1

EM Waves

This Lecture

!More on EM waves

!EM spectrum

!Polarization

From previous Lecture!Displacement currents

!Maxwell’s equations

!EM Waves

Friday Honor Lecture

Prof. D. OertelDept. of PhysiologyInterpreting Nerve Signals

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The 4 Maxwell’s Equations

!

E " dA =q

#0

S$ (Gauss' Law) B " dA = 0

S$

E " ds = %d&B

dt (Faraday - Henry) B " ds

L$ = µ

0I +

L$ µ

0#0

d&E

dt (Ampere - Maxwell law)

! Currents create a magnetic field

! A changing electric field can create a magnetic field

charged particles create an electric field

An electric field can be created by a changing magnetic field

Consequence: induced current

There are no magnetic monopoles

Lorentz force

!

F = q(E + v "B)

• E and B are perpendicular oscillating vectors

•The direction of propagation is perpendicular to E and B

!

Facts on EM waves

! EM waves are solutions of Maxwell’s equations.

! In empty space: sinusoidal wave propagating along x with velocity

! E = Emax cos (kx – "t)

! B = Bmax cos (kx – "t)

E ! B = 0

E x B direction of c

Transverse waves

An electromagnetic wave propagates in the –y direction. The electric field at a point in space is momentarily oriented in the +x direction. The magnetic field at that point is momentarily oriented in the

(a) –x direction

(b) +y direction

(c) +z direction

(d) –z direction

Quick Quiz on EM waves

x

y

z

c

E

B

xz

y

E

B

Faraday’s law: Loops use B not E!

Only the loop in the xy plane will have a magnetic flux through it as the wave passes. The flux will oscillate with time and induce an emf.

loop in xy plane

loop in xz plane

loop in

yz

plane

A B C

Which orientation will have the largest induced emf?

Quick Quiz pn EM waves

EM Waves from an Antenna

! 2 rods connected to AC generator; charges oscillate between the rods (a)

! As oscillations continue, the rods become less charged, the field near the charges decreases and the field produced at t = 0 moves away from the rod (b)

! The charges and field reverse (c)

! The oscillations continue (d)

Sources of EM waves: oscillating charges, accelerated/decellerated charges, electron transitions between energy levels in atoms, nuclei and molecules

EM waves from antennas

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- Electromagnetic radiation is greatest when

charges accelerate at right angles to the line of

sight.

- Zero radiation is observed when the charges

accelerate along the line of sight.

- These observations apply to electromagnetic

waves of all frequencies.

Relation between E and B

! E = Emax cos (kx – "t)

! B = Bmax cos (kx – "t)

! First derivatives:

!

"E

"x= #kEmax sin(kx #$t)

"B

"t=$Bmax sin(kx #$t)

"E

"x= #

"B

"t

This relation comes from Maxwell’s equations!

From:

Speed of em waves

(speed of light in vacuum)

The EM Spectrum

X-rays: ~10-12 -10-8 m

source: deceleration of high-energy

electrons striking a metal target

Diagnostic tool in medicine

Source: atoms and molecules Human eyeVisible range from red (700 nm) to violet (400 nm)

Radio: ! ~ 10 - 0.1 m

Sources: charges accelerating through conducting wires Radio and TV

Microwaves: ! ~10-4 -0.3 m

sources: electronic devices

radar systems, MW ovens

Infrared: ! ~ 7 x 10-7-10-3 m

Sources: hot objects and molecules

UV !~ 6 x 10-10 - 4 x 10-7 m

Most UV light from the sun is absorbed in the stratosphere by ozone

Gamma rays: !~ 10-14- 10-10 m

Source: radioactive nuclei, cause serious damage to living tissues

Poynting vector

! Rate at which energy flows through a unit area perpendicular to direction of wave propagation

! This is the power per unit area (J/s.m2 = W/m2)

! Its direction is the direction of propagation of the EM wave

! Magnitude:

! This is time dependent

! Its magnitude varies in time

! Its magnitude reaches a maximum at the same instant as E and B do

!

S =EB

µ0

=E2

cµ0

E/B=c

Energy density of E and B field

! In a parallel plate capacitor:

!

C ="0A

d

!

U =1

2CV

2=1

2

"0A

dE2d2# u

E=U

Ad=1

2"0E2 True for any geometry

When a battery is connected to a circuit the current does not jump instantaneously from 0 to the final value !/R because there is an induced emfopposing to battery action. By calculating the work done in a solenoid against induced emf arising from increasing B-flux as current goes from 0 to I we can derive the energy density of the B-field associated to the current

True for any geometry

!

uB

=B2

2µ0

Energy carried by EM waves

!

uE

=1

2"0E2 = u

B=B2

2µ0

=E2

2c2µ

0

! Total instantaneous energy density of EM waves

u =uE + uB = 1/2 $oE2 + B2 /(2µo)

! Since B = E/c and

In a given volume, the energy is shared equally by the two fields!

uE

= uB

! Let’s consider a cylinder with axis along x of area A and length L and the time for the wave to travel L is !t=L/c

! The average power is:

!

