This is one of the webpages of Libarid A. Maljian at the Department of Physics at CSLA at NJIT.
New Jersey Institute of Technology
College of Science and Liberal Arts
Department of Physics
Introductory Astronomy and Cosmology
Phys 202
Spring 2018
First Examination additional lecture notes
Everything in the universe is
composed of (made of) atoms. (By
“everything” we mean everything tangible.)
Atoms are composed of (made of) even smaller particles. The center of the atom is called the nucleus. (The center of anything is often called its
nucleus. For example, the center of a
biological cell is called the cellular nucleus.
The center of an entire galaxy is called the galactic nucleus. The center of an atom should really be called
the atomic nucleus, but we will often be lazy and just say nucleus.) Around the atomic nucleus are electrons. The atomic nucleus is positively charged, and
electrons are negatively charged. In
fact, it is the attraction between the positive nucleus and the negative
electrons that holds the atom together.
(Like charges repel and unlike charges attract. In other words, positive and positive repel,
negative and negative repel, and positive and negative attract.) The atomic nucleus is composed of even
smaller particles: protons and neutrons.
The protons are positively charged.
In fact, it is because of the protons that the entire atomic nucleus has
a positive charge. The neutrons have
zero electrical charge. In other words,
neutrons are neutral. This is why they are
called neutrons!
The number of protons in the
nucleus is the single most important number of the atom. It is so important that it is called the atomic
number. The atomic number, which is
always the number of protons in the nucleus, is so important that an atom is
named solely based on the atomic number.
For example, every atom in the universe with twelve protons in
its nucleus is considered to be a magnesium atom. As another example, every atom in the
universe with seven protons in its nucleus is considered to be a nitrogen
atom. We are not saying that the number
of neutrons or the number of electrons is irrelevant. They are quite important. We are saying that the atomic number is
always the number of protons, and the name of the atom is based only upon the
atomic number (the number of protons).
If we change the number of
electrons, we change the charge of the atom.
Why? Imagine an atom where the
number of electrons balances the number of protons. Since protons are positive and electrons are
negative, the atom is neutral overall.
Now imagine we add extra electrons to the atom. Since electrons are negative, the atom will
no longer be neutral overall; it will be negative overall. Imagine we removed electrons from the atom in
the first place. Now the atom will be
positive overall. A charged atom is
called an ion. We have now made
it clear that changing the number of electrons results in ions. Some examples will help illustrate this. Consider the sodium atom with the symbol
Na. The atomic number of sodium is
eleven, meaning that every sodium atom in the universe has eleven protons. We will make this clear with a subscript
before the atom’s symbol like this: 11Na. If the sodium atom were neutral, it would
have eleven electrons as well, but suppose we add three more electrons. Since electrons are negative, we now have an
ion with a charge of negative three. We
write the charge as a superscript after the name of the atom like this: 11Na3–. (Even though
the charge is read “negative three,” the superscript is written in the strange
way “3–.”) As another
example, consider the aluminum atom with the symbol Al. The atomic number of aluminum is thirteen,
meaning that every aluminum atom in the universe has thirteen protons. We make this clear with a subscript before
the atom’s symbol like this: 13Al.
If the aluminum atom were neutral, it would have thirteen electrons as
well, but suppose we remove two of the electrons. We now have an ion with a charge of positive
two. We write the charge as a
superscript after the name of the atom like this: 13Al2+. (Even though the charge is read “positive
two,” the superscript is written in the strange way “2+.”) A positive ion is called a cation, and a
negative ion is called an anion.
If we change the number of
neutrons, we do not get ions.
Why? Neutrons are neutral. So, adding or removing neutrons does not
change the charge at all. What is being
changed however is the mass of the atom.
The atomic mass of an atom is the number of protons plus the number
of neutrons. (You may be offended that
we are not counting the electrons when calculating the mass of the atom. It turns out that an electron is almost two
thousand times less massive than a proton or a neutron. Thus, as far as the atomic mass is concerned,
the electrons do not count. A proton and
a neutron have roughly equal amounts of mass, which is why we count them
equally.) When we change the number of
neutrons, we are changing the atomic mass of the atom. Two atoms with the same atomic number but
different atomic mass are called isotopes. We have now made it clear that changing the
number of neutrons results in isotopes.
Some examples will help illustrate this.
