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Electromagnetic Waves
CHAPTER 2
ELECTROMAGNETIC WAVES
BEHAVING AS PARTICLES
A plane electromagnetic wave travelling in the +x direction
Problem 1
(a) Verify that
equation
Problem 1 (cont.)
is a solution to the wave
provided that  = kc.
(Hint: Use the Euler formula
(b) (cont.)
What must be the relationship between the
wavelength l and frequency f so that
is a solution of this wave equation?
)
(b) The wave equation in a medium such as glass with
index of refraction n is given by:
Problem 2
Blackbody radiation
The average intensity of an electromagnetic wave in
vacuum is:
where Em is the amplitude of the electric field and e0 =
8.851012 C2/Nm2 is the permittivity of free space.
The electric field in an EM wave in vacuum has a peak
value of 0.0218 V/m. What is the average power at
which this wave carries energy across unit area?
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Blackbody radiation (Supplementary
notes 1)

Classical law of equipartition of energy states that if
a system is in thermal equilibrium at the temp. T,
Blackbody radiation (Supplementary
notes 2)

where a degree of freedom is a mode of energy
possession.
Since each standing EM wave in a cavity is created
by an oscillating atom in the cavity wall, the average
energy of the wave is E  kT
Conceptual question 1
By considering how to enable objects of unequal
temperature to reach the same common temperature,
explain why the electromagnetic waves enclosed in a
cavity has the same temperature as that of the cavity
walls.
n  1,2,3,
for an oscillating atom at frequency f.
Hence, for the oscillating atoms in the cavity wall,
Average energy
1
 kT
Degree of freedom 2

Planck’s suggestion:
E  nhf
E 
 EP ( E ) 
 P( E ) e
hf
hf / kT
1
for the Boltzmann distribution P( E )  e E / kT / kT
Note that P(E)dE is the probability for the system to have
energy in the interval of E to E+dE.
Conceptual question 2

What assumptions did Planck make in dealing with
the problem of blackbody radiation? Discuss the
consequences of these assumptions.
Problem 3
Problem 4
Estimate the surface temperature of the Sun using the
following information. The Sun’s radius is Rs = 7.0108
m. The average Earth-Sun distance is d = 1.51011 m.
The power per unit area (at all frequencies) from the
Sun is measured at the Earth to be Id = 1400 W/m2.
Take the approximation that the Sun is a blackbody.
(a) Show that the Planck formula can be expressed in
terms of the wavelength l as:
(b) Show that the Wiens displacement law can be
derived from the spectral energy density dU/dl in
part (a).
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Photoelectric effect
Problem 5
An object of mass m = 2 kg is attached to a “massless”
spring of force constant k = 25 N/m. The spring is
stretched A = 0.40 m from its equilibrium position and
released.
(a) Find the total energy and frequency of oscillation
according to classical calculations.
(b) Assume that the energy of the system is quantized.
Find the quantum number, n, for the system.
(c) How much energy would be carried away by 1quantum change?
Conceptual question 3
Consider the following
photoelectric effect:
characteristics
f
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Conceptual question 3 (cont.)
of
the
(i) The generation of photoelectrons.
(ii) The existence of a threshold frequency.
(iii) The photoelectric current increases with increasing
light intensity.
(cont.)
(v) The photoelectric current decreases slowly as V
becomes much more negative.
(vi) The stopping potential is independent of the light
intensity.
(vii) The photoelectric current appears almost instantly
when the light is turned on.
(iv) The photoelectric current is independent of anodecathode potential difference V for V > 0.
Which of these cannot be explained by classical physics?
Conceptual question 4
Conceptual question 4 (cont.)
The figure on the left shows
a typical current-versuspotential difference graph
for a photoelectric effect
experiment.
(cont.)
On the figure, draw and
label curves for the following three situations:
(ii) The light frequency is increased
(iii) The cathode work function is increased.
It is assumed that no other parameters of the
experiment are changed in each case.
(i) The light intensity (i.e. power per area) is increased.
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Problem 6
Problem 7
Consider a potassium surface that is at a distance d = 75
cm away from bulb of power P = 100 W. Suppose that
the energy radiated by the bulb is 5% of the input
power. Treating each potassium atom as a circular disk
of radius r = 0.05 nm, determine the time required for
an atom to absorb an amount of energy equal to its
work function of f = 2.0 eV (1 eV = 1.6021019 J),
according to the wave picture of light.
When light of wavelength l1 = 450nm is incident on
potassium, photoelectrons with stopping potential Vs,1 =
0.52 V are emitted. If the wavelength is changed to l2
300 nm, the stopping potential is Vs,2 = 1.90 V. Using
only these numbers together with the values of the
speed of light c and the electron charge e,
(a) find the work function f of potassium and
Conceptual question 5
Conceptual question 6
Suppose we produce X-rays not by smashing electrons
into targets but by smashing protons, which are far
more massive (but each with the same magnitude of
charge as an electron). If the same accelerating
potential difference were used for both, how would the
cutoff wavelengths of the two X-ray spectra compare?
Explain your answer.
Why is it important to produce X-ray tubes with high
accelerating voltages that are also able to withstand
high electron currents?
Problem 8
Problem 9
An electron with kinetic energy of K = 20 keV is
brought to rest in a collision with a heavy nucleus.
An electron accelerated to kinetic energy E = 50 keV in
an X-ray tube has two successive collisions in being
brought to rest in the target, emitting two
bremsstrahlung photon in the process. The second
photon has wavelength 0.095nm longer than the first
one.
(a) Calculate the frequency f of the photon produced by
the collision. Assume the energy transfers to the
heavy nucleus is negligible.
(b) Show that momentum is not conserved in such
collision.
(b) compute a value of Planck's constant h.
(a) What are the wavelengths of the two photons?
(b) What is the kinetic energy of the electron after the
emission of the first photon?
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Relativistic Energy and Momentum

