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(J1) Relativity
PHY 20 2 Student Code: nature
Special Relativity
The Development of Special Relativity
Maxwell’s Equations were developed in the mid-1800s. They transformed the electric and magnetic
theories into the single electromagnetic theory, which also embodied a theory of light, and the equations
were fantastically successful. But there was a problem – in some situations, such as when objects move
close to the speed of light, the equations contradicted Newton’s Laws.
In 1905, Einstein proposed that Maxwell’s Equations had the advantage over Newton’s Laws, and his
Theory of Special Relativity left Maxwell’s Equations unchanged but did modify Newton’s Laws.
In doing this, Einstein relied on a seemingly simple postulate – the speed of light in vacuum is a constant.
But the effects of this assumption were (and are) at times dramatic.
Today, Maxwell’s Equations are still widely and successfully used.
(The theory has been further developed into a quantum mechanical version, but the classical theory of
Maxwell continues to work well in most situations.)
Constant Speed of Light
The speed of light (in vacuum) is a constant (independent of the motion of the source of the light and the
motion of the observer of the light).
𝑐 = 3 × 10
8
m/s
PQGUA
If you were on a spaceship travelling at 0. 50 𝑐
= 1. 5 × 10
8
m/s
away from a star, and you measured the
speed of the light from the star, what speed would you find?
a. The speed of light would be doubled to 6 × 10
8
m/s.
b. The speed of light would be further reduced to 0. 75 × 10
8
m/s.
c. The speed of light would be the same as always: 3 × 10
8
m/s.
A Matter of Perspective
The odd effects described by Special Relativity are most evident for objects moving close to the speed of
light from YOUR perspective. In fact, for objects at rest (or moving at “ordinary” speeds) relative to YOU,
everything seems normal. Even if you were on a spaceship zipping across the universe, every regular
thing onboard the ship would seem perfectly normal to you.
PQGUB
Suppose you are travelling away from the Earth at 0.50c (= 1. 5 × 10
8
m/s). From your perspective,
would your height, waistline, or heartbeat change?
a. Lengths would increase; time would slow down.
b. Lengths would decrease; time would speed up.
c. Everything would be the same.
Time Dilation
The clocks of fast-moving objects (with a speed near the speed of light) run noticeably slower than clocks
at rest. If Δ𝑡′ is the amount of time that has passed for a clock at rest and Δ𝑡 is the amount of time that has
passed for the fast-moving clock, then
′
2
Note. Everything is a clock! (Or least has an internal clock.)
Note. When 𝑣 is small compared to the speed of light, so 𝑣 ⁄𝑐 is much less than 1 , then
𝑣
𝑐
2
≈ 1 , and
′
≈ 𝛥𝑡, so time passes at essentially the same rate for both the stationary object and the slowly moving
object.
It can be useful to define an addition quantity, 𝛽, which is the speed of an object relative to the speed of
light:
Then
Length Contraction
For a fast-moving object, the length of the object (only the length in the direction of motion is affected)
seems to be shorter compared to the length when the object is at rest.
′
2
PQGUE
A spaceship with length Δ𝑥 = 459ˍm (measured when the spaceship was stationary) passed by the Earth.
What length Δ𝑥′ would the people on Earth say the spaceship was as it passed the Earth at 0.959𝑐?
A. 120.2ˍm D. 114.5ˍm
B. 141ˍm E. 130.1ˍm
C. 136.8ˍm F. 125.6ˍm
Assist:
𝛽 = ______________ and Δ𝑥
′
2
⋅ Δ𝑥 = ______________
PQGUF
A steel beam with length Δ𝑥 = 85.4ˍm is moving past the Earth. Observers on the Earth measure the steel
beam to be only Δ𝑥
′
= 22.1ˍm long. How fast was the beam traveling?
A. 347000000ˍm/s D. 289800000ˍm/s
B. 336900000ˍm/s E. 305800000ˍm/s
C. 275000000ˍm/s F. 320700000ˍm/s
Assist:
′
2
Now solve for 𝛽…
′
2
Then
Mass and Rest Energy
Perhaps the most famous equation in all of science is 𝐸 = 𝑚𝑐
2
. This equation calculates the amount of
energy produced when a mass 𝑚 is converted into energy. (This occurs during a nuclear explosion, for
instance, although only a very small fraction of the mass of the bomb is converted into energy!)
Note. The reverse also occurs—energy can be converted into mass.
The equation seems simplistic, but a significant detail is hidden—the mass of an object depends on its
speed!
When the mass is at rest, then this equation represents the rest energy 𝐸 0
and the rest mass 𝑚
0
Then…
0
0
2
PQGUG
What is the rest energy of a 10.6ˍkg rock?