Consider the carbon atom with the symbol C. The atomic number of carbon is six, meaning
that every carbon atom in the universe has six protons. We make this clear with a subscript before
the atom’s symbol like this: 6C, but carbon has three isotopes:
carbon-twelve, carbon-thirteen, and carbon-fourteen. An isotope is named based on its atomic
mass. Thus, the numbers twelve,
thirteen, and fourteen are the atomic masses of these isotopes of carbon. We make this clear with a superscript before
the atom’s symbol like this: 
for carbon-twelve, 
for carbon-thirteen, and 
for
carbon-fourteen. Notice that carbon has
six protons no matter what, but the carbon-fourteen isotope has eight neutrons,
since six plus eight equals fourteen.
The carbon-thirteen isotope has seven neutrons, since six plus seven
equals thirteen. The carbon-twelve
isotope has six neutrons, since six plus six equals twelve.
We
can put all of this together with the following examples. Consider the neon atom with the symbol
Ne. Now suppose we write 
. This neon atom has
ten protons, eleven neutrons, twelve electrons, an atomic number of ten, an
atomic mass of twenty-one, and a charge of negative two. As another example, consider the boron atom
with the symbol B. (There are borons in this class!)
Now suppose we write 
. This boron atom has
five protons, four neutrons, two electrons, an atomic number of five, an atomic
mass of nine, and a charge of positive three.
The most important atom in
this course is hydrogen, since most of the atoms in the universe are hydrogen
atoms. The symbol for the hydrogen atom
is H. The atomic number of hydrogen is
one, meaning that every hydrogen atom in the universe has one proton in its
nucleus. We make this clear with a
subscript before the atom’s symbol like this: 1H,
but hydrogen has three isotopes: hydrogen-one which is written 
,
hydrogen-two which is written 
,
and hydrogen-three which is written 
. Hydrogen is so important that these three isotopes
have additional names besides hydrogen-one, hydrogen-two, and
hydrogen-three. Hydrogen-one is also
called protium. It is also called
“ordinary hydrogen” since most of the hydrogen atoms in the universe are this
isotope. Hydrogen-two is also called
deuterium. It is also called “heavy
hydrogen” since it is twice as massive as “ordinary hydrogen.” (When an oxygen atom chemically bonds to two
“ordinary hydrogen” atoms, the result is a molecule of “ordinary water.” When an oxygen atom chemically bonds to two
“heavy hydrogen” atoms, the result is a molecule of “heavy water.”) Hydrogen-three is also called tritium. Where do the names protium,
deuterium, and tritium come from? The
atomic number of hydrogen is one, meaning that every hydrogen atom in the
universe has one proton in its nucleus.
This means that the hydrogen-one isotope (or protium
or “ordinary hydrogen”) has no neutrons in its nucleus, since one plus zero is
one. In other words, its nucleus is a
single proton all by itself. This is the
simplest nucleus in the universe. Since
the nucleus is a proton, when we put an electron around it to build the entire
atom, we name the entire atom protium, since its
nucleus is a proton. The hydrogen-two
isotope (or deuterium or “heavy hydrogen”) has one neutron in its nucleus,
since one plus one is two. In other
words, its nucleus is a proton and a neutron stuck to each other. A proton and a neutron stuck to each other is
called a deuteron. Since the nucleus is
a deuteron, when we put an electron around it to build the entire atom, we name
the entire atom deuterium, since its nucleus is a deuteron. The hydrogen-three isotope (or tritium) has
two neutrons in its nucleus, since one plus two is three. In other words, its nucleus is a proton and
two neutrons all stuck to one another. A
proton and two neutrons all stuck to one another is called a triton. Since the nucleus is a triton, when we put an
electron around it to build the entire atom, we name the entire atom tritium,
since its nucleus is a triton. The
helium atom with the symbol He has an atomic number of two, meaning that every
helium atom in the universe has two protons in its nucleus. We make this clear with a subscript before
the atom’s symbol like this: 2He. Most of the helium atoms in the universe are
the helium-four isotope which is written 
. Helium-four is also called “ordinary helium”
since most of the helium atoms in the universe are this isotopes. The nucleus of helium-four is composed of two
protons and two neutrons, since two plus two is four. In other words, the nucleus of helium-four is
two protons and two neutrons all stuck to one another. Two protons and two neutrons all stuck to one
another is called an alpha particle. To
summarize, the nucleus of the protium atom is a
proton, the nucleus of the deuterium atom is a deuteron, the nucleus of the
tritium atom is a triton, and the nucleus of the “ordinary helium” atom is an
alpha particle.
