Lecture 7
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Announcement
Phys.342 Exam 1 02/19. Wednesday 8:00-10:00pm in ME 1061
Phys. 342 Exam 2
04/02. Wednesday 8:00-10:00pm in Phys. 203
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Homework set #3
Due date Feb.07. 2014.
Tipler 6th Edition, Modern Physics, page 147
Chapter 3. Problems: 3.15
3.18
3.19
3.20
3.22
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Photoelectric Effect
When Electromagnetic radiation is incident on metallic surfaces
electrons may be ejected.
This phenomenon is called Photoelectric Effect.
First observed by Heinrich Hertz 1887.
e
Radiation in – Electron out
Metal
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We have seen that blackbody radiation in equilibrium with matter was
explained by the quantization of the oscillator energy levels in matter.
Albert Einstein (1905) conjectured that the energy of electromagnetic
radiation which these oscillators emit and absorb are also quantized.
A quantum of electromagnetic radiation is called a photon and denoted
by “ ”.
By conservation of energy the photon energy is given by the difference
in energy levels En 1 En of the oscillator:
E photon
En
1
En
(n 1)h
nh
h
hf
hc
where h is Planck’s constant.
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Quantization of energy in electromagnetic radiation explains
features of the photoelectric effect.
All atoms are built from a central nucleus surrounded by
electrons in concentric spherical shells.
In metals the electrons located in the outer shells are free
to move from atom to atom.
These electrons behave like a gas with a continuous spectrum
of energy levels.
If an electron absorbs a photon with sufficiently large energy,
the electron may gain enough energy to be freed from the metal.
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The photon has been absorbed by the electron and it disappeared
and its energy was transferred to the electron.
For an incident photon with a given energy, an electron emerges
with a maximum kinetic energy if happens to reside in the outer
most energy level (orbit).
If an electron is in a slightly lower energy level absorbs the
photon, then the electron emerges with less kinetic energy
EKmax
mv 2
(
) max
2
h
where is a property of the metal called the work function. The
work function is the amount of energy by which an outermost
electron is bound in the metal. The typical size of is a few
electron volts (eV).
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The schematic diagram of the apparatus used by Lenard is
shown below
V
- +
e
Cathode
Anode
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Let V denote the potential difference between the anode and cathode.
If the battery is connected so that the anode is more positive than
the cathode, the electrons will be accelerated toward the anode and
current will be observed in the circuit.
If the anode is made more negative than the cathode the electron will
decelerate.
At some negative voltage ( Vs ) no electrons will reach the anode.
When V is negative, the electrons are repelled from the anode, thus
only those electrons can reach the anode whose kinetic energy
EK mv 2 / 2 grater than e V .
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1 2
mv
2
eV0
bright light
same
frequency
dim light
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2
4
6
8 10
V
V0
The potential V0 is called the stopping potential. Note the fact that
the stopping potential is independent of intensity. Apparently
increasing the rate of energy, falling on the cathode, does not increase
the maximum kinetic energy of the emitted electrons contrary to
classical expectation.
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If however one shines two sources of light with the same
intensity but different frequencies f 2 f1 we obtain the
following result
f2
f 2 f1
f1
V
5
V02
V01
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Classical Time Lag
Classically there is a time lag between the turning on of the light
source and the appearance of photo electrons.
The incident light energy is distributed uniformly over the
illuminated surface.
The time required for an area the size of an atom to acquire enough
energy to allow the emission of electron can be calculated from
the intensity of the incident radiation.
No time lag has ever been observed.
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Assume a λ=400nm and I =10-2 W/m 2 intensity radiation incident
on potassium.
The work function of potassium is 2.22eV and assume the
atomic radius of potassium r =10-10 m.
The energy falling on an area
r 2 during t sec is
E 10 2 W / m 2 ( r 2 )t
Knowing that 1eV
t
1.6 10
19
(3.14 10
22
J / s )t
J after conversation
2.22eV 1.6 10 19 J / eV
3.14 10 22 J / s
1.13 103 s 18.8 min
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According to the classical prediction, no atom would be expected
to emit an electron until 18.8 min after the light source was turned on.
According to the photon model of light, each photon has enough
energy to eject an electron immediately.
Because of the low intensity, there are few photons incident per second,
so the chance of any particular atom absorbing a photon and emitting
an electron in any given time interval is small.
However, there are so many atoms in the cathode that some emit
electrons immediately.
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Experimental Results
• The photoelectric current (or the number of photoelectrons)
increases as the intensity of the light increases. However,
there does not appear to be a minimum intensity below
which the current is not produced.
• The photoelectric current is produced instantaneously upon
illumination.
• The reversed voltage required to stop all the photoelectrons
(stop potential) is independent of the intensity of the incident
light.
• For a given cathode material there exists a minimum
frequency of the incident light, below which no photocurrent
is produced.
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Classical Interpretation
The electrons in the metal can acquire sufficient energy from light to
escape. Consequently, the higher the light intensity, the more the
electrons are produced and thus the larger the photoelectric current.
However, it cannot explain the rest of the observed properties:
• There is a minimum intensity below which no photoelectrons
are produced.
• There is no time delay between the illumination of the
cathode by light and the appearance of photocurrent.
• The stop potential is independent of the intensity of the
incident light.
• There is a threshold on the frequency of the light, below
which no photocurrent can be produced.
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Einstein’s Interpretation
• A beam of monochromatic light is composed of small
quanta, now known as photons.
• The energy of a photon is uniquely determined by the
frequency of the light: E = hn, where h is Planck’s constant.
• The interaction between light and matter is equivalent to
that between photons and matter, and the photons act like
particles during the interaction.
• Unlike in the classical theory, the electrons in an atom can
acquire energy from the incident light only in units of
photon energy quantum, and, for all practical purposes, one
electron can absorb only one photon during the interaction.
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Work Function
The work function of a metal is defined as the amount of energy
that is required to remove an electron from the surface of the
metal. So, it is essentially the binding energy of electrons in the
metal.
• The work function obviously depend on the type of metal,
but it is typically on the order of a few electron volts.
• The work function represents the minimum energy that a
photon must possess to liberate an electron from the
surface of a metal. Therefore, it is the physical origin of the
frequency threshold for photoelectric processes, in the
context of the interpretation proposed by Einstein.
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Kinetic Energy of Photoelectrons
If the photocathode material has a work function f, and the
frequency of the incident light is n, Einstein’s interpretation
implies
eV0
K max
h
where V0 is the stop potential and Kmax is the maximum kinetic
energy of photoelectrons.
We conclude that
•
There is no requirement on the intensity for photoelectric
processes to occur.
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Confronting Experiment
• A photoelectron is produced as soon as a photon is absorbed,
so no time delay is expected here.
• The maximum kinetic energy of photoelectrons does not
depend on the intensity of the incident light; neither does
the stop potential naturally.
• The need for a lower threshold on the frequency of the
incident light is clear: no photoelectrons can be produced if
hn < f.
• Prediction: plotting the stop potential against the frequency
of the light should yield a straight line whose slope is
independent of the nature of the cathode material.
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Experimental Confirmation
As discussed, the stop potential
and the frequency of the light is
related by
eV0
V0
h
So, the slope of the line should
simply be h/e and the intercept
gives the work function.
n
Millikan confirmed the prediction of the universal slope
experimentally. From the slope, the Planck’s constant was
measured very precisely and the result agreed with that
from the studies of blackbody radiation.
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