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Maxwell’s equations and EM wavesThis LectureMore on Motional EMF and Faraday’s lawDisplacement currentsMaxwell’s equationsEM Waves

From previous LectureTime dependent fields and Faraday’s Law

Radar Remote Sensing of the Atmosphere

Stan BriczinskiApril 4, 2008

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Calculation of Electric/Magnetic Field for a moving charge

• Coulomb's Law:

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dE =14πε0

dqr2ur

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dB = µ04π

I ds×urr2

• Biot-Savart

+

×I

Nowcharge moves inside page

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Gauss’ law in magnetostaticsNet magnetic flux through any closed surface is always zero (the number of lines that exit and enter the closed surface is equal)

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B• dA∫ = 0

No magnetic ‘charge’, so right-hand side=0 for B-field

Basic magnetic element is the dipole

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E • dA∫ =Qenclosed

εo

Compare to Gauss’ law for E-field

Lorentz force in E and B-fields If a charge moves in the presence of E-field

and B-field

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r F = q

r E + q

r v ×

r B

velocity selector

If we switch on an E-field parallel to B it will accelerate q and loops become more stretched as q gains speed

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Summary on static E- and B-fields

Ampere’s law true for static (no time dependence) electric fields.

What happens to the Ampere’s law if fields depend on time?

Gauss’ law Ampere’s law

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B• dA∫ = 0

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E• dA∫ =Qenclosed

εo€

B• ds∫ = µoIMagnetostatics

Electrostatics

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E• ds∫ = ?0

Surface

Surface

Line

Line

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Time-dependent B-field

Not only charges produce E-field a changing B-field also produces an E-field. This is not like

a Coulomb field produced by a charge starting or stopping in the charge but it is a ‘circular’ E-field

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E• ds∫ = 0 becomes E• ds∫ = 0 − dΦB

dt

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⇒ E ≠ −∇V

Faraday’s law

Remember last lecture Faraday’s Law

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ε = E• ds∫ = −ddtΦB = −

ddt

B∫ • dA

Magnetic flux through surface bounded by path

Integral over closed path

E is not conservative!!! €

E• ds

Lenz’s law

This ‘circular’ E-field is able to move charges around the loop around the changing B-field

Quick Quiz A rectangular loop of wire is pulled at a constant velocity from a region

with B = 0 into a region of a uniform B-field. The induced current; A) will be zero B) will be some constant value that is not zero C) will increase linearly with time

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vL

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I =ε

R=dΦB

dt= BL dx

dt= BLv

What happens when the loop exits the B-field region?

Iind

x

Bind

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Faraday’s and Lenz’s law

vS N

B flux increases when magnet moves since B from N to SBind opposite to B

B flux decreases when magnet moves, Bind parallel to B and I changes direction

B

B

Bind

Bind

vS N

Magnetic flux change: the magnitude of B changes in time

Other ways to change B-flux

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2) the area crossed by B lines changes with t (motional emf)

3) The angle θ between B and normal to loop changes with t

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ΦB = BAcosθ

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Quick Quiz: motional emf

A conducting bar moves with velocity v. Which statement is true?

A) + accumulates at the top, and - at the bottomB) no complete circuit, therefore, no charge accumulationC) - accumulates at the top, and + at the bottom

THERE IS AN ELECTRIC FIELD ACROSS THE BAR AND A POTENTIAL DIFFERENCE

AT EQUILIBRIUM qE = qvB or E = vB

Potential difference = v B L

L v

-

+

+-

v

FB=-e vxB

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Induced current in a moving rod

Equivalentcircuit

R resistance of circuit

force due to induced current on bar (drag force opposing to bar motion)!

Moving bar acts as a battery:

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FB = I r l ×

r B

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Ampere’s laws with time-dep fields

Not only charges produce E-field, a changing B-field produces a non conservative E-field

a changing E-field produces a B-field

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E• ds∫ = 0 becomes E• ds∫ = 0 − dΦB

dt

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B• ds∫ = µoI becomes B• ds∫ = µoI + µoεodΦE

dt€

⇒ E ≠ −∇V

Faraday’s law

Ampere-Maxwell’s law

?