Electrons do not orbit
an atomic nucleus like planets orbit the Sun.
In fact, the electrons do not orbit at all; they exist in an abstract
quantum-mechanical state that we will not explain deeply in this course. For now, we simply state that there are
definite energy levels within an atom.
Some levels are at lower energies, and other levels are at higher
energies. If an electron wishes to
change its energy from a lower level to a higher level, it must absorb a
photon, a particle of light. However,
any photon will not be satisfactory. The
energy of the photon absorbed must be exactly equal to the difference in energy
between the two levels. If an electron
wishes to change its energy from a higher level to a lower level, it must emit
(spit out) a photon, but not any photon.
The energy of the photon emitted must be exactly equal to the difference
in energy between the two levels.
Therefore, an atom can only absorb or emit photons of very specific
energies (or very specific frequencies or very specific wavelengths). The list of all the allowed photon energies
(or frequencies or wavelengths) an atom is permitted to absorb is called the
absorption spectrum of the atom, and the list of all the allowed photon
energies (or frequencies or wavelengths) an atom is permitted to emit is called
the emission spectrum of the atom. Since
different atoms have different energy levels, every atom has its own unique
spectrum, different from the spectra of all other atoms. Therefore, the spectrum of an atom is rather
like its fingerprint, enabling us to uniquely identify an atom. A spectacular example of this is the
discovery of the Sun’s composition. How
do we know which atoms compose the Sun?
In the early 1800s, Joseph von Fraunhofer discovered missing wavelengths in the Sun’s
light. These absorption lines are called
Fraunhofer lines in his honor. By measuring the wavelengths of these
absorption lines and consulting a table of absorption spectra, we can determine
which atoms absorbed these various wavelengths and thus determine the
composition of the Sun. We discover that
the Sun is composed of all the atoms on the Periodic Table of Elements, but not
in equal amounts. Only two atoms account
for close to one hundred percent of the Sun’s mass; all the other atoms on the
Periodic Table of Elements account for only a tiny percentage of the Sun’s mass. What are these two elements that account for
close to one hundred percent of the Sun’s mass?
We discover from the Fraunhofer lines in
sunlight that hydrogen atoms account for roughly seventy-five percent
(three-quarters) of the Sun’s mass. What
about the remaining twenty-five percent (one-quarter) of the Sun’s mass? The wavelengths of the remaining absorption
lines were not found in any atom’s tabulated absorption spectrum! It seems that one-quarter of the Sun’s mass
is composed of a new atom never before discovered! This atom was called helium, named after
Helios the personification of the Sun in ancient Greek mythology. In the early 1900s,
helium was discovered on Earth as the product of certain nuclear reactions, and
for many decades we find helium everywhere on Earth (in blimps and in party
balloons for example). Nevertheless,
helium was first discovered from its absorption lines in the Sun’s light.
What is temperature? What do we mean when we say something is
hot? What do we mean when we say
something is cold? The temperature of an
object is a measure of the average energy of the atoms that compose that
object. In this course, we may assume
that the average energy of atoms corresponds to their average speed. In other words, the atoms of a hotter object
are moving relatively faster, whereas the atoms of a cooler object are moving
relatively slower. There are two scales
of temperature in common use: degrees fahrenheit and
degrees celsius.
However, neither degrees fahrenheit nor
degrees celsius are acceptable units of
temperature. What is wrong with these
two scales? The zero is in the wrong
place in both of these scales. What do
we mean by this? If temperature of an
object is a measure of the average speed of its atoms, then the coldest
possible temperature of our universe is the temperature at which all the atoms
of an object completely stop moving.
After all, you cannot be moving any slower than not moving at all! The temperature at which all atoms completely
stop moving is commonly called absolute zero.
