29. Maxwell’s Equations

2

Topics

Laws of Electric & Magnetic Fields

James Clerk Maxwell

Maxwell’s Equations

3

Laws of Electric & Magnetic Fields

4

James Clerk Maxwell1831 – 1879

In 1865, Maxwell published a paper entitled: A Dynamical Theory of the Electromagnetic Field,Philosophical Transactions of the Royal Society of London 155, 459-512 (1865). This is one of the greatest scientific papers ever written.

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Maxwell’s Equations

inside

0Closed Surface

QE dA

ε× =∫

rr

Closed Surface

0B dA× =∫rr

Closed Loop

mdE dr

dt

ϕ× = −∫r r

0 0

Closed Loop

0eB dr I

d

dtµ µ ϕε× = +∫

r r

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Displacement Current

Maxwell realized that Ampere’s law is notvalid when the current is discontinuous asis true of the current through a parallelplate capacitor:

Encircled

Closed Lo

0

op

B dr Iµ× =∫r r

wikimedia.org

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Displacement Current

He concluded that when the charge within an enclosed surface is changing it is necessary to add to Ampere’s law another current called the displacement current: ID

inside0D

edQI

t dtd

dϕε= =

wikimedia.orgD

Closed Loop

0 ( )B dr I Iµ× = +∫r r

The 2nd Unification of Forces

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The 2nd Unification of Forces

µ0 is the magneticconstant

7 20 4 10 N/Aµ π −= ×

ε0 is the electricconstant

12 2 20 8.854 10 C /(N m )ε −= × ×

70 0

12

1

2

2 2

27 2

4 10 [ ]

8.854 10 [ ]

1.1

N/A

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C /(N m )

s10 [ ]/m

µ ε π −

−

−

= ×

× ×= ×

×

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The 2nd Unification of Forces

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0 0

12.998 10 m/s

µ ε= ×

From7

02

0211.113 1 s /m0 [ ]µ ε −= ×

we can write

which is the speed of light in vacuum!

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Light

“We can scarcely avoid the conclusion that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena.” (1866)

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5th Unification?

4th Unification?

3rd Unification

2nd Unification

1st Unification

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Summary

Maxwell’s EquationsGauss’s Law for EGauss’s Law for BFaraday’s LawAmpere’s Generalized Law

Electromagnetic Waves

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Topics

Maxwell’s Wave Equations

Waves – Recap

Electromagnetic Waves

Electromagnetic Radiation

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Maxwell’s Wave Equations

2 2

2 2 2

1E E

x c t

∂ ∂=∂ ∂

r rWave equation for E

2 2

2 2 2

1B B

x c t

∂ ∂=∂ ∂

r rWave equation for B

These equations describe electric andmagnetic waves traveling in the x direction

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Maxwell’s Wave Equations

yzBE

x t

∂∂ =∂ ∂

Relationship betweenEz and By

yzEB

t x

∂∂ = −∂ ∂

Relationship betweenBz and Ey

Maxwell showed that the different components of the electric and magnetic fields are related:

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Waves – Recap

( ) sin( )Ay x kx=

( ) sin ( )y x k x tA v= −

Stationary wave

Wave traveling in x direction

Wave number

2k

λπ=

2kv

T

πω = =

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Electromagnetic Waves

p( , ) sin( )yE x t kE x tω= −

Consider an electric wave, traveling in the positive x direction, but oscillating in the y direction:

We can find Bz from

yzEB

t x

∂∂ = −∂ ∂

Ep

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Electromagnetic Waves

p( , ) sin( )zB x t kB x tω= −This leads to the result

where

p p( / )kB Eω=

p pE cB=Bp

zthat is,

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y

xz

Electromagnetic Waves

p

p

ˆsin( )

ˆsin( )

E kx t j

B kx

E

B t k

ω

ω

= −

= −

r

r

Electromagnetic waves always travel in the direction ofthe Poynting vector:

0

E BS

µ×=

r rr

Units: W/m2

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Electromagnetic Waves

But the direction of the electric and magnetic fields themselves, that is, their polarization, can change

y

xz

Linear polarization

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Polarizers

1

2

3 90o

20 cosS S θ=

Law of Malus

Only a componentEpcosθ of the electric field along the polarization axis can get through

Electromagnetic Radiation

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The ElectromagneticSpectrum

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Spectral Response

http://landsat.gsfc.nasa.gov/education/compositor

Himalyan balsam

human bee butterfly

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Electromagnetic Radiation

An electromagnetic wave carries energy and momentum.

The average power per unit area iscalled the intensity of the wave

The momentum per unit time (that is, force)per unit area is called the radiation pressure

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Electromagnetic Radiation

The radiation pressure, Prad, is given by

rad

SP

c=

where the average intensity is given by

p p

0 0

1

2

E BEBS

µ µ= =

which can be written interms of energy density:

E BS cu cu= =

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The Pressure of Sunshine

Solar Luminosity L = 3.8 x 1026 W

Astronomical Unit r = 1.5 x 1011 m

Intensity S = L / 4π r2

Pressure P = S / c

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The Pressure of Sunshine

Intensity S = L / 4π r2

= 1370 W/m2

Pressure P = S / c= 4.6 µN/m2

31 Credit: Michael Carroll, The Planetary Society

Interstellar Travel

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Summary

Maxwell’s Equations2nd Unification of forcesElectromagnetic wavesUniversal speed c = 3 x 108 m/s

Electromagnetic WavesGamma rays to radio wavesCarry energy and momentumExert pressure