Module 3
Chapters 1-3
A. Classical Physics of the 1890s
Although the Greek scholars Aristotle and Eratosthenes performed measurements
and calculations that today we would call physics, the discipline of physics has its roots
in the work of Galileo and Newton and others in the scientific revolution of the sixteenth
and seventeenth centuries. The knowledge and practice of physics grew steadily for 200
to 300 years until another revolution in physics took place, which is the subject of this
book. Physicists distinguish classical physics, which was mostly developed before 1895,
from modern physics, which is based on discoveries made after 1895. The precise year is
unimportant, but monumental changes occurred in physics around 1900.
Scientists and engineers of the late nineteenth century were indeed rather smug.
They thought they had just about everything under control (see the quotes from
Michelson and Kelvin at the beginning of the chapter). The best scientists of the day were
highly recognized and rewarded. Public lectures were frequent. Some scientists had easy
access to their political leaders, partly because science and engineering had benefited
their war machines, but also because of the many useful technological advances. Basic
research was recognized as important because of the commercial and military
applications of scientific discoveries. Although there were only primitive automobiles
and no airplanes in 1895, advances in these modes of transportation were soon to follow.
A few people already had telephones, and plans for widespread distribution of electricity
were under way.
Based on their success with what we now call macroscopic classical results,
scientists felt that given enough time and resources, they could explain just about
anything. They did recognize some difficult questions they still couldn’t answer; for
example, they didn’t clearly understand the structure of matter— that was under intensive
investigation. Nevertheless, on a macroscopic scale, they knew how to build efficient
engines. Ships plied the lakes, seas, and oceans of the world. Travel between the
countries of Europe was frequent and easy by train. Many scientists were born in one
country, educated in one or two others, and eventually worked in still other countries.
The most recent ideas traveled relatively quickly among the centers of research. Except
for some isolated scientists, of whom Einstein is the most notable example, discoveries
were quickly and easily shared. Scientific journals were becoming accessible.
The laws of mechanics were developed over hundreds of years by many
researchers. Important contributions were made by astronomers because of the great
interest in the heavenly bodies. Galileo (1564–1642) may rightfully be called the first
great experimenter. His experiments and observations laid the groundwork for the
important discoveries to follow during the next 200 years. Isaac Newton (1642–1727)
was certainly the greatest scientist of his time and one of the best the world has ever seen.
His discoveries were in the fields of mathematics, astronomy, and physics and include
gravitation, optics, motion, and forces.
We owe to Newton our present understanding of motion. He understood clearly
the relationships among position, displacement, velocity, and acceleration. He understood
how motion was possible and that a body at rest was just a special case of a body having
constant velocity. It may not be so apparent to us today, but we should not forget the
tremendous unification that Newton made when he pointed out that the motions of the
planets about our sun can be understood by the same laws that explain motion on Earth,
like apples falling from trees or a soccer ball being kicked toward a goal.
Electromagnetism developed over a long period of time. Important contributions
were made by Charles Coulomb (1736–1806), Hans Christian Oersted (1777–1851),
Thomas Young (1773–1829), André Ampère (1775–1836), Michael Faraday (1791–
1867), Joseph Henry (1797–1878), James Clerk Maxwell (1831– 1879), and Heinrich
Hertz (1857–1894). Maxwell showed that electricity and magnetism were intimately
connected and were related by a change in the inertial frame of reference. His work also
led to the understanding of electromagnetic radiation, of which light and optics are
special cases.
Thermodynamics deals with temperature T, heat Q, work W, and the internal
energy of systems U. The understanding of the concepts used in thermodynamics—such
as pressure P, volume V, temperature, thermal equilibrium, heat, entropy, and especially
energy—was slow in coming. We can understand the concepts of pressure and volume as
mechanical properties, but the concept of temperature must be carefully considered. The
internal energy of a system of noninteracting point masses depends only on the
temperature.
The change in the internal energy DU of a system is equal to the heat Q added to
the system plus the work W done on the system. It is not possible to convert heat
completely into work without some other change taking place. Equivalent forms of the
second law may appear different, but instead describe what kinds of energy processes can
or cannot take place. For example, it is not possible to build a perfect engine or a perfect
refrigerator. It is not possible to build a perpetual motion machine. Heat does not
spontaneously flow from a colder body to a hotter body without some other change
taking place. The second law forbids all these from happening.
Two other laws of thermodynamics are sometimes expressed. One is called the
zeroth law, and it is useful in understanding temperature. It states that if two thermal
systems are in thermodynamic equilibrium with a third system, they are in equilibrium
with each other. We can state it more simply by saying that two systems at the same
temperature as a third system have the same temperature as each other. This concept was
not explicitly stated until the twentieth century. The third law of thermodynamics
expresses that it is not possible to achieve an absolute zero temperature.
In 1811 the Italian physicist Amedeo Avogadro (1776–1856) proposed that equal
volumes of gases at the same temperature and pressure contained equal numbers of
molecules. This hypothesis was so far ahead of its time that it was not accepted for many
years. The famous English chemist John Dalton opposed the idea because he apparently
misunderstood the difference between atoms and molecules. Considering the rudimentary
nature of the atomic theory of matter at the time, this was not surprising. Daniel Bernoulli
(1700–1782) apparently originated the kinetic theory of gases in 1738, but his results
were generally ignored. Many scientists, including Newton, Laplace, Davy, Herapath,
and Waterston, had contributed to the development of kinetic theory by 1850. Theoretical
calculations were being compared with experiments, and by 1895 the kinetic theory of
gases was widely accepted. The statistical interpretation of thermodynamics was made in
the latter half of the nineteenth century by Maxwell, the Austrian physicist Ludwig
Boltzmann (1844–1906), and the American physicist J. Willard Gibbs (1839–1903).
B. Waves and Particles
We first learned the concepts of velocity, acceleration, force, momentum, and
energy in introductory physics by using a single particle with its mass concentrated in one
small point. In order to adequately describe nature, we add twoand three-dimensional
bodies and rotations and vibrations. However, many aspects of physics can still be treated
as if the bodies are simple particles. In particular, the kinetic energy of a moving particle
is one way that energy can be transported from one place to another.
But we have found that many natural phenomena can be explained only in terms
of waves, which are traveling disturbances that carry energy. This description includes
standing waves, which are superpositions of traveling waves. Most waves, like water
waves and sound waves, need an elastic medium in which to move. Curiously enough,
matter is not transported in waves—but energy is. Mass may oscillate, but it doesn’t
actually propagate along with the wave. Two examples are a cork and a boat on water. As
a water wave passes, the cork gains energy as it moves up and down, and after the wave
passes, the cork remains. The boat also reacts to the wave, but it primarily rocks back and
forth, throwing around things that are not fixed on the boat. The boat obtains considerable
kinetic energy from the wave.
Waves and particles were the subject of disagreement as early as the seventeenth
century, when there were two competing theories of the nature of light. Newton
supported the idea that light consisted of corpuscles (or particles). He performed
extensive experiments on light for many years and finally published his book Opticks in
1704. Geometrical optics uses straight-line, particle-like trajectories called rays to explain
familiar phenomena such as reflection and refraction. Geometrical optics was also able to
explain the apparent observation of sharp shadows. The competing theory considered
light as a wave phenomenon. Its strongest proponent was the Dutch physicist Christian
Huygens (1629–1695), who presented his theory in 1678. The wave theory could also
explain reflection and refraction, but it could not explain the sharp shadows observed.
