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Maxwells equations (sec. 32.1)Plane EM waves & speed of light (sec. 32.2)The EM spectrum (sec. 32.6)

Electromagnetic Waves Ch. 32

C 2012 J. F. Becker

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Learning Goals- we will learn: ch 32

Maxwells Equationsthe four fundamentalequations of EM theory.

How the speed of lightis related to thefundamental constants of electricity and

magnetism (eo and mo). How to describe propagating and standing

EM waves.

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MAXWELLS EQUATIONS

C 2004 Pearson Educational / Addison Wesley

The relationships between electricand magnetic fields and their

sources can be stated compactly infour equations, calledMaxwells equations.

Together they form a completebasis for the relation of Eand B

fields to their sources.

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A capacitor being charged by a current ichas adisplacement current equal to iCbetween the plates,

with displacement current iD= eA dE/dt. Thischanging Efield can be regarded as the source of themagnetic fieldbetween the plates.

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A capacitor being charged by a current iC has adisplacement current equal to iC in magnitude between

the plates, with

DISPLACEMENT CURRENTiD= eA dE/dt

From C = eA / d and DV= E d wecan useq = C V to get

q = (eA / d) (E d ) = eE A = e F E andfrom iC= dq / dt= eA dE / dt = e dF E / dt= iD

We have now seen thatachanging Efield can produce a Bfield

and from Faradays Lawachanging Bfield can produce an E field(or emf)

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MAXWELLS EQUATIONS

C 2004 Pearson Educational / Addison Wesley

The relationships between electricand magnetic fields and their

sources can be stated compactly infour equations, calledMaxwells equations.

Together they form a completebasis for the relation of Eand B

fields to their sources.

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An electromagnetic wave front. The plane representingthe wave front (yellow) moves to the right with speed c.The Eand Bfields are uniform over the region behind

the wave front but are zero everywhere in front of it.

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Gaussian surface for an electromagneticwave propagating through empty space.

The total electric flux and

total magnetic flux throughthesurfaceare both zero.

Both E and B are _ tothe direction of

propagation.

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Applying Faradays Law to a plane wave.

E dl= -d/dt{FB}= - d/dtB dALH: Eodl= -EaRH: In time dtthe wave frontmoves to the right a distance c dt.The magnetic fluxthrough therectangle in the xy-plane

increases by an amount d FBequalto the flux through the shadedrectanglein the xy-plane witharea ac dt, that is,d F

B

=B ac dt; d FB

/dt = B ac-d FB/dt = -B acand (LH = RH):-Ea = -B ac. So

E=

Bc

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Applying Amperes Law to a plane wave: iC= 0

LH: Bodl= BaRH: In time dtthe wave front

moves to the right a distancec dt. The electric fluxthroughthe rectangle in the xz-planeincreases by an amount dFE equalto E times the areaac dtof theshaded rectangle, that is,d FE = E ac dt. Thusd FE/dt= E ac, and (LH = RH):Ba = mo eoEac B = mo eoEcand fromE= Bc and B = m

oeoEc

we must have c = 1 / (mo eo)1/2

B dl= mo iC +mo eod FE/dt

= 3.00 (10)8 m/sec

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Faradays Lawapplied to a rectanglewith height aand widthDx parallel to the

xy-plane.

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Amperes Lawapplied to a rectanglewith height aand widthDx parallel to the

xz-plane.

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Representation of the electric and magnetic fields in apropagating wave. One wavelength is shown at

time t= 0. Propagation direction is Ex B.

WAVE PROPAGATIONSPEED

c= 1 / (mo eo)1/2c= 3.00 (10)8 m/sec

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ENERGY AND MONENTUM IN EM WAVES

Energy density: u = eoE2 /2 + B2 /2 m

o(Ch 30)

Using B = E/c = E (moeo)1/2 we get

u = eoE2 /2 + E2 (m

oeo)/2 m

o

u = eoE2 /2 + eoE2 /2 = eoE2

u = eoE2 (half in E and half in B) (eqn 32.25)

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ENERGY FLOW IN EM WAVES

dU = u dV = eoE2 (Ac dt)

Define the Poynting vectorS = energy flow / time x area

S = dU / dt A = eoE2

(Ac) / A = eo c E2

or

S = eoc E2 = e

oE2/ (m

oeo)1/2= (e

o/m

o)1/2 E2 = EB / m

o

And define the Poynting vector:

S= Ex B/ mo

With units of Joule/sec meter2 or Watt/meter2

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Wave front at time dtafter it passes through a stationaryplane with area A. The volume between the plane and the wave

front contains an amount of electromagnetic energy uAc dt.

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A standingelectromagnetic wave does not propagatealong the x-axis; instead, at every point on the x-axis

theE

andB

fields simply oscillate.

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Examples of standingelectromagnetic waves

Microwave ovenshave a standing wave with l=12.2 cm, a wavelength that is strongly absorbedby water in foods. Because the wave has nodes

(zeros) every 6.1 cm the food must be rotatedwith cooking to avoid cold spots!

Lasersare made of cavities of length Lwith

highly reflecting mirrors at each end to reflectwaves with wavelengths that satisfy L = m l 2where m = 1, 2, 3,

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THE ELECTROMAGNETIC SPECTRUMThe frequencies and wavelengths found in nature extend over

a wide range. The visible wavelengths extend from

approximately 400 nm (blue)to 700 nm (red).

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One cycle in the production of an electro-magnetic waveby an oscillating electric dipole antenna. The red

arrows represent the E field. (Bnot shown.)

PREPARATION FOR FINAL EXAM

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PREPARATION FOR FINAL EXAMAt a minimum the following should be reviewed:

Gauss's Law - calculation of the magnitude of the electric field caused bycontinuous distributions of charge starting with Gauss's Law and completing all the

steps including evaluation of the integrals.

Ampere's Law- calculation of the magnitude of the magnetic field caused byelectric currents using Ampere's Law (all steps including evaluation of the integrals).

Faraday's Law and Lenz's Law- calculation of induced voltage and current,

including the direction of the induced current.

Calculation of integralsto obtain values of electric field, electric potential, andmagnetic field caused by continuous distributions of electric charge and currentconfigurations (includes the Law of Biot and Savart for magnetic fields).

Maxwell's equations- Maxwell's contribution and significance.

DC circuits- Ohm's Law, Kirchhoff's Rules, series-parallel Rs, RC ckts, power.

Series RLC circuits- phasors, phase angle, current, power factor, average power.

Vectors- as used throughout the entire course.

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