However, absolute zero temperature is not zero degrees fahrenheit nor is it zero degrees celsius. Atoms are still moving at zero degrees fahrenheit, and atoms are still moving at zero degree celsius. There are
still negative temperatures on both of these scales (commonly called
temperatures below zero) where the atoms move slower still. The absolute zero of temperature when all
atoms completely stop moving is exactly negative 273.15 degrees celsius or exactly negative 459.67 degrees fahrenheit. A
correct unit of temperature must assign the number zero to the absolute zero of
temperature. The simplest way to correct
degrees celsius is to add 273.15 to all degrees celsius. What does
this accomplish? Since absolute zero is
negative 273.15 degrees celsius, then adding 273.15
would yield zero, and all other temperatures would be positive. The simplest way to correct degrees fahrenheit is to add 459.67 to all degrees fahrenheit. What
does this accomplish? Since absolute
zero is negative 459.67 degrees fahrenheit, then
adding 459.67 would yield zero, and all other temperatures would be
positive. When we correct the celsius scale by adding 273.15 to it, we get correct units
of temperature called kelvins. When we
correct the fahrenheit scale by adding 459.67 to it,
we get correct units of temperature called rankines. To summarize, absolute zero is negative
273.15 degrees celsius or negative 459.67 degrees fahrenheit on these unacceptable temperature scales, but
absolute zero is zero kelvins or zero rankines using
acceptable units of temperature. We will
use kelvins throughout this course. It
is somewhat difficult growing accustomed to kelvins. For example, most humans consider 270 kelvins
to be uncomfortably cold, most humans consider 300 kelvins to be a comfortable
room temperature, and most humans consider 330 kelvins to be uncomfortably hot.
The Third Law of
Thermodynamics states that it is impossible to cool an object to absolute zero
temperature in a finite number of steps. It follows that everything in the universe is
at some temperature above absolute zero.
Therefore, every object in the universe has its atoms moving at some
average speed. Since atoms are made of
protons, neutrons, and electrons and since protons and electrons are charged,
every object in the universe radiates electromagnetic waves from its moving
atoms. The amount of energy radiated at
various wavelengths from a hot, dense object often follows the blackbody spectrum,
which is a continuous spectrum with its primary radiation within a band of the
electromagnetic spectrum determined by the temperature of the object. In particular, hotter temperatures correspond
to higher photon energies which are at higher frequencies and shorter
wavelengths, while cooler temperatures correspond to lower photon energies
which are at lower frequencies and longer wavelengths. In other words, a hot, dense object’s primary
radiation is displaced as its temperature changes. At very low temperatures (close to absolute
zero), objects radiate primarily in the microwave part of the electromagnetic
spectrum. At a few hundred kelvins (such
as room temperatures), objects radiate primarily in the infrared part of the
electromagnetic spectrum. At one or two
thousand kelvins, objects radiate primarily red visible light. At three or four thousand kelvins, objects
radiate primarily orange visible light.
At five or six thousand kelvins, objects radiate primarily yellow
visible light. At roughly ten thousand
kelvins, objects radiate primarily blue visible light. At hundreds of thousands of kelvins, objects
radiate primarily in the ultraviolet part of the electromagnetic spectrum. At millions of kelvins, objects radiate
primarily in the X-ray part of the electromagnetic spectrum. At tens of millions of kelvins, objects
radiate primarily in the gamma-ray part of the electromagnetic spectrum. Notice how hotter temperatures displace the
primary radiation to higher and higher photon energies or higher and higher
frequencies or shorter and shorter wavelengths.
This can easily be demonstrated by heating metal. First, the metal radiates red. As the metal is made hotter, it radiates
orange. If the metal is made hotter
still, it radiates yellow. This can also
be demonstrated with a flame on a stovetop.
At the lowest setting, the flame radiates red. At a higher setting, the flame radiates
orange. At an even higher setting, the
flame radiates yellow, and the hottest part of the flame is blue. The Sun is a yellow star, and from that color
we can correctly estimate that the surface temperature of the Sun is roughly
six thousand kelvins. Stars throughout
the universe that are red in color are cooler than our Sun, stars that are blue
in color are hotter than our Sun, and stars yellow in color are approximately
the same temperature as our Sun. We must
emphasize that we are talking about the color that an object radiates because
it is hot enough to be emitting that color.
Many objects have various different colors even though they are all at
room temperature, such as red ink, yellow paint, green grass, and blue
jeans. But these objects are not
radiating these colors. These colors are
being reflected, while all other colors are being absorbed. We must be careful to make a distinction
between the color of an object simply because it is reflecting that color versus
the color of an object because it is actually hot enough to be radiating that
color.