Experimental physics of the 1600s and 1700s was not able to discern between the two
competing theories. Huygens’s poor health and other duties kept him from working on
optics much after 1678. Although Newton did not feel strongly about his corpuscular
theory, the magnitude of his reputation caused it to be almost universally accepted for
more than a hundred years and throughout most of the eighteenth century.
Finally, in 1802, the English physician Thomas Young (1773–1829) announced
the results of his two-slit interference experiment, indicating that light behaved as a wave.
Even after this singular event, the corpuscular theory had its supporters. During the next
few years Young and, independently, Augustin Fresnel (1788–1827) performed several
experiments that clearly showed that light behaved as a wave. By 1830 most physicists
believed in the wave theory—some 150 years after Newton performed his first
experiments on light.
Conservation laws are the guiding principles of physics. The application of a few
laws explains a vast quantity of physical phenomena. We listed the conservation laws of
classical physics in Section 1.1. They include energy, linear momentum, angular
momentum, and charge. Each of these is extremely useful in introductory physics. We
use linear momentum when studying collisions, and the conservation laws when
examining dynamics. We have seen the concept of the conservation of energy change. At
first we had only the conservation of kinetic energy in a force-free region. Then we added
potential energy and formed the conservation of mechanical energy. In our study of
thermodynamics, we added internal energy, and so on. The study of electrical circuits
was made easier by the conservation of charge flow at each junction and the conservation
of energy throughout all the circuit elements.
In our study of modern physics we will find that mass is added to the conservation
of energy, and the result is sometimes called the conservation of mass–energy, although
the term conservation of energy is still sufficient and generally used. When we study
fundamental particles we will add the conservation of baryons and the conservation of
leptons. Closely related to conservation laws are invariance principles. Some parameters
are invariant in some interactions or in specific systems but not in others. Examples
include time reversal, parity, and distance. We will study the Newtonian or Galilean
invariance and find it lacking in our study of relativity; a new invariance principle will be
needed. In our study of nuclear and elementary particles, conservation laws and
invariance principles will often be used.
In introductory physics, we often begin our study of forces by examining the
reaction of a mass at the end of a spring, because the spring force can be easily calibrated.
We subsequently learn about tension, friction, gravity, surface, electrical, and magnetic
forces. Despite the seemingly complex array of forces, we presently believe there are
only three fundamental forces. All the other forces can be derived from them. These three
forces are the gravitational, electroweak, and strong forces. Some physicists refer to the
electroweak interaction as separate electromagnetic and weak forces because the
unification occurs only at very high energies. The approximate strengths and ranges of
the three fundamental forces are listed in Table 1.1. Physicists sometimes use the term
interaction when referring to the fundamental forces because it is the overall interaction
among the constituents of a system that is of interest.
The primary component of the electroweak force is electromagnetic. The other
component is the weak interaction, which is responsible for beta decay in nuclei, among
other processes. In the 1970s Sheldon Glashow, Steven Weinberg, and Abdus Salam
predicted that the electromagnetic and weak forces were in fact facets of the same force.
Their theory predicted the existence of new particles, called W and Z bosons, which were
discovered in 1983. We discuss bosons and the experiment in Chapter 14. For all
practical purposes, the weak interaction is effective in the nucleus only over distances the
size of 10215 m. Except when dealing with very high energies, physicists mostly treat
nature as if the electromagnetic and weak forces were separate. Therefore, you will
sometimes see references to the four fundamental forces (gravity, strong,
electromagnetic, and weak).
The strong force is so strong that it easily binds two protons inside a nucleus even
though the electrical force of repulsion over the tiny confined space is huge. The strong
force is able to contain dozens of protons inside the nucleus before the electrical force of
repulsion becomes strong enough to cause nuclear decay. We study the strong force
extensively in this book, learning that neutrons and protons are composed of quarks, and
that the part of the strong force acting between quarks has the unusual name of color
force. Physicists strive to combine forces into more fundamental ones. Centuries ago the
forces responsible for friction, contact, and tension were all believed to be different.
Today we know they are all part of the electroweak force. Two hundred years ago
scientists thought the electrical and magnetic forces were independent, but after a series
of experiments, physicists slowly began to see their connection. This culminated in the
1860s in Maxwell’s work, which clearly showed they were but part of one force and at
the same time explained light and other radiation. Figure 1.6 is a diagram of the
unification of forces over time. Newton certainly had an inspiration when he was able to
unify the planetary motions with the apple falling from the tree.
C. The Atomic Theory of Matter
Today the idea that matter is composed of tiny particles called atoms is taught in
grade school and expounded throughout later schooling. We are told that the Greek
philosophers Democritus and Leucippus proposed the concept of atoms as early as 450
bc. The smallest piece of matter, which could not be subdivided further, was called an
atom, after the Greek word atomos, meaning “indivisible.” Not many new ideas were
proposed about atoms until the seventeenth century, when scientists started trying to
understand the properties and laws of gases. The work of Boyle, Charles, and Gay-Lussac
presupposed the interactions of tiny particles in gases. Chemists and physical chemists
made many important advances. In 1799 the French chemist Joseph Proust (1754–1826)
proposed the law of definite proportions, which states that when two or more elements
combine to form a compound, the proportions by weight (or mass) of the elements are
always the same. Water (H2O) is always formed of one part hydrogen and eight parts
oxygen by mass.
In 1811 the Italian physicist Amedeo Avogadro proposed the existence of
molecules, consisting of individual or combined atoms. He stated without proof that all
gases contain the same number of molecules in equal volumes at the same temperature
and pressure. Avogadro’s ideas were ridiculed by Dalton and others who could not
imagine that atoms of the same element could combine. If this could happen, they argued,
then all the atoms of a gas would combine to form a liquid. The concept of molecules and
atoms was indeed difficult to imagine, but finally, in 1858, the Italian chemist Stanislao
Cannizzaro (1826–1910) solved the problem and showed how Avogadro’s ideas could be
used to find atomic masses. Today we think of an atom as the smallest unit of matter that
can be identified with a particular element. A molecule is a combination of two or more
atoms of either like or dissimilar elements. Molecules can consist of thousands of atoms.
In 1827 the English botanist Robert Brown (1773–1858) observed with a
microscope the motion of tiny pollen grains suspended in water. The pollen appeared to
dance around in random motion, while the water was still. At first the motion (now called
Brownian motion) was ascribed to convection or organic matter, but eventually it was
observed to occur for any tiny particle suspended in liquid. The explanation according to
the atomic theory is that the molecules in the liquid are constantly bombarding the tiny
grains. A satisfactory explanation was not given until the twentieth century (by Einstein).
Although it may appear, according to the preceding discussion, that the atomic
theory of matter was universally accepted by the end of the nineteenth century, that was
not the case. Certainly most physicists believed in it, but there was still opposition. A
principal leader in the antiatomic movement was the renowned Austrian physicist Ernst
Mach. Mach was an absolute positivist, believing in the reality of nothing but our own
sensations. A simplified version of his line of reasoning would be that because we have
never seen an atom, we cannot say anything about its reality. The Nobel Prize–winning
German physical chemist Wilhelm Ostwald supported Mach philosophically but also had
more practical arguments on his side. In 1900 there were difficulties in understanding
radioactivity, x rays, discrete spectral lines, and how atoms formed molecules and solids.
Ostwald contended that we should therefore think of atoms as hypothetical constructs,
useful for bookkeeping in chemical reactions.
