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Lecture 4
Physics 1502: Lecture 26
Today’s Agenda
• Announcements:
• Midterm 2: NOT Nov. 6
– About Monday Nov. 16 …
• Homework 07: due Friday this week
• Electromagnetic Waves
– Maxwell’s Equations - Revised
– Energy and Momentum in Waves
f( x
f( x )
x
x
z
y
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Lecture 4
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.
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Lecture 4
Plane Wave Derivation
Step 1
Assume we have a plane wave propagating in z (ie E, B
not functions of x or y)
does this
Example:
Step 2
Apply Faraday’s Law to infinitesimal loop in x-z plane
x
Ex
y
Ex
z1
By
ΔZ
z2
Δx
z
Plane Wave Derivation
Step 3
Apply Ampere’s Law to an infinitesimal loop in the y-z
plane:
x
Ex
z1
y
Step 4
By
ΔZ
z2
By
z
Δy
Combine results from steps 2 and 3 to eliminate By
!!
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Lecture 4
Plane Wave Derivation
• We derived the wave eqn for Ex:
• We could have also derived for By:
• How are Ex and By related in phase and magnitude?
– Consider the harmonic solution:
where
(Result from step 2)
• By is in phase with Ex
• B0 = E0 / c
Review of Waves from last semester
• The one-dimensional wave equation:
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.
• A specific solution for harmonic waves traveling in the +x
direction is:
h λ
A
x
A = amplitude
λ = wavelength
f = frequency
v = speed
k = wave number
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Lecture 4
E & B in Electromagnetic Wave
• Plane Harmonic Wave:
where:
y
x
z
Note: the direction of propagation
where
is given by the cross product
are the unit vectors in the (E,B) directions.
Nothing special about (Ey,Bz); eg could have (Ey,-Bx)
Note cyclical relation:
Lecture 26, ACT 1
• Suppose the electric field in an e-m wave is given by:
5A
– In what direction is this wave traveling ?
(a) + z direction
(c) +y direction
(b) -z direction
(d) -y direction
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Lecture 4
Lecture 26, ACT 2
• Suppose the electric field in an e-m wave is given
by:
5B
• Which of the following expressions describes
the magnetic field associated with this wave?
(a) Bx = -(Eo/c)cos(kz + ωt)
(b) Bx = +(Eo/c)cos(kz - ωt)
(c) Bx = +(Eo/c)sin(kz - ωt)
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 µ0 & ε0, we get:
• This value is identical to the measured speed of light!
– We identify light as an electromagnetic wave.
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Lecture 4
The EM Spectrum
10-14
10-10
Long Radio Waves
TV and FM Radio
AM Radio
Short Wave Radio
Microwaves
Visible Light
Infrared
Ultraviolet
X Rays
Gamma Rays
• These EM waves can take on any wavelength from
angstroms to miles (and beyond).
• We give these waves different names depending on the
wavelength.
10-6
10-2 1 102
Wavelength [m]
106
1010
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
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Lecture 4
Energy in EM Waves / review
• Electromagnetic waves contain energy which is stored in E
and B fields:
=
• Therefore, the total energy density in an e-m wave = u, where
• The Intensity of a wave is defined as the average power
transmitted per unit area = average energy density times wave
velocity:
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.
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Lecture 4
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
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Lecture 4
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(ωt)
+
+
-
E
E
+
+
-
• time t=0
x
• time t=π/2ω
• time t=π/ω
one half cycle
later
z
y
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Lecture 4
dipole radiation pattern
proportional to sin(ωt)
• 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
Receiving E-M Radiation
receiving antenna
y
x
Speaker
z
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.
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Lecture 4
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?
a) Along S
b) Along B
C) Along E
Loop Antennas
Magnetic 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
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Lecture 4
Lecture 26, ACT 5
• 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
loop antenna in order to best receive this signal?
a) â Along S
b) â Along B
C) â Along E
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