Consider any wave propagating
in a certain medium that encounters a second medium. This wave is called the incident wave. At the boundary between the two media, a part
of the wave will bounce back into the first medium while the rest of the wave
will be transmitted into the second medium.
The wave that bounces back into the first medium is called the reflected
wave, and the wave that is transmitted into the second medium is called the
refracted wave. (The meanings of the
words reflection and refraction will be made clear in a moment.) Any line perpendicular to the boundary
between the two media is called the normal to the boundary, since the word
normal in physics and engineering means perpendicular. The angle between the incident wave and the
normal is called the angle of incidence and is written θ1. The angle between the reflected wave and the
normal is called the angle of reflection and is written θ3. The angle between the refracted wave and the
normal is called the angle of refraction and is written θ2. The Law of Reflection states θ1 = θ3
in all cases. In other words, the angle
of incidence is equal to the angle of reflection in all cases for all
waves. Reflection is the bouncing of a
part of a wave off of another medium with no change in angle with respect to
the normal. The Law of Refraction states
sin(θ1)/v1 = sin(θ2)/v2, where v1 is the speed of the wave in the first medium,
and v2
is the speed of the refracted wave in the second medium. Refraction is the bending of a wave due to a
change in speed of the wave. According
to the Law of Refraction, a wave is refracted (bent) toward the normal if v2
< v1
(if the transmitted wave propagates slower than the incident wave); conversely,
a wave is refracted (bent) away from the normal if v2 > v1 (if
the transmitted wave propagates faster than the incident wave).
Since light is a wave, light
must obey the Law of Reflection and the Law of Refraction. A device that reflects light is called a
mirror. A device that refracts light is
called a lens. Most metals reflect light
very well. Therefore, a mirror can be
manufactured by coating a piece of glass with a metal (often aluminum) and
polishing the metal. Any piece of glass
may be regarded as a lens, since light will refract (bend) as it is transmitted
from the air into the glass and will refract (bend) again as it is transmitted
from within the glass back into the air.
In the following discussion of mirrors and lenses, we will assume that
we are in the paraxial approximation. In
this approximation, all light rays incident upon a mirror or lens remain near
the symmetry axis of the mirror or lens.
This can be guaranteed by requiring that the mirror or lens has a large
radius of curvature.
A curved mirror with its
center of curvature and its focal point facing toward the light incident upon
it is called a concave mirror. In this
case, the focal point is said to be “in front of” the mirror. A curved mirror with its center of curvature
and its focal point facing away from the light incident upon it is called a
convex mirror. In this case, the focal
point is said to be “behind” the mirror.
In the paraxial approximation, light incident upon a concave mirror will
reflect and converge at the focal point that is in front of the mirror. For this reason, a concave mirror is also
called a converging mirror. Also in the
paraxial approximation, light incident upon a convex mirror will reflect and
diverge away from the focal point that is behind the mirror. For this reason, a convex mirror is also
called a diverging mirror. A curved lens
that is thicker in its middle than it is at its edge is called a convex lens. A curved lens that is thinner in its middle
than it is at its edge is called a concave lens. In the paraxial approximation, light incident
upon a convex lens will refract and converge at the focal point that is on the
opposite side of the lens as the incident rays.
For this reason, a convex lens is also called a converging lens. Also in the paraxial approximation, light
incident upon a concave lens will refract and diverge away from the focal point
that is on the same side of the lens as the incident rays. For this reason, a concave lens is also
called a diverging lens. Notice that
mirrors and lenses are completely opposite in character. A concave mirror is converging, but a concave
lens is diverging. A convex mirror is
diverging, but a convex lens is converging.
Even the geometry of convergence or divergence (as the case may be) is
opposite in character. In particular,
light rays converge to a focus on the same side as the incident rays for a
converging mirror, but light rays converge to a focus on the opposite side as
the incident rays for a converging lens.
Similarly, light rays diverge away from a focus on the opposite side as
the incident rays for a diverging mirror, but light rays diverge away from a focus
on the same side as the incident rays for a diverging lens.
Links
New Jersey Institute of Technology
College of Science and Liberal Arts at NJIT
Department of Physics at CSLA at NJIT
Libarid A. Maljian at web.njit.edu
Libarid A. Maljian at the Department of Physics at CSLA at NJIT
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