Overwhelming evidence for the existence of atoms was finally presented in the
first decade of the twentieth century. First, Einstein, in one of his three famous papers
published in 1905 (the others were about special relativity and the photoelectric effect),
provided an explanation of the Brownian motion observed almost 80 years earlier by
Robert Brown. Einstein explained the motion in terms of molecular motion and presented
theoretical calculations for the random walk problem. A random walk is a statistical
process that determines how far from its initial position a tiny grain may be after many
random molecular collisions. Einstein was able to determine the approximate masses and
sizes of atoms and molecules from experimental data.
Finally, in 1908, the French physicist Jean Perrin (1870–1942) presented data
from an experiment designed using kinetic theory that agreed with Einstein’s predictions.
Perrin’s experimental method of observing many particles of different sizes is a classic
work, for which he received the Nobel Prize for Physics in 1926. His experiment utilized
four types of measurements. Each was consistent with the atomic theory, and each gave a
quantitative determination of Avogadro’s number—the first accurate measurements that
had been made. By 1908 the atomic theory was well accepted.
We choose 1895 as a convenient time to separate the periods of classical and
modern physics, although this is an arbitrary choice based on discoveries made in 1895–
1897. The thousand or so physicists living in 1895 were rightfully proud of the status of
their profession. The precise experimental method was firmly established. Theories were
available that could explain many observed phenomena. In large part, scientists were
busy measuring and understanding such physical parameters as specific heats, densities,
compressibility, resistivity, indices of refraction, and permeabilities. The pervasive
feeling was that, given enough time, everything in nature could be understood by
applying the careful thinking and experimental techniques of physics. The field of
mechanics was in particularly good shape, and its application had led to the stunning
successes of the kinetic theory of gases and statistical thermodynamics.
In hindsight we can see now that this euphoria of success applied only to the
macroscopic world. Objects of human dimensions such as automobiles, steam engines,
airplanes, telephones, and electric lights either existed or were soon to appear and were
triumphs of science and technology. However, the atomic theory of matter was not
universally accepted, and what made up an atom was purely conjecture. The structure of
matter was unknown. There were certainly problems that physicists could not resolve.
Only a few of the deepest thinkers seemed to be concerned with them. Lord Kelvin, in a
speech in 1900 to the Royal Institution, referred to “two clouds on the horizon.” These
were the electromagnetic medium and the failure of classical physics to explain
blackbody radiation. We mention these and other problems here. Their solutions were
soon to lead to two of the greatest breakthroughs in human thought ever recorded—the
theories of quantum physics and of relativity.
The waves that were well known and understood by physicists all had media in
which the waves propagated. Water waves traveled in water, and sound waves traveled in
any material. It was natural for nineteenthcentury physicists to assume that
electromagnetic waves also traveled in a medium, and this medium was called the ether.
Several experiments, the most notable of which were done by Albert Michelson, had
sought to detect the ether without success. An extremely careful experiment by
Michelson and Morley in 1887 was so sensitive, it should have revealed the effects of the
ether. Subsequent experiments to check other possibilities were also negative. In 1895
some physicists were concerned that the elusive ether could not be detected. Was there an
alternative explanation?
The other difficulty with Maxwell’s electromagnetic theory had to do with the
electric and magnetic fields as seen and felt by moving bodies. What appears as an
electric field in one reference system may appear as a magnetic field in another system
moving with respect to the first. Although the relationship between electric and magnetic
fields seemed to be understood by using Maxwell’s equations, the equations do not keep
the same form under a Galilean transformation [see Equations (2.1) and (2.2)], a situation
that concerned both Hertz and Lorentz. Hertz unfortunately died in 1894 at the young age
of 36 and never experienced the modern physics revolution. The Dutch physicist Hendrik
Lorentz (1853–1928), on the other hand, proposed a radical idea that solved the
electrodynamics problem: space was contracted along the direction of motion of the
body. George FitzGerald in Ireland independently proposed the same concept. The
Lorentz–FitzGerald hypothesis, proposed in 1892, was a precursor to Einstein’s theory
advanced in 1905.
In 1895 thermodynamics was on a strong footing; it had achieved much success.
One of the interesting experiments in thermodynamics concerns an object, called a
blackbody, that absorbs the entire spectrum of electromagnetic radiation incident on it.
An enclosure with a small hole serves as a blackbody, because all the radiation entering
the hole is absorbed. A blackbody also emits radiation, and the emission spectrum shows
the electromagnetic power emitted per unit area. The radiation emitted covers all
frequencies, each with its own intensity. Precise measurements were carried out to
determine the spectrum of blackbody radiation, shown in Figure 1.8. Blackbody radiation
was a fundamental issue, because the emission spectrum is independent of the body itself
—it is characteristic of all blackbodies.
During the years 1895–1897 there were four discoveries that were all going to
require deeper understanding of the atom. The first was the discovery of x rays by the
German physicist Wilhelm Röntgen (1845–1923) in November 1895. Next came the
accidental discovery of radioactivity by the French physicist Henri Becquerel (1852–
1908), who in February 1896 placed uranium salt next to a carefully wrapped
photographic plate. When the plate was developed, a silhouette of the uranium salt was
evident—indicating the presence of a very penetrating ray.
The third discovery, that of the electron, was actually the work of several
physicists over a period of years. Michael Faraday, as early as 1833, observed a gas
discharge glow—evidence of electrons. Over the next few years, several scientists
detected evidence of particles, called cathode rays, being emitted from charged cathodes.
In 1896 Perrin proved that cathode rays were negatively charged. The discovery of the
electron, however, is generally credited to the British physicist J.QJ. Thomson (1856–
1940), who in 1897 isolated the electron (cathode ray) and measured its velocity and its
ratio of charge to mass.
The final important discovery of the period was made by the Dutch physicist
Pieter Zeeman (1865–1943), who in 1896 found that a single spectral line was sometimes
separated into two or three lines when the sample was placed in a magnetic field. The
(normal) Zeeman effect was quickly explained by Lorentz as the result of light being
emitted by the motion of electrons inside the atom. Zeeman and Lorentz showed that the
frequency of the light was affected by the magnetic field according to the classical laws
of electromagnetism.
The unresolved issues of 1895 and the important discoveries of 1895–1897 bring
us to the subject of this book, Modern Physics. In 1900 Max Planck completed his
radiation law, which solved the blackbody problem but required that energy be quantized.
In 1905 Einstein presented his three important papers on Brownian motion, the
photoelectric effect, and special relativity. While the work of Planck and Einstein may
have solved the problems of the nineteenth-century physicists, they broadened the
horizons of physics and have kept physicists active ever since.
D. The Apparent Need for Ether
One of the great theories of physics appeared early in the twentieth century when
Albert Einstein presented his special theory of relativity in 1905. We learned in
introductory physics that Newton’s laws of motion must be measured relative to some
reference frame. A reference frame is called an inertial frame if Newton’s laws are valid
in that frame. If a body subject to no net external force moves with constant velocity, then
the coordinate system attached to that body defines an inertial frame. If Newton’s laws
are valid in one reference frame, then they are also valid in a reference frame moving at a
uniform velocity relative to the first system. This is known as the Newtonian principle of
relativity or Galilean invariance.
In the late nineteenth century Albert Einstein was concerned that although
Newton’s laws of motion had the same form under a Galilean transformation, Maxwell’s
equations did not. Einstein believed so strongly in Maxwell’s equations that he showed
there was a significant problem in our understanding of the Newtonian principle of
relativity. In 1905 he published ideas that rocked the very foundations of physics and
science. He proposed that space and time are not separate and that Newton’s laws are
only an approximation. This special theory of relativity and its ramifications are the
subject of this chapter. We begin by presenting the experimental situation historically—
showing why a problem existed and what was done to try to rectify the situation. Then
we discuss Einstein’s two postulates on which the special theory is based. The
interrelation of space and time is discussed, and several amazing and remarkable
predictions based on the new theory are presented.
