Physics 1402: Lecture 26Today’s Agenda

• Announcements:

• Midterm 2: NOT Nov. 6– About Monday Nov. 16 …

• Homework 07: due Friday this weekHomework 07: due Friday this week

• Electromagnetic Waves– Maxwell’s Equations - Revised

– Energy and Momentum in Waves

f( )x

x

f( x

x

z

y

Maxwell’s Equations• These equations describe all of Electricity and

Magnetism.

• They are consistent with modern ideas such as relativity.

• They describe light !

Maxwell’s Equations - Revised• In free space, outside the wires of a circuit, Maxwell’s equations

reduce to the following.

• These can be solved (see notes) to give the following differential equations for E and B.

• These are wave equations. Just like for waves on a string. But here the field is changing instead of the displacement of the string.

Step 1 Assume we have a plane wave propagating in z (ie E, B not functions of x or y)

Plane Wave Derivation

x

z

y

z1 z2

Ex Ex

Z

x

By

Step 2 Apply Faraday’s Law to infinitesimal loop in x-z plane

Example: does this

Plane Wave Derivation

x

z

y

z1 z2

By

Z

yBy

Ex

Step 3 Apply Ampere’s Law to an infinitesimal loop in the y-z plane:

Step 4 Combine results from steps 2 and 3 to eliminate By

Plane Wave Derivation• We derived the wave eqn for Ex:

• By is in phase with Ex

• B0 = E0 / c

• How are Ex and By related in phase and magnitude?

(Result from step 2)

• We could have also derived for By:

– Consider the harmonic solution:

where

Review of Waves from last semester

• The one-dimensional wave equation:

• A specific solution for harmonic waves traveling in the +x direction is:

has a general solution of the form:

where h1 represents a wave traveling in the +x direction and h2 represents a wave traveling in the -x direction.

h

x

A

A = amplitude = wavelengthf = frequencyv = speedk = wave number

E & B in Electromagnetic Wave• Plane Harmonic Wave:

where:

y

x

z

Nothing special about (Ey,Bz); eg could have (Ey,-Bx)

2

Note: the direction of propagation is given by the cross product

where are the unit vectors in the (E,B) directions.

Note cyclical relation:

Lecture 26, ACT 1• Suppose the electric field in an e-m wave is given by:

– In what direction is this wave traveling ?5A

(a) + z direction (b) -z direction

(c) +y direction (d) -y direction

Lecture 26, ACT 2• Suppose the electric field in an e-m wave is given

by:

• Which of the following expressions describes the magnetic field associated with this wave?

(a) Bx = -(Eo/c)cos(kz + t) (b) Bx = +(Eo/c)cos(kz - t) (c) Bx = +(Eo/c)sin(kz - t)

5B

Velocity of Electromagnetic Waves• The wave equation for Ex: (derived from Maxwell’s Eqn)

• Therefore, we now know the velocity of electromagnetic waves in free space:

• Putting in the measured values for 0 & 0, we get:

• This value is identical to the measured speed of light! – We identify light as an electromagnetic wave.

The EM Spectrum

• These EM waves can take on any wavelength from angstroms to miles (and beyond).

• We give these waves different names depending on the wavelength.

Wavelength [m]10-14 10-10 10-6 10-2 1 102 106 1010

Gam

ma

Ray

s

Infr

ared

Mic

row

aves

Sh

ort

Wav

e R

adio

TV

an

d F

M R

adio

AM

Rad

io

Lo

ng

Rad

io W

aves

Ult

ravi

ole

t

Vis

ible

Lig

ht

X R

ays

Lecture 26, ACT 3• Consider your favorite radio station. I will

assume that it is at 100 on your FM dial. That means that it transmits radio waves with a frequency f=100 MHz.

• What is the wavelength of the signal ?

A) 3 cm B) 3 m C) ~0.5 m D) ~500 m

Energy in EM Waves / review• Electromagnetic waves contain energy which is stored in E

and B fields:

• The Intensity of a wave is defined as the average power transmitted per unit area = average energy density times wave velocity:

• Therefore, the total energy density in an e-m wave = u, where

=

Momentum in EM Waves• Electromagnetic waves contain momentum.

• The momentum transferred to a surface depends on the area of the surface. Thus Pressure is a more useful quantity.

• If a surface completely absorbs the incident light, the momentum gained by the surface is,

• We use the above expression plus Newton’s Second Law in the form F=dp/dt to derive the following expression for the Pressure,

• If the surface completely reflects the light, conservation of momentum indicates the light pressure will be double that for the surface that absorbs.

The Poynting Vector• The direction of the propagation of the electromagnetic wave is

given by:

• This wave carries energy. This energy transport is defined by the Poynting vector S as:

– The direction of S is the direction of propagation of the wave

– The magnitude of S is directly related to the energy being transported by the wave:

• The intensity for harmonic waves is then given by:

The Poynting Vector

• Thus we get some useful relations for the Poynting vector.

1. The direction of propagation of an EM wave is along the Poynting vector.

2. The Intensity of light at any position is given by the magnitude of the Poynting vector at that position, averaged over a cycle.

I = Savg

3. The light pressure is also given by the average value of the Puynting vector as,

P = S/c Absorbing surface

P = 2S/c Reflecting surface

Generating E-M Waves

• Static charges produce a constant Electric Field but no Magnetic Field.

• Moving charges (currents) produce both a possibly changing electric field and a static magnetic field.

• Accelerated charges produce EM radiation (oscillating electric and magnetic fields).

• Antennas are often used to produce EM waves in a controlled manner.

A Dipole Antenna• V(t)=Vocos(t)

x

zy

• time t=0

++

--

E

• time t=/2

E

• time t=/ one half cycle later

--

++

dipole radiation pattern

• oscillating electric dipole generates e-m radiation that is polarized in the direction of the dipole

• radiation pattern is doughnut shaped & outward traveling– zero amplitude directly above and below dipole– maximum amplitude in-plane

proportional to sin(t)

Receiving E-M Radiation

receiving antenna

One way to receive an EM signal is to use the same sort of antenna.• Receiving antenna has charges which are

accelerated by the E field of the EM wave. • The acceleration of charges is the same thing as an

EMF. Thus a voltage signal is created.

Speakery

x

z

Lecture 26, ACT 4

• Consider an EM wave with the E field POLARIZED to lie perpendicular to the ground.

y

x

z

In which orientation should you turn your receiving dipole antenna in order to best receive this signal?

C) Along Ea) Along S b) Along B

Loop AntennasMagnetic Dipole Antennas

• The electric dipole antenna makes use of the basic electric force on a charged particle

• Note that you can calculate the related magnetic field using Ampere’s Law.

• We can also make an antenna that produces magnetic fields that look like a magnetic dipole, i.e. a loop of wire.

• This loop can receive signals by exploiting Faraday’s Law.

For a changing B field through a fixed loop

Lecture 26, ACT 5• Consider an EM wave with the E field

POLARIZED to lie perpendicular to the ground.

y

x

zIn which orientation should you turn your receiving loop antenna in order to best receive this signal?

a) â Along S b) â Along B C) â Along E