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

• Biot-Savart Law for magnetostatics

• Ratio of proportionality constants

Maxwell’s Equations• Gauss’ Law for electricity (Coulomb’s Law)

• Gauss’ Law for magnetism (no magnetic charges)

• Faraday’s Law of Induction

• Ampere’s Law

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

• Integral form

• 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

• Integral form

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

• Integral form

• 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

• 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

Ampere’s Law Revisited• A changing electric flux and an enclosed current result in a

magnetic field.• Differential form

• Integral form

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

where represents or • The same equation can be solved for and .• These can be solved to yield

• This is the equation for a wave propagating at speed

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

Upstream time:

Michelson-Morley Experiment

Downstream time:

or

Michelson-Morley Experiment

Cross stream time:

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 Michelson-

Morley 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.”