As the concepts of relativity became used more often in everyday research and
development, it became essential to understand the transformation of momentum, force,
and energy. Here we study relativistic dynamics and the relationship between mass and
energy, which leads to one of the most famous equations in physics and a new
conservation law of mass–energy. Finally, we return to electromagnetism to investigate
the effects of relativity. We learn that Maxwell’s equations don’t require change, and
electric and magnetic effects are relative, depending on the observer. We leave until
Chapter 15 our discussion of Einstein’s general theory of relativity.
Thomas Young, an English physicist and physician, performed his famous
experiments on the interference of light in 1802. A decade later, the French physicist and
engineer Augustin Fresnel published his calculations showing the detailed understanding
of interference, diffraction, and polarization. Because all known waves (other than light)
require a medium in which to propagate (water waves have water, sound waves have, for
example, air, and so on), it wasQnaturally assumed that light also required a medium, even
though light was apparently able to travel in vacuum through outer space. This medium
was called the luminiferous ether, or just ether for short, and it must have some amazing
properties. The ether had to have such a low density that planets could pass through it,
seemingly for eternity, with no apparent loss of orbit position. Its elasticity must be
strong enough to pass waves of incredibly high speeds!
Albert Michelson (1852–1931) performed perhaps the most significant American
physics experiment of the 1800s. Michelson, who was the first U.S. citizen to receive the
Nobel Prize in Physics (1907), was an ingenious scientist who built an extremely precise
device called an interferometer, which measures the phase difference between two light
waves. Michelson used his interferometer to detect the difference in the speed of light
passing through the ether in different directions. The basic technique is shown in Figure
2.2. Initially, it is assumed that one of the interferometer arms (AC) is parallel to the
motion of the Earth through the ether.
Light leaves the source S and passes through the glass plate at A. Because the
back of A is partially silvered, part of the light is reflected, eventually going to the mirror
at D, and part of the light travels through A on to the mirror at C. The light is reflected at
the mirrors C and D and comes back to the partially silvered mirror A, where part of the
light from each path passes on to the telescope and eye at E. The compensator is added at
B to make sure both light paths pass through equal thicknesses of glass. Interference
fringes can be found by using a bright light source such as sodium, with the light filtered
to make it monochromatic, and the apparatus is adjusted for maximum intensity of the
light at E. We will show that the fringe pattern should shift if the apparatus is rotated
through 908 such that arm AD becomes parallel to the motion of the Earth through the
ether and arm AC is perpendicular to the motion.
The measurement so shattered a widely held belief that many suggestions were
made to explain it. What if the Earth just happened to have a zero motion through the
ether at the time of the experiment? Michelson and Morley repeated their experiment
during night and day and for different seasons throughout the year. It is unlikely that at
least sometime during these many experiments, the Earth would not be moving through
the ether. Michelson and Morley even took their experiment to a mountaintop to see if the
effects of the ether might be different. There was no change.
Of the many possible explanations of the null ether measurement, the one taken
most seriously was the ether drag hypothesis. Some scientists proposed that the Earth
somehow dragged the ether with it as the Earth rotates on its own axis and revolves
around the sun. However, the ether drag hypothesis contradicts results from several
experiments, including that of stellar aberration noted by the British astronomer James
Bradley in 1728. Bradley noticed that the apparent position of the stars seems to rotate in
a circular motion with a period of one year. The angular diameter of this circular motion
with respect to the Earth is 41Qseconds of arc. This effect can be understood by an
analogy. From the viewpoint of a person sitting in a car during a rainstorm, the raindrops
appear to fall vertically when the car is at rest but appear to be slanted toward the
windshield when the car is moving forward. The same effect occurs for light coming
from stars directly above the Earth’s orbital plane.
E. Einstein’s Postulates
At the turn of the twentieth century, the Michelson–Morley experiment had laid to
rest the idea of finding a preferred inertial system for Maxwell’s equations, yet the
Galilean transformation, which worked for the laws of mechanics, was invalid for
Maxwell’s equations. This quandary represented a turning point for physics. Albert
Einstein (1879–1955) was only two years old when Michelson reported his first null
measurement for the existence of the ether. Einstein said that he began thinking at age 16
about the form of Maxwell’s equations in moving inertial systems, and in 1905, when he
was 26 years old, he published his startling proposal* about the principle of relativity,
which he believed to be fundamental.
Working without the benefit of discussions with colleagues outside his small
circle of friends, Einstein was apparently unaware of the interest concerning the null
result of Michelson and Morley.† Einstein instead looked at the problem in a more formal
manner and believed that Maxwell’s equations must be valid in all inertial frames. With
piercing insight and genius, Einstein was able to bring together seemingly inconsistent
results concerning the laws of mechanics and electromagnetism with two postulates (as
he called them; today we would call them laws).
The first postulate indicates that the laws of physics are the same in all coordinate
systems moving with uniform relative motion to each other. Einstein showed that
postulate 2 actually follows from the first one. He returned to the principle of relativity as
espoused by Newton. Although Newton’s principle referred only to the laws of
mechanics, Einstein expanded it to include all laws of physics—including those of
electromagnetism. We can now modify our previous definition of inertial frames of
reference to be those frames of reference in which all the laws of physics are valid.
We must be careful when comparing the same event in two systems moving with
respect to one another. Time comparison can be accomplished by sending light signals
from one observer to another, but this information can travel only as fast as the finite
speed of light. It is best if each system has its own observers with clocks that are
synchronized. How can we do this? We place observers with clocks throughout a given
system. If, when we bring all the clocks together at one spot at rest, all the clocks agree,
then the clocks are said to be synchronized. However, we have to move the clocks
relative to each other to reposition them, and this might affect the synchronization. A
better way would be to have a light flash half-way between each pair of clocks at rest and
make sure the pulses arrive simultaneously at each clock. This will require many
measurements, but it is a safe way to synchronize the clocks. We can determine the time
of an event occurring far away from us by having a colleague at the event, with a clock
fixed at rest, measure the time of the particular event, and send us the results, for
example, by text, email, telephone, or even by mail. If we need to check our clocks, we
can always send light signals to each other over known distances at some predetermined
time.
F. Time Dilation and Length Contraction
The Lorentz transformations have immediate consequences with respect to time
and length measurements made by observers in different inertial frames. We shall
consider time and length measurements separately and then see how they are related to
one another. Thus the time interval measured in the moving system K9 is greater than the
time interval measured in system K where the sparkler is at rest. This effect is known as
time dilation and is a direct result of Einstein’s two postulates.
The time dilation result is often interpreted by saying that moving clocks run slow
by the factor y-1, and sometimes this is a useful way to remember the effect. The moving
clock in this case can be any kind of clock. It can be the time that sand takes to pass
through an hourglass, the time a sparkler stays lit, the time between heartbeats, the time
between ticks of a clock, or the time spent in a class lecture. In all cases, the actual time
interval on a moving clock is greater than the proper time as measured on a clock at rest.
The proper time is always the smallest possible time interval between two events.
The preceding results naturally seem a little strange to us. In relativity we often
carry out thought (or gedanken from the German word) experiments, because the actual
experiments would be somewhat impractical. Consider the following gedanken
experiment. Now consider what might happen to the length of objects in relativity. Let an
observer in each system K and K9 have a meterstick at rest in his or her own respective
system. Every observer measures a meterstick at rest in his or her own system to have the
same length, namely one meter. The length as measured at rest is called the proper length.