Displacement current

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Ampere’s law says we can considerany surface bounded by close line CBUT current through S1 is I ≠ 0and current through S2 I=0 Is there something wrong?There is an electric flux through S2

A is the area of the capacitor plates

C

Displacement current = conduction current

Id is given by the variation of E

I conduction current in wire

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ΦE = E• dA = E• dA =qε0

∫S1 +S2

∫S2

Ampere’s - Maxwell Law

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B• ds∫ = µoI becomes B• ds∫ = µoI + µoεodΦE

dt

Magnetic fields are produced both by conduction currents and by time-varying electric fields

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+µ0IdTot current is conduction+ displacement

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The 4 Maxwell’s Equations

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E ⋅ dA =qε0S∫ (Gauss' Law) B ⋅ dA = 0

S∫

E ⋅ ds = −dΦB

dt (Faraday - Henry) B ⋅ ds

L∫ = µ0I + L∫ µ0ε0

dΦE

dt (Ampere - Maxwell law)

Currents create a B-field A changing E-field can create

a B-field

charged particles create an electric field

An E-field can be created by a changing B-fieldConsequence: induced current

There are no magnetic monopoles

Lorentz force

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F = q(E + v ×B)

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James Clerk Maxwell and the theory of electromagnetism

Electricity and magnetism are deeply related

1865 Maxwell: mathematical theory showing relationship between electric and magnetic phenomena

Maxwell’s equations predict existence of electromagnetic waves propagating through space at velocity of light

Permittivity of free space:

Permeability of free space:

= 2.99792458 x 108 m/s

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Reminders: classification of waves

Longitudinal wave: medium particles move in direction parallel to direction of propagation of wave (eg. sound waves)

Sound wave:sinusoidal variation of pressure in air

A mechanical wave is a disturbance created by a vibrating object that travels through a medium from one location to another

Transverse wave: medium particles move in direction perpendicular to direction of wave (waves on a pond, EM waves)

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Reminders: waves

Velocity of waves: λ=vT ⇒ v = λ/T=λf

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Reminders on waves

Traveling waves on a string along x obey the wave equation:

y=wave function

General solution : y(x,t) = f1(x-vt) + f2(x+vt) y = displacement

Traveling wave: superposition of sinusoidal waves (produced by a source that oscillates with simple harmonic motion): y(x,t) = A sin(kx-ωt) y(x,t) = sin(kx+ ωt) A = amplitudek = 2π/λ = wave number λ = wavelength f = frequency T = 1/f = periodω = 2πf=2π/T angular frequency

λ

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∂ 2y(x, t)∂x 2

=1v 2∂ 2y(x, t)∂t 2

pulse traveling along +x pulse traveling along -x

v

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λ

Fields exist even with q=0 and I=0!

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E ⋅ dA = 0S∫ (Gauss' Law) B ⋅ dA = 0

S∫

E ⋅ ds = −dΦB

dt (Faraday - Henry) B ⋅ ds

L∫ = L∫ µ0ε0

dΦE

dt (Ampere - Maxwell law)

Transverse wave composed of E and B fields propagating in empty space with velocity c emerges from Maxwell equations!

From these equations we get EM wave equations traveling in vacuum!

E x B direction of c

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λ

Solutions of Maxwell’s equations

The simplest solution to partial differential equations is sinusoidal wave propagating along x: E = Emax cos (kx – ωt) B = Bmax cos (kx – ωt)

The speed of the electromagnetic wave is

E · B = 0E and B are perpendicular at all times

Hertz confirms Maxwell’s predictions

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• Heinrich Hertz was thefirst to generate anddetect electromagneticwaves in a laboratorysetting

• The most importantdiscoveries were in 1887

• He also showed otherwave aspects of light

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Hertz’s Experiment (1887) Transmitter T: two spherical electrodes separated by

narrow gap charged until air gap ionized=>sparks The discharge has oscillatory behavior of frequency f

~ 4 x 107 Hz This ‘oscillating dipole’ emits E and B plane waves When resonance frequency of T and R match sparks

also in R = receiver Hertz’s hypothesis: the energy transmitted from T to

R is carried by wavesT

R

measured λ by having standing waves using a metal screen: nodes at distance λ/2

knowing f he measured c!) Radiation has wave properties: interference,

diffraction, reflection, refraction and polarization

Shown below is the E-field of an EM wave broadcast at 30 MHz and traveling to the right.

What is the direction of the magnetic field during the first λ/2?

What is the wave length?

Quick Quiz on EM Waves

E

x

1) Into the page 2) out of the page

10.0 20.0

1) 10 m 2) 5 m

λ/2

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=3×108m /s30 ×106 /s

=10m