Pav

=U

av

"t=uavAL

"t= u

avAc

! And the intensity (average P/area)

Intensity and Poynting vector

E x B

!

u = uE

+ uB

= "0E2 =

B2

µ0

=EB

µ0c

!

S =EB

µ0

=E2

cµ0

! Wave intensity I = time average over one or more cycle <sin2(kx - "t)> = 1/2 then <E2> = Emax

2/2 and <B2> = Bmax2/2

power per unit area

!

Iav

=Pav

A= u

avc

!

Iav

= uavc =

EmaxBmax

2µ0

I & E2

E/B=c

Radiation pressure and momentum

! EM waves transport momentum# pressure on a surface

! Complete absorption on a surface: total transported energy U in time interval %t # total momentum p = U / c and prad=Sav/c

! Radiation Pressure = force per unit area

! Perfectly reflecting surface: momentum of incoming and reflected light p = U/c # total transferred momentum p = 2U/c

and prad = 2Sav/c

! Direct sunlight pressure ~5 x 10-6 N/m2

!

F ="p

"t=

(energy absorbed)/"t

c=Power

c

!

prad =F

A=Power /A

c=I

cfrom

momentum p = U / c

!

I =Power

c= S

ave

Quick Quiz on radiation pressure

To maximize the radiation pressure on the sails of a spacecraft using solar sailing, the sheets must be

(a) very black to absorb as much sunlight as possible

(b) very shiny, to reflect as much sunlight as possible

Which is the value of the radiation pressure in the above case?

(a) P = 2S/c

(b) P = S/c

Polarization of Light Waves (34.8)

! Linearly polarized waves: E-field oscillates at all times in the plane of polarization

Linearly polarized light: E-field has one spatial orientation

Unpolarized light: E-field in random directions. Superpositionof waves with E vibrating in many different directions

Circular and elliptical polarization

! Circularly polarized light: superposition of 2 waves of equal amplitude with orthogonal linear polarizations, and 90˚ out of phase. The tip of E describes a circle (counterclockwise = RH and clockwise=LH depending on y component ahead or behind)

! The electric field rotates in time with constant magnitude.

! If amplitudes differ # elliptical polarization

Producing polarized light

! Polarization by selective absorption: material that transmits waves whose E-field vibrates in a plain parallel to a certain direction and absorbs all others

This polarizationabsorbed

This polarizationtransmitted transmission axis

Polaroid sheet (E. Land 1928)Long-chain hydrocarbon

molecules

DEMO with MW generator and metal grid

MW generator

Metal grid

pick up antenna connected to Ammeter

If the wires of the grid are parallel to the plane of polarization the grid absorbs

the E-component (electrons oscillate in the wire).

The same thing happens to a polaroid: the component parallel

to the direction of the chains of hydrocarbons is absorbed.

If the grid is horizontal the Ammeter will measure a

not null current since the wave reaches the antenna

pick-up

This

polarization

absorbed

This polarization

transmitted transmission axis

Polarization by selective absorption

transmission axis

Polaroid sheet

Long-chain hydrocarbon molecules

If linearly polarized light of intensity I0 passes through a polarizing

filter with transmission axis at an angle ' along y

Einc = E0sin' i + E0 cos' j

After the polarizer Etransm = E0cos' j

So the intensity transmitted isItransm = E0

2 cos2' = (0cos2'

y

x

'

E0cos'

A polarizer is used to produce polarized light of intensity I0

and an analyzer rotated at an angle ': transmission 100%

when ' = 0 and zero when ' =

90°

Detecting polarized light: Malus’ law

! Ideal polarizer transmits waves with E parallel to transmission axis and absorbs those with E ) axis

! Relative orientation of axis of polarizer and analyzer determines intensity of transmitted light.

! Transmitted intensity: I = I0cos2' I0 = intensity of polarized beam on analyzer

(Malus’ law)

Allowed componentparallel to analyzer axis

Polaroid sheets

Quick Quiz on Polarization

A

B

Law of Malus Example

1) Light transmitted by first polarizer is vertically polarized. I1 = I0

cos245=I0/2

2) Angle between it and second polarizer is '=45º. I2= I1 cos2 (45º)

=0.5I1=0.25 I0

3) Light transmitted through second polarizer is polarized 45º

from vertical. Angle between it and third polarizer is '=45º. I3= I2

cos2(45º) = 0.125I0

I2= I1cos2(45)

I1= 0.5 I0

Relative orientation of polarizers

! Transmitted amplitude is Eocos'

(component of polarization along polarizer axis)

! Transmitted intensity is Iocos2'

( square of amplitude)

! Perpendicular polarizers give zero intensity.