A spaceship launched from a space station quickly reaches its cruising speed of
0.60c with respect to the space station when a band of asteroids is observed straight ahead
of the ship. Mary, the commander, reacts quickly and orders her crew to blast away the
asteroids with the ship’s proton gun to avoid a catastrophic collision. Frank, the admiral
on the space station, listens with apprehension to the communications because he fears
the asteroids may eventually destroy his space station as well. Will the high-energy
protons of speed 0.99c be able to successfully blast away the asteroids and save both the
spaceship and space station? If 0.99c is the speed of the protons with respect to the
spaceship, what speed will Frank measure for the protons?
The Lorentz transformation does not allow a material object to have a speed
greater than c. Only massless particles, such as light, can have speed c. If the crew
members of the spaceship spot the asteroids far enough in advance, their reaction times
should allow them to shoot down the uncharacteristically swiftly moving asteroids and
save both the spaceship and the space station. Although no particle with mass can carry
energy faster than c, we can imagine a signal being processed faster than c. Consider the
following gedanken experiment.
G. Experimental Verification
We have used the special theory of relativity to describe some unusual
phenomena. The special theory has also been used to make some startling predictions
concerning length contraction, time dilation, and velocity addition. In this section we
discuss only a few of the many experiments that have been done to confirm the special
theory of relativity. Because the classical calculation does not agree with the
experimental result, we should consider a relativistic calculation. The muons are moving
at a speed of 0.98c with respect to us on Earth, so the effects of time dilation will be
dramatic.
An interesting test of the velocity addition relations was made by T. Alväger and
colleagues* at the CERN nuclear and particle physics research facility on the border of
Switzerland and France. The experimental measurement was accomplished by measuring
the time taken for the g rays to travel between two detectors placed about 30 m apart and
was in excellent agreement with the relativistic prediction, but not the Galilean one. We
again have conclusive evidence of the need for the special theory of relativity.
Although we have mentioned only three rather interesting experiments, physicists
performing experiments with nuclear and particle accelerators have examined thousands
of cases that verify the correctness of the concepts discussed here. Quantum
electrodynamics (QED) includes special relativity in its framework, and QED has been
tested to one part in 1012. Lorentz symmetry requires the laws of physics to be the same
for all observers, and Lorentz symmetry is important at the very foundation of our
description of fundamental particles and forces. Lorentz symmetry, together with the
principles of quantum mechanics that are discussed in much of the remainder of this
book, form the framework of relativistic quantum field theory.
One of the most interesting topics in relativity is the twin (or clock) paradox.
Almost from the time of publication of Einstein’s famous paper in 1905, this subject has
received considerable attention, and many variations exist. Let’s summarize the paradox.
Suppose twins, Mary and Frank, choose different career paths. Mary (the Moving twin)
becomes an astronaut and Frank (the Fixed twin) a stockbroker. At age 30, Mary sets out
on a spaceship to study a star system 8Qly from Earth. Mary travels at very high speeds to
reach the star and returns during her life span. According to Frank’s understanding of
special relativity, Mary’s biological clock ticks more slowly than his own, so he claims
that Mary will return from her trip younger than he. The paradox is that Mary similarly
claims that Frank is moving rapidly with respect to her, so that when she returns, Frank
will be the younger twin. To complicate the paradox further one could argue that because
nature cannot allow both possibilities, it must be true that symmetry prevails and that the
twins will still be the same age. Which is the correct solution?
The important fact here is that Frank’s clock is in an inertial system† during the
entire trip; however, Mary’s clock is not. As long as Mary is traveling at constant speed
away from Frank, both of them can argue that the other twin is aging less rapidly.
However, when Mary slows down to turn around, she leaves her original inertial system
and eventually returns in a completely different inertial system. Mary’s claim is no longer
valid, because she does not remain in the same inertial system. There is also no doubt as
to who is in the inertial system. Frank feels no acceleration during Mary’s entire trip, but
Mary will definitely feel acceleration during her reversal time, just as we do when we
step hard on the brakes of a car.
The acceleration at the beginning and the deceleration at the end of her trip
present little problem, because the fixed and moving clocks could be compared if Mary
were just passing by Frank each way. It is Mary’s acceleration at the star system that is
the key. If we invoke the two postulates of special relativity, there is no paradox. The
instantaneous rate of Mary’s clock is determined by her instantaneous speed, but she
must account for the acceleration effect when she turns around. A careful analysis of
Mary’s entire trip using special relativity, including acceleration, will be in agreement
with Frank’s assessment that Mary is younger.
H. Spacetime
When describing events in relativity, it is sometimes convenient to represent
events on a spacetime diagram as shown in Figure 2.21. For convenience we use only one
spatial coordinate x and specify position in this one dimension. We use ct instead of time
so that both coordinates will have dimensions of length. Spacetime diagrams were first
used by H. Minkowski in 1908 and are often called Minkowski diagrams. We have
learned in relativity that we must denote both space and time to specify an event. This is
the origin of the term fourth dimension for time.
The events for A and B in Figure 2.21 are denoted by the respective coordinates
(xA, ctA) and (xB, ctB), respectively. The line connecting events A and B is the path
from A to B and is called a worldline. A spaceship launched from x 5 0, ct 5 0 with
constant velocity v has the worldline shown in Figure 2.22: a straight line with slope c/v.
For example, a light signal sent out from the origin with speed c is represented on a
spacetime graph with a worldline that has a slope c/c 5 1, so that line makes an angle of
458 with both the x and ct axes. Any real motion in the spacetime diagram cannot have a
slope of less than 1 (angle with the x axis , 458), because that motion would have a speed
greater than c. The Lorentz transformation does not allow such a speed.
Invariant quantities have the same value in all inertial frames. They serve a
special role in physics because their values do not change from one system to another.
For example, the speed of light c is invariant. The four-vector formalism gives us
equations that produce form-invariant quantities under appropriate Lorentz
transformations. It allows the mathematical construction of relativistic physics to be
somewhat easier. However, the penalty is that we would have to stop and learn matrix
algebra and perhaps even about tensors and, eventually, spinors. At this point in our study
there is little to be gained in understanding about relativity. Another disadvantage in
utilizing fourvectors at this point is that there is no general agreement among authors as
to terminology.
You may have already studied the Doppler effect of sound in introductory
physics. It causes an increased frequency of sound as a source such as a train (with
whistle blowing) approaches a receiver (our eardrum) and a decrease in frequency as the
source recedes. A change in sound frequency also occurs when the source is fixed and the
receiver is moving. The change in frequency of the sound wave depends on whether the
source or receiver is moving. On first thought it seems that the Doppler effect in sound
violates the principle of relativity, until we realize that there is in fact a special frame for
sound waves. Sound waves depend on media such as air, water, or a steel plate to
propagate. For light, however, there is no such medium. It is only relative motion of the
source and receiver that is relevant, and we expect some differences between the
relativistic Doppler effect for light waves and the normal Doppler effect for sound. It is
not possible for a source of light to travel faster than light in a vacuum, but it is possible
for a source of sound to travel faster than the speed of sound. Similarly, in a medium such
as water in which light travels slower than c, a light source can travel faster than the
speed of light in that medium.
Elements absorb and emit characteristic frequencies of light due to the existence
of particular atomic levels. We will learn more about this later. Scientists have observed
these characteristic frequencies in starlight and have observed shifts in the frequencies.