Compton Scattering
To calculate the momentum and energy of a particle
with speed v  c, we must use relativistic formulas:
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Hyperphysics
Conceptual question 7
Conceptual question 8
Must Compton scattering take place only between Xrays and free electrons? Can radiation in the visible
region (say, a green light) undergo Compton scattering
with a free electron?
Why is the Compton effect unimportant in the
transmission of television and radio waves?
Problem 10
Problem 11
A 0.3 MeV X-ray photon makes a head-on collision
with an electron initially at rest.
An X-ray source of unknown wavelength is directed at
a carbon sample. An electron is scattered with a speed
of u = 4.5107 m/s at an angle of q = 60 relative to the
motion of the original X-ray photon. Determine the
wavelength of the X-ray source.
(a) Using conservation of energy and momentum, find
the recoil velocity of the electron u in terms of the
speed of light c.
(b) Check the velocity obtained in (a) agrees with the
value determined from the Compton's formula.
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Conceptual question 9
Problem 12
An isolated atom can emit a photon and the atom's
internal energy drops. This process is called
spontaneous emission. Can an isolated electron emit a
photon? Why or why not?
A gamma-ray photon changes into a proton-antiproton
pair. After creation, each of the pair moves off
at 0.6c perpendicular to the motion of the photon.
(a) Ignore momentum conservation,
wavelength l of the photon.
find
the
(b) Assume that this interaction occurs as the photon
encounters a lead plate and that a lead nucleus
participates in momentum conservation. What
fraction of the photon's energy must be absorbed by
the lead nucleus?
Problem 13
Conceptual question 10
An electron with kinetic energy of Ke = 5 MeV
undergoes annihilation with a positron that is at rest,
producing two photons. One of the photons travels in
the direction of the incident electron. Calculate the
energy of each photon.
A beam of photons passes through a block of matter.
What are the three ways discussed in this chapter that
the photons can lose energy in interacting with the
material?
Conceptual question 11
Conceptual question 12
You have a monoenergetic source of X-rays of energy
84 keV, but for an experiment you need 70 keV X-rays.
How would you convert the X-ray energy from 84 to
70 keV?
The intensity of a beam of light is increased but the
frequency of the light is unchanged. As a result,
(i) the photons travel faster.
(ii) each photon has more energy.
(iii) the photons are larger.
(iv) there are more photons per second.
Which of these (perhaps more than one) are true?
Explain your answer.
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