
𝑟
4𝜋𝜖0 𝑟
• Biot-Savart Law for magnetostatics
𝑖𝒍 × 𝒓 𝜇0 𝑖𝒍 × 𝒓
𝑩 = 𝑘2 3 =
𝑟
4𝜋 𝑟 3
• Ratio of proportionality constants
𝑘1
1
=
= 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑠𝑞𝑢𝑎𝑟𝑒𝑑
𝑘2 𝜇0 𝜖0
Maxwell’s Equations
• Gauss’ Law for electricity (Coulomb’s Law)
𝜌
𝛻∙𝑬=
𝜖0
• Gauss’ Law for magnetism (no magnetic charges)
𝛻∙𝑩=0
• Faraday’s Law of Induction
𝜕𝑩
𝛻×𝑬=−
𝜕𝑡
• Ampere’s Law
𝜕𝑬
𝛻 × 𝑩 = 𝜇0 𝑱 + 𝜇0 𝜖0
𝜕𝑡
Flux
• Measure of “stuff” passing through an area
• Water passing through a cross-section of pipe
• Electric/magnetic flux: number of field lines passing through a
surface area
• Flux integral: sum of infinitely many tiny surfaces
Gauss’ Law for Electricity
• The electric flux through a closed surface is proportional to
the charge enclosed inside the surface.
• Differential form
𝜌
𝛻∙𝑬=
𝜖0
• Integral form
𝑞
𝑬 ∙ 𝑑𝑨 =
𝜖0
• Coulomb’s Law is a special case of Gauss’ Law
Gauss’ Law for Magnetism
• The magnetic flux through a closed surface is zero.
• There are no “magnetic charges,” so magnets always come in
North-South pairs.
• Differential form
𝛻∙𝑩=0
• Integral form
𝑩 ∙ 𝑑𝑨 = 0
Faraday’s Law
• A change in magnetic flux through a loop results in a voltage
through the loop.
• Voltage and electric field are related by
• Therefore, a magnetic field that varies over time produces an
electric field.
• Differential form
• Integral form
Ampere’s Law
• The magnetic field around a closed loop is proportional to the
current passing through the loop.
• Differential form
𝛻 × 𝑩 = 𝜇0 𝑱
• Integral form
𝑩 ∙ 𝑑𝒔 = 𝜇0 𝑖
• Ampere’s Law was not quite correct in its original form.
The Displacement Current
• Mathematical inconsistency or physical consequence?
• Mathematical inconsistency between right-hand sides of
Faraday’s Law and Ampere’s Law
• Faraday’s Law has a flux integral, but Ampere’s Law does not
𝑑
𝑬 ∙ 𝑑𝒔 = −
𝑩 ∙ 𝑑𝑨
𝑩 ∙ 𝑑𝒔 = 𝜇0 𝑖
𝑑𝑡
• Maxwell noticed this discrepancy and found a way to add a
term that contained the time derivative of electric flux
The Displacement Current
• A material can become polarized when exposed to an electric
field
• An electric field that changes in time will cause the degree of
polarization to change.
• Moving charges generate a magnetic field
• Therefore, an electric field that varies over time produces a
magnetic field.
• Displacement current term includes the time derivative of the
electric flux
𝜕𝑬
𝑑
𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑙: 𝜇0 𝜖0
𝐼𝑛𝑡𝑒𝑔𝑟𝑎𝑙: 𝜇0 𝜖0
𝑬 ∙ 𝑑𝑨
𝜕𝑡
𝑑𝑡
Ampere’s Law Revisited
• A changing electric flux and an enclosed current result in a
magnetic field.
• Differential form
𝜕𝑬
𝛻 × 𝑩 = 𝜇0 𝑱 + 𝜇0 𝜖0
𝜕𝑡
• Integral form
𝑑
𝑩 ∙ 𝑑𝒔 = 𝜇0 𝑖 + 𝜇0 𝜖0
𝑬 ∙ 𝑑𝑨
𝑑𝑡
Role of the Aether
• Maxwell’s model of the aether was key to his development of
the displacement current.
• Dipoles in the aether could be polarized, making it possible for
an electric field to be transmitted across space.
Consequences of Maxwell’s Theory
• Maxwell’s first two equations are properties of the electric
and magnetic fields.
• The last two equations, Faraday’s Law and Ampere’s Law show
that electric and magnetic fields generate each other:
• A varying magnetic field generates an electric field, and a
varying electric field generates a magnetic field.
• What is the solution to Maxwell’s equations?
Electromagnetic Waves
• The electric and magnetic fields must propagate at right
angles to each other, and the direction of propagation must be
at a right angle to both fields.
• The speed of the waves is
• This speed happens to agree with the speed of light measured
experimentally!