One reason for these shifts is the Doppler effect, and the frequency changes are used to
determine the speed of the emitting object with respect to us. This is the source of the
redshifts of starlight caused by objects moving away from us. These data have been used
to ascertain that the universe is expanding. The farther away the star, the higher the
redshift. This observation is what led Harlow Shapley and Edwin Hubble to the idea that
the universe started with a Big Bang.
I. Relativistic Momentum
We need to take a careful look at our previous definition of linear momentum to
see whether it is still valid at high speeds. According to Newton’s second law, for
example, an acceleration of a particle already moving at very high speeds could lead to a
speed greater than the speed of light. That would be in conflict with the Lorentz
transformation, so we expect that Newton’s second law might somehow be modified at
high speeds. Because physicists believe the conservation of linear momentum is
fundamental, we begin by considering a collision that has no external forces. Frank
(Fixed or stationary system) is at rest in system K holding a ball of mass m. Mary
(Moving system) holds a similar ball in system K9 that is moving in the x direction with
velocity v with respect to system K.
Rather than abandon the conservation of linear momentum, let us look for a
modification of the definition of linear momentum that preserves both it and Newton’s
second law. We follow a procedure similar to the one we used in deriving the Lorentz
transformation; we assume the simplest, most reasonable change that may preserve the
conservation of linear momentum. Some physicists like to refer to the mass in Equation
(2.48) as the rest mass m0 and call the term m 5 gm0 the relativistic mass.
In this manner the classical form of linear momentum, muS, is retained. The mass
is then imagined to increase at high speeds. Most physicists prefer to keep the concept of
mass as an invariant, intrinsic property of an object. We adopt this latter approach and
will use the term mass exclusively to mean rest mass. Although we may use the terms
mass and rest mass synonymously, we will not use the term relativistic mass. The use of
relativistic mass too often leads the student into mistakenly inserting the term into
classical expressions where it does not apply.
We now turn to the concepts of energy and force. When forming the new theories
of relativity and quantum physics, physicists resisted changing the wellaccepted ideas of
classical physics unless absolutely necessary. In this same spirit we also choose to keep
intact as many definitions from classical physics as possible and let experiment dictate
when we are incorrect. In practice, the concept of force is best defined by its use in
Newton’s laws of motion, and we retain here the classical definition of force as used in
Newton’s second law.
These last few equations suggest the equivalence of mass and energy, a concept
attributed to Einstein. Nuclear reactions are certain proof that mass and energy are
equivalent. The concept of motion as being described by kinetic energy is preserved in
relativistic dynamics, but a particle with no motion still has energy through its mass. In
order to establish the equivalence of mass and energy, we must modify two of the
conservation laws that we learned in classical physics.
Mass and energy are no longer two separately conserved quantities. We must
combine them into one law of the conservation of mass–energy. We will see ample proof
during the remainder of this book of the validity of this basic conservation law. Physicists
believe that linear momentum is a more fundamental concept than kinetic energy. There
is no conservation of kinetic energy, whereas the conservation of linear momentum in
isolated systems is inviolate as far as we know. A more fundamental result for the total
energy in Equation (2.65) might include momentum rather than kinetic energy.
J. Computations in Modern Physics
We were taught in introductory physics that the international system of units is
preferable when doing calculations in science and engineering. This is generally true, but
in modern physics we sometimes use other units that are more convenient for atomic and
subatomic scales. In this section we introduce some of those units and demonstrate their
practicality through several examples. The eV unit is used more often in modern physics
than the SI unit J. The term eV is often used with the SI prefixes where applicable.
The equivalence of mass and energy becomes apparent when we study the
binding energy of atoms and nuclei that are formed from individual particles. For
example, the hydrogen atom is formed from a proton and electron bound together by the
electrical (Coulomb) force. A deuteron is a proton and neutron bound together by the
nuclear force. The potential energy associated with the force keeping the system together
is called the binding energy.
We have been concerned mostly with the kinematical and dynamical aspects of
the special theory of relativity strictly from the mechanics aspects. However, recall that
Einstein first approached relativity through electricity and magnetism. He was convinced
that Maxwell’s equations were invariant (have the same form) in all inertial frames.
Einstein was convinced that magnetic fields appeared as electric fields observed in
another inertial frame. That conclusion is the key to electromagnetism and relativity.
Maxwell’s equations and the Lorentz force law are invariant in different inertial
frames. In fact, with the proper Lorentz transformations of the electric and magnetic
fields (from relativity theory) together with Coulomb’s law (force between stationary
charges), Maxwell’s equations can be obtained. We will not attempt that fairly difficult
mathematical task here, nor do we intend to obtain the Lorentz transformation of the
electric and magnetic fields. These subjects are studied in more advanced physics classes.
However, we will show qualitatively that the magnetic force that one observer sees is
simply an electric force according to an observer in another inertial frame. The electric
field arises from charges, whereas the magnetic field arises from moving charges.
Electricity and magnetism were well understood in the late 1800s. Maxwell
predicted that all electromagnetic waves travel at the speed of light, and he combined
electricity, magnetism, and optics into one successful theory. This classical theory has
withstood the onslaught of time and experimental tests.* There were, however, some
troubling aspects of the theory when it was observed from different Galilean frames of
reference. In 1895 H. A. Lorentz “patched up” the difficulties with the Galilean
transformation by developing a new transformation that now bears his name, the Lorentz
transformation. However, Lorentz did not understand the full implication of what he had
done. It was left to Einstein, who in 1905 published a paper titled “On the
Electrodynamics of Moving Bodies,” to fully merge relativity and electromagnetism.
Einstein did not even mention the famous Michelson–Morley experiment in this classic
1905 paper, which we take as the origin of the special theory of relativity, and the
Michelson–Morley experiment apparently played little role in his thinking. Einstein’s
belief that Maxwell’s equations describe electromagnetism in any inertial frame was the
key that led Einstein to the Lorentz transformations. Maxwell’s assertion that all
electromagnetic waves travel at the speed of light and Einstein’s postulate that the speed
of light is invariant in all inertial frames seem intimately connected.
K. Discovery of the X Ray and the Electron
In the 1890s scientists and engineers were familiar with the “cathode rays” that
were generated from one of the metal plates in an evacuated tube across which a large
electric potential had been established. The origin and constitution of cathode rays were
not known. The concept of an atomic substructure of matter was widely accepted because
of its use in explaining the results of chemical experiments. Therefore, it was surmised
that cathode rays had something to do with atoms. It was known, for example, that
cathode rays could penetrate matter, and their properties were of great interest and under
intense investigation in the 1890s.
In 1895 Wilhelm Röntgen was studying the effects of cathode rays passing
through various materials and noticed a nearby phosphorescent screen glowing vividly in
the darkened room. Röntgen soon realized he was observing a new kind of ray, one that,
unlike cathode rays, was unaffected by magnetic fields and was far more penetrating than
cathode rays. These x rays, as he called them, were apparently produced by the cathode
rays bombarding the glass walls of his vacuum tube. Röntgen studied their transmission
through many materials and even showed that he could obtain an image of the bones in a
hand when the x rays were allowed to pass through as shown in Figure 3.1. This
experiment created tremendous excitement, and medical applications of x rays were
quickly developed. For this discovery, Röntgen received the first Nobel Prize for Physics
in 1901.
