Physics 202, Lecture 21

Today’s Topics

!  More Wave Review !  Standing waves

!  Electromagnetic Waves (EM Waves) !  The Hertz Experiment !  Review of the Laws of Electromagnetism !  Maxwell’s equations !  Propagation of E and B

!  The Linear Wave Equation

1

Parameters For A Sinusoidal Wave "  Snapshot with fixed t:

wavevector = k wavelength !=2"/k "  Snapshot with fixed x: angular frequency = ! frequency f =#/2"$ Period T=1/f Amplitude =A$"  Wave Speed v=#/k ! v=!f, or ! v=!/T "  Phase angle difference

between two positions "% =-k&x

Snapshot:Fixed t

Snapshot:Fixed x 2

Standing Waves " When two waves of the same amplitude, same

frequency but opposite direction meet standing waves occur.

!

y1 = Asin(kx "#t)

!

y2 = Asin(kx +"t)

!

y = y1 + y2 = 2Asin(kx)cos("t)" Points of destructive interference (nodes)

!

kx = n"; x =n"k

=n#2

; n =1,2,3...

" Points of constructive interference (antinodes)

!

kx = (2n "1) #2

; x = (2n "1) $4

; n =1,2,3...3

Standing Waves (cont)

" Nodes and antinodes will occur at the same positions, giving impression that wave is standing

4

Standing Waves (cont) " Standing waves with a string of given length L are

produced by waves of natural frequencies or resonant frequencies:

!

" =2Ln

; n =1,2,3...

!

f =v"

=nv2L

=Tµ

n2L

5

Demo: Hertz Experiment

In 1887, Heinrich Hertz first demonstrated that EM fields can transmit over space.

6

Review: Gauss’s Law / Coulomb’s Law " The relation between the electric flux through a

closed surface and the net charge q enclosed within that surface is given by the Gauss’s Law

!

E • dA =q"0

#

7

Gauss’s Law for Magnetism " The Gauss’s Law for the electric flux is a reflection

of the existence of electric charge. In nature we have not found the equivalent, a magnetic charge, or monopole

" We can express this result differently: if any closed surface as many lines enter the enclosed volume as they leave it

!

B• dA = 0"

8

Review: Faraday’s Law " The emf induced in a “circuit” is proportional to the

time rate of change of magnetic flux through the “circuit” or closed path.

" Since

" Then

A

B

'$

!

"B = B• dA#

nominal direction of ($

!

" = #d$B

dt

!

" = E • d!#

!

E • d! = "d#B

dt$9

Review: Ampere’s Law " A magnetic field is produced by an electric current is

given by the Ampere’s Law

But Ampere’s Law can be ambiguous: currents enclosed by surfaces 1 and 2 are different!

!

B• d! = µ0I"

10

But note time-dependent electric flux though surface 2.

Maxwell’s modification of Ampere’s Law

" A time-dependent electric flux induces a magnetic field (in analogy to Faraday’s law)

B!"• d#"= µ0 I + !0

d"E

dt#$%

&'($)

!

"E = E • dA#

11

Ampere’s law

“Maxwell’s displacement current”

Maxwell Equations

!

E • dA =q"0

#

!

E • d! = "d#B

dt$

!

F = qE + qv "B

!

B• d! = µ0I + "0µ0d#E

dt$!

B• dA = 0"

!Gauss’s Law/ Coulomb’s Law

!Faraday’s Law

!Gauss’s Law of Magnetism, no magnetic charge

!Ampere Maxwell Law

Also, Lorentz force Law !

These are the foundations of the electromagnetism 12

EM Fields in Space "  Maxwell equations when there is no charge and current:

! =• 0AE d

!

E • d! = "d#B

dt$

!

B• d! = "0µ0d#E

dt$! =• 0AB d

tB

xE zy

!!

"=!

!

tE

xB yz

!

!"=

!!

00#µdifferential forms:

(single polarization)

x

y

z 2

2

002

2

tE

xE yy

!

!=

!

!"µ 2

2

002

2

tB

xB zz

!!

=!!

"µ

E

B

13

Linear Wave Equation " Linear wave equation

2

2

22

2 1ty

vxy

!!

=!!

certain physical quantity

Wave speed

" Sinusoidal wave

)22sin( !"#"

+$= ftxAy

A:Amplitude

f: frequency %:Phase

General wave: superposition of sinusoidal waves

v=!f k=2"/!$#=2"f

!:wavelength

14

Electromagnetic Waves "  EM wave equations:

" Plane wave solutions: E= Emaxcos(kx-#t+%) B= Bmaxcos(kx-#t+%) " Properties:

#  No medium is necessary. #  E and B are normal to each other #  E and B are in phase #  Direction of wave is normal to both E and B (EM waves are transverse waves) #  Speed of EM wave: #  E/B = Emax/Bmax=c #  Transverse wave: two polarizations possible

2

2

002

2

tE

xE yy

!

!=

!

!"µ 2

2

002

2

tB

xB zz

!!

=!!

"µ

same %, set to be 0

smc /109972.21 8

00

!=="µ

x

y

z

E

B c

15

The EM Wave

x

y

z

E

B c

Two polarizations possible (showing one) 16