Electromagnetic Waves
• Maxwell’s equations can be combined to yield the differential
equation
𝜕2
𝜕2
− 𝜇0 𝜖0 2 𝑓 𝑥, 𝑡 = 0
𝜕𝑥 2
𝜕𝑡
where 𝑓 𝑥, 𝑡 represents 𝐸𝑥 or 𝐵𝑥
• The same equation can be solved for 𝑦 and 𝑧.
• These can be solved to yield
1
1
𝑓 𝑥, 𝑡 = 𝑔 𝑥 −
𝑡 +ℎ 𝑥+
𝑡
𝜇0 𝜖0
𝜇0 𝜖0
• This is the equation for a wave propagating at speed
1
.
𝜇0 𝜖0
Electromagnetic Waves
• The solution to the equations of electromagnetism is a wave
propagating at the speed measured for light.
• Supported the wave theory of light, and suggested that light
was specifically an electromagnetic wave.
Newton v. Maxwell
• Can electromagnetism be understood using Newtonian
mechanics?
• Mechanical models of the aether
• Newton’s laws depend on acceleration so no reference frame
is favored over any other.
• Maxwell’s equations depend on velocity, so perhaps one
“absolute” frame is favored.
• Can the absolute reference frame, or frame of the aether, be
detected?
Michelson-Morley Experiment
Greatest negative experiment in the history of physics?
Michelson-Morley Experiment
• Aether is the rest frame with
respect to which c=3x10-8 m/s
• Our apparatus is moving
through this frame with a speed
of v
• Examine the difference
between the speed of light
travelling upstream and
travelling downstream, and
their difference would be
twice the speed of the aether
• However measured by
sending light down to a mirror
and back, which would offset
the effects
Michelson-Morley Experiment
• “Suppose we have a river of width w (say, 100 feet), and two
swimmers who both swim at the same speed v feet per second
(say, 5 feet per second). The river is flowing at a steady rate,
say 3 feet per second. The swimmers race in the following
way: they both start at the same point on one bank. One
swims directly across the river to the closest point on the
opposite bank, then turns around and swims back. The other
stays on one side of the river, swimming upstream a distance
(measured along the bank) exactly equal to the width of the
river, then swims back to the start. Who wins?”
Michelson-Morley Experiment
Swimmer 1: going
upstream travels at
2 ft/s, takes 50
seconds:
Downstream:
speed of 8 ft/s,
12.5 s
Total: 62.5 s
Swimmer 2:
crossing rate
of 4 ft/s,
takes 25 s
each way
Total: 50 s
Michelson-Morley Experiment
http://galileoandeinstein.physics.virginia.edu/more_stuff/flashlets/mmexpt6.htm
Michelson-Morley Experiment
Upstream time:
l
c-v
Downstream time:
l
c+v
2l 2 2 2l
(c - v ) =
c
c
1
v2
(1- 2 )
c
or
2l
v2
(1+ 2 )
c
c
Michelson-Morley Experiment
Cross stream time:
2l
c2 - v2
2l
1
2l
v2
@ (1+ 2 )
c 1- v 2 / c 2 c
2c
Michelson-Morley Experiment
• Two equations differ by (2l/c)*(v2/2c2), a difference an
observer would not be able to detect
• Get around this using inference properties of lightwaves
• Difference in phases would produce an interference fringe
pattern
• Rotating the apparatus, the path difference decreases until it
is 0 when both are inclined at 45 degrees to v
• Constructive interference observed
• As it is rotated through 90 degrees, the change in time
difference is twice the original
Michelson-Morley Experiment
• Parallel wave should fall behind the perpendicular
beam
• For a setup with l≅11m, expect 0.4 fringe shifts
• But no fringe shift was detected!
Lorentz-FitzGerald Contraction
Hypothesis
• George FitzGerald, Trinity College Dublin
• 1892: If the length of the parallel arm of the MichelsonMorley apparatus is shortened, the speed of the earth relative
to the aether would not be detected.
• In general, an object moving parallel to v is contracted in the
direction of motion.
Lorentz-FitzGerald Contraction
Hypothesis
•
•
•
•
Hendrik Lorentz, University of Leiden
1902 Nobel Prize for electromagnetic radiation
Maxwell’s equations in a moving frame
Derived a transformation between the moving frame and the
frame at rest
• Showed that the electric force is less in the moving frame than
in the rest frame
• All interactions are fundamentally electromagnetic, so
molecules might transform in the same way
• Result: objects contract in the direction of motion
Discarding the Aether
• Henri Poincare, French mathematician
• 1899: “…optical phenomena depend only on the relative
motions of the material bodies, luminous sources, and optical
apparatus concerned.”
• 1904: Postulated “principle of relativity”
• Laws of physics must be the same for all inertial reference
frames
• No velocity can be greater than the speed of light
• Same as Einstein’s postulates in 1905
Discarding the Aether
• After special relativity was published, most scientists accepted
that there was no aether.
• 1916: Lorentz’ Theory of Electrons
• Einstein’s theory was a work of genius
• Aether still existed and was “endowed with a certain degree of
substantiality, however different it may be from all ordinary
matter.”