Thomson was able to prove in 1897 that the charged particles emitted from a
heated electrical cathode were in fact the same as cathode rays. The main features of
Thomson’s experiment are shown in the schematic apparatus of Figure 3.2. The rays
from the cathode are attracted to the positive potential on aperture A (anode) and are
further collimated by aperture B to travel in a straight line and strike a fluorescent screen
in the rear of the tube, where they can be visually detected by a flash of light. A voltage
across the deflection plates sets up an electric field that deflects charged particles.
Previously, in a similar experiment, Hertz had observed no effect on the cathode rays due
to the deflecting voltage. Thomson at first found the same result, but on further
evacuating the glass tube, he observed the deflection and proved that cathode rays had a
negative charge. The previous experiment, in a poorer vacuum, had failed because the
cathode rays had interacted with and ionized the residual gas. Thomson also studied the
effects of a magnetic field upon the cathode rays and proved convincingly that the
cathode rays acted as negatively charged particles (electrons) in both electric and
magnetic fields, for which he received the Nobel Prize for Physics in 1906.
After Thomson’s measurement of e/m and the confirmation of the cathode ray as
a charge carrier (called electron), several investigators attempted to determine the actual
magnitude of the electron’s charge. In 1911 the American physicist Robert A. Millikan
(1868–1953) reported convincing evidence for an accurate determination of the electron’s
charge. Millikan’s classic experiment began in 1907 at the University of Chicago. The
experiment consisted of visual observation of the motion of uncharged and both
positively and negatively charged oil drops moving under the influence of electrical and
gravitational forces. The essential parts of the apparatus are shown in Figure 3.4. As the
drops emerge from the nozzle, frictional forces sometimes cause them to be charged.
Millikan’s method consisted of balancing the upward force of the electric field between
the plates against the downward force of the gravitational field.
L. Line Spectra
In contrast to the smooth, continuous radiation spectrum obtained from thermal
bodies, chemical elements produce unique wavelengths (colors) when burned in a flame
or when excited in an electrical discharge, a fact already known in the early 1800s.
Prisms had been used to investigate these early sources of spectra, and optical
spectroscopy became an important area of experimental physics, primarily because of the
modern development of diffraction gratings by Henry Rowland* (1848–1901) of Johns
Hopkins University in the 1880s.
The resulting pattern of light bands and dark areas on the screen is called a line
spectrum. By 1860 Bunsen and Kirchhoff realized that the wavelengths of these line
spectra would allow identification of the chemical elements and the composition of
materials. It was discovered that each element had its own characteristic wavelengths (see
examples shown in Appendix 9). The field of spectroscopy flourished because finer and
more evenly ruled gratings became available, and improved experimental techniques
allowed more spectral lines to be observed and catalogued. Particular attention was paid
to the sun’s spectrum in hopes of understanding the origin of sunlight. The helium atom
was actually “discovered” by its line spectra from the sun before it was recognized on
Earth.
Many scientists believed that the increasing number of spectral lines suggested a
complicated internal structure of the atom, and that by carefully investigating the
wavelengths for many elements, the structure of atoms and matter could be understood.
That belief was eventually partially realized. For much of the nineteenth century,
scientists attempted to find some simple underlying order for the characteristic
wavelengths of line spectra. Hydrogen appeared to have an especially simple-looking
spectrum, and because some chemists thought hydrogen atoms might be the constituents
of heavier atoms, hydrogen was singled out for intensive study. Finally, in 1885, Johann
Balmer, a Swiss schoolteacher, succeeded in obtaining a simple empirical formula that fit
the wavelengths of the four lines then known in the hydrogen spectrum and several
ultraviolet lines that had been identified in the spectra of white stars.
The word atom means “not further divisible.” Today some scientists believe, as
these ancient philosophers did, that matter must eventually be indivisible. However, as
we have encountered new experimental facts, our ideas about the fundamental,
indivisible “building blocks” of matter have changed. Whatever the elementary units of
matter may turn out to be, we suppose there are some basic units of mass–energy of
which matter is composed. This idea is hardly foreign to us: we have already seen that
Millikan’s oil-drop experiment showed the quantization of electric charge.
In nature we see other examples of quantization. The measured atomic weights
are not continuous—they have only discrete values, which are close to integral multiples
of a unit mass. Molecules are formed from an integral number of atoms. The water
molecule is made up of exactly two atoms of hydrogen and one of oxygen. The fact that
an organ pipe produces one fundamental musical note with overtones is a form of
quantization arising from fitting a precise number (or fractions) of sound waves into the
pipe.
Line spectra provide a prime example of quantization. We have learned that the
hydrogen line spectra have precise wavelengths that can be described empirically by
simple equations. We will see in the next chapter that Niels Bohr used some simple
assumptions based on the new quantum theory to model the atom and successfully
predict these wavelengths. By the end of the nineteenth century radiation spectra had
been well studied. There certainly didn’t appear to be any quantization effects observed
in blackbody radiation spectra emitted by hot bodies. However, the explanation of
blackbody radiation spectra was to have a tremendous influence on the discovery of
quantum physics.
M. Blackbody Radiation
It has been known for many centuries that when matter is heated, it emits
radiation. We can feel heat radiation emitted by the heating element of an electric stove
as it warms up. As the heating element reaches 550°C, its color becomes dark red, turning
to bright red around 700°C. If the temperature were increased still further, the color
would progress through orange, yellow, and finally white. We can determine
experimentally that a broad spectrum of wavelengths is emitted when matter is heated.
This process was of great interest to physicists of the nineteenth century. They measured
the intensity of radiation being emitted as a function of material, temperature, and
wavelength. All bodies simultaneously emit and absorb radiation. When a body’s
temperature is constant in time, the body is said to be in thermal equilibrium with its
surroundings. In order for the temperature to be constant, the body must absorb thermal
energy at the same rate as it emits it. This implies that a good thermal emitter is also a
good absorber.
Physicists generally try to study first the simplest or most idealized case of a
problem to gain the insight needed to analyze more complex situations. For thermal
radiation the simplest case is a blackbody, which has the ideal property that it absorbs all
the radiation falling on it and reflects none. The simplest way to construct a blackbody is
to drill a small hole in the wall of a hollow container as shown in Figure 3.8. Radiation
entering the hole will be reflected around inside the container and then eventually
absorbed. Only a small fraction of the entering rays will be reemitted through the hole. If
the blackbody is in thermal equilibrium, then it must also be an excellent emitter of
radiation. Blackbody radiation is theoretically interesting because of its universal
character: the radiation properties of the blackbody (that is, the cavity) are independent of
the particular material of which the container is made. Physicists can study the previously
mentioned properties of intensity versus wavelength (called spectral distribution) at fixed
temperatures without having to understand the details of emission or absorption by a
particular kind of atom. The question of precisely what the thermal radiation actually
consisted of was also of interest, although it was assumed, for lack of evidence to the
contrary (and correctly, it turned out!), to be electromagnetic radiation.
In the 1880s the German Max Planck, who was an expert on the second law of
thermodynamics, rejected Boltzmann’s statistical version of thermodynamics and even
doubted the atomic theory of matter or “atomism.” Planck was appointed Professor of
Physics at the University of Berlin in 1889, and his views began to change. He was not
quite ready to accept atomism, but he set out in 1895 to examine the irreversibility of
radiation processes. He thought he had shown that laws of electromagnetism
distinguished between past and present, but Boltzmann showed in 1897 that there could
be no difference. Planck then began to consider blackbody radiation. Planck tried various
functions of wavelength and temperature until he found a single formula that fit the
measurements of I(l,T) over the entire wavelength range. It is not clear that Planck was
even aware of Lord Rayleigh’s result. Planck was simply looking for a formula that fit
the known blackbody spectral distribution. Planck reported his formula in October 1900,
but he realized a month later it was nothing but an inspired guess. By then Planck had
accepted Boltzmann’s view. Planck followed Hertz’s work using oscillators to confirm
the existence of Maxwell’s electromagnetic waves, and lacking detailed information
about the atomic composition of the cavity walls, Planck assumed that the radiation in the
cavity was emitted (and absorbed) by some sort of “oscillators” that were contained in the
walls. When adding up the energies of the oscillators, he assumed (for convenience) that
each one had an energy that was an integral multiple of hf, where f is the frequency of the
oscillating wave and h is a constant.
Perhaps the most compelling, and certainly the simplest, evidence for the
quantization of radiation energy comes from the only acceptable explanation of the
photoelectric effect. While Heinrich Hertz was performing his famous experiment in
1887 that confirmed Maxwell’s electromagnetic wave theory of light, he noticed that
when ultraviolet light fell on a metal electrode, a charge was produced that separated the
leaves of his electroscope. Although Hertz recognized this discovery of what would
become known as the photoelectric effect, it was of little use to him at the time, and he
left the exploitation of the effect to others, particularly Philipp Lenard. The photoelectric
effect is one of several ways in which electrons can be emitted by materials. By the early
1900s it was known that electrons are bound to matter. The valence electrons in metals
are “free”—they are able to move easily from atom to atom but are not able to leave the
surface of the material.
It is not surprising that electromagnetic radiation interacts with electrons within
metals and gives the electrons increased kinetic energy. Because electrons in metals are
weakly bound, we expect that light can give electrons enough extra kinetic energy to
allow them to escape. We call the ejected electrons photoelectrons. The minimum extra
kinetic energy that allows electrons to escape the material is called the work function f.
The work function is the minimum binding energy of the electron to the material (see
Table 3.3 for work function values for several elements).
Experiments carried out around 1900 showed that photoelectrons are produced
when visible and/or ultraviolet light falls on clean metal surfaces. Photoelectricity was
studied using an experimental apparatus shown schematically in Figure 3.11. Incident
light falling on the emitter (also called the photocathode or cathode) ejects electrons.
Some of the electrons travel toward the collector (also called the anode), where either a
negative (retarding) or positive (accelerating) applied voltage V is imposed by the power
supply. The current I measured in the ammeter (photocurrent) arises from the flow of
photoelectrons from emitter to collector. Consider the situation where the emitter and
collector are made of the same material and therefore have the same work functions.
As stated previously, classical theory allows electromagnetic radiation to eject
photoelectrons from matter. However, classical theory predicts that the total amount of
energy in a light wave increases as the light intensity increases. Therefore, according to
classical theory, the electrons should have more kinetic energy if the light intensity is
increased. However, according to experimental result 1 and Figure 3.12, a characteristic
retarding potential 2V0 is sufficient to stop all photoelectrons for a given light frequency
f, no matter what the intensity. Classical electromagnetic theory is unable to explain this
result. Similarly, classical theory cannot explain result 2, because the maximum kinetic
energy of the photoelectrons depends on the value of the light frequency f and not on the
intensity.
Finally, classical theory would predict that for extremely low light intensities, a
long time would elapse before any one electron could obtain sufficient energy to escape.
We observe, however, that the photoelectrons are ejected almost immediately. For
example, experiments have shown that a light intensity equivalent to the illumination
produced over a 1-cm2 area by a 100-watt incandescent bulb at a distance of 1000 km is
sufficient to produce photoelectrons within a second.
Albert Einstein was intrigued by Planck’s hypothesis that the electromagnetic
radiation field must be absorbed and emitted in quantized amounts. Einstein took
Planck’s idea one step further and suggested that the electromagnetic radiation field itself
is quantized and that “the energy of a light ray spreading out from a point source is not
continuously distributed over an increasing space but consists of a finite number of
energy quanta which are localized at points in space, which move without dividing, and
which can only be produced and absorbed as complete units.”* We now call these energy
quanta of light photons.
In other words, Einstein proposed that in addition to its well-known wavelike
aspect, amply exhibited in interference phenomena, light should also be considered to
have a particle-like aspect. Einstein suggested that the photon (quantum of light) delivers
its entire energy hf to a single electron in the material. To leave the material, the struck
electron must give up an amount of energy f to overcome its binding in the material. The
electron may lose some additional energy by interacting with other electrons on its way to
the surface. Whatever energy remains will then appear as kinetic energy of the electron as
it leaves the emitter.
We should now reexamine the experimental results of the photoelectric effect to
see whether Einstein’s quantum interpretation can explain all the data. The first and
second experimental results (which indicate that the kinetic energies of the photoelectrons
depend on the light frequency, but not the light intensity) can be explained. A potential
slightly more positive than 2V0 will not be able to repel all the electrons, and, for a close
geometry of the emitter and collector, practically all the electrons will be collected when
the retarding voltage is near zero. For very large positive potentials all the electrons will
be collected, and the photocurrent levels off.
N. X-Ray Production
In the photoelectric effect, a photon gives up all of its energy to an electron,
which may then escape from the material in which it was bound. Can the inverse process
occur? Can an electron (or any charged particle) give up its energy and create a photon?
The answer is yes, but the process must be consistent with the laws of physics. Recall that
photons must be created or absorbed as whole units. A photon cannot give up half its
energy; it must give up all its energy. If in some physical process only part of the
photon’s energy were required, then a new photon would be created to carry away the
remaining energy.
Unlike a photon, an electron may give up part or all of its kinetic energy and still
be the same electron. When an electron interacts with the strong electric field of the
atomic nucleus and is consequently accelerated, the electron radiates electromagnetic
energy. According to classical electromagnetic theory, it should do so continuously. In
the quantum picture we must think of the electron as emitting a series of photons with
varying energies; this is the only way that the inverse photoelectric effect can occur. An
energetic electron passing through matter will radiate photons and lose kinetic energy.
The process by which photons are emitted by an electron slowing down is called
bremsstrahlung, from the German word for “braking radiation.”
The x rays are produced by the bremsstrahlung effect in an apparatus shown
schematically in Figure 3.18. Current passing through a filament produces copious
numbers of electrons by thermionic emission. These electrons are focused by the cathode
structure into a beam and are accelerated by potential differences of thousands of volts
until they impinge on a metal anode surface, producing x rays by bremsstrahlung (and
other processes) as they stop in the anode material. Much of the electron’s kinetic energy
is lost by heating the anode material and not by bremsstrahlung. The x-ray tube is
evacuated so that the air between the filament and anode will not scatter the electrons.
The x rays produced pass through the sides of the tube and can be used for a large
number of applications, including medical diagnosis and therapy, fundamental research in
crystal and liquid structure, and engineering diagnoses of flaws in large welds and
castings.
When a photon enters matter, it is likely to interact with one of the atomic
electrons. According to classical theory, the electrons will oscillate at the photon
frequency because of the interaction of the electron with the electric and magnetic field of
the photon and will reradiate electromagnetic radiation (photons) at this same frequency.
This is called Thomson scattering. However, in the early 1920s Arthur Compton
experimentally confirmed an earlier observation by J. A. Gray that, especially at
backward-scattering angles, there appeared to be a component of the emitted radiation
(called a modified wave) that had a longer wavelength than the original primary
(unmodified) wave. Classical electromagnetic theory cannot explain this modified wave.
Compton then attempted to understand theoretically such a process and could find only
one explanation: Einstein’s photon particle concept must be correct.