A. 8.610E+17ˍJ D. 8.112E+17ˍJ
B. 7.748E+17ˍJ E. 1.003E+18ˍJ
C. 9.540E+17ˍJ F. 9.118E+17ˍJ
(J02) GW_Quantum
PHY 202 Student Code: nature
Quantum Mechanics
Atomic Spectra
By the end of the 1800’s, many scientists thought there was little else to add to science! With Newton’s
Laws and Maxwell’s Equations, science was remarkably successful.
But there were at least a few phenomena that had not yet been satisfactorily explained by science.
For instance, when excited by an electric current, elements in gaseous form emit a discrete spectrum of
light (or electromagnetic radiation). Each element has its own unique spectrum, serving as sort of a
fingerprint for the element. Just as with white light, the light from the glowing gas must pass through a
prism (or spectroscope) to reveal the details.
While then-current theories could not explain the spectrum, the Swiss mathematician Johann Jakob
Balmer came up with a formula that matched the spectrum for hydrogen, the simplest element.
Ballmer Series Formula
𝜆 = 3. 6456 × 10
− 7
2
𝑚 = 3, 4, 5, 6… gives the wavelengths of the various lines in the spectrum. (Assume SI units are being
used.)
(A success of early quantum theory was the derivation of Ballmer’s Formula by Niels Bohr, who
postulated that atomic electrons orbited nuclei in only certain special, wave-like orbits.)
QLGOW
Find the wavelength of the 𝑚 = 6 spectral line in the Ballmer Series.
A. 5.829E-08ˍm D. 7.363E-08ˍm
B. 6.060E- 08 ˍm E. 6.836E-08ˍm
C. 7 .070E-08ˍm F. 6.438E-08ˍm
Particles of Light
The work of Max Planck (concerning blackbody radiation, in 1900) and Albert Einstein (concerning the
photoelectric effect, in 1905) indicated that light, although commonly thought to exist as waves, had
some characteristics of particles. In particular, the energy of a particle of light (later named the photon) is
discrete.
The Energy of an Electron
An electron has energy given by…
ℎ = 6. 63 × 10
− 34
J/s is Planck’s Constant, and 𝑓 is the frequency of the associated light wave.
QLHEB
(Yes, we’ve worked this problem before!) Find the energy of a photon whose associated wave has
frequency 79700ˍHz.
A. 5.078E-29ˍJ D. 4.854E-29ˍJ
B. 4.370E-29ˍJ E. 4.123E-29ˍJ
C. 4.538E-29ˍJ F. 5.284E-29ˍJ
QLHEI
Consider a 22ˍwatts light bulb with an average light wavelength 540ˍnanometers. About how many
photons of light are emitted each second by the light bulb?
A. 6.965E+19 D. 6.620E+
B. 5.703E+19 E. 6.275E+
C. 7.294E+19 F. 5.973E+
Note:
𝑃 = 𝐸 ⋅ 𝑡 (power, energy, time)
𝑐 = 𝑓 ⋅ 𝜆 (speed of light, frequency, wavelength)
photon
= ℎ𝑓 (photon energy, Planck’s Constant, frequency)
Electronvolts
Adhering to the standardized SI units can avoid a number of issues. Nevertheless, for individuals who
work frequently in a particular domain, other units can be more convenient, which in turn avoids other
issues. For instance, energies associated with the atomic scale can be more conveniently expressed in
electronvolts rather than joules. The electronvolt can be defined as the energy acquired by an electron
when it “falls” through a one volt potential difference. As such, the electronvolt is approximately…
1 eV = 1. 6 × 10
− 19
C ⋅ 1 V
= 1. 6 × 10
− 19
C ⋅ 1 J/C
= 1. 6 × 10
− 19
J
RFQEK
Convert 2.83E-16ˍJ from joules to electronvolts.
- Energy
A. 2079ˍeV D. 183 8ˍeV
B. 1710ˍeV E. 2014ˍeV
C. 1769ˍeV F. 1940ˍeV
Uncertainty Principle for Position and Momentum
It may seem reasonable that measuring one quantity may alter the value of another quantity. If, for
instance, the position of a particle is measured, then the momentum (or velocity) may be altered by the
measurement. In fact, in quantum mechanics this result is inescapable. That is, no matter how carefully
the position is measured, the momentum will be altered. (The reverse is also true: measuring momentum
alters the position.)
In effect, the position and momentum of a particle cannot both be perfectly known at the same time. As
an extreme example, if the position of a particle is measured exactly, then its momentum is not known at
all. Similarly, if the momentum of the particle is measured exactly, then the position is totally unknown.
This idea is summarized in the Uncertainty Principle.
Using the example of position and momentum, if Δ𝑥 is the uncertainty in the position of the particle and
Δ𝑝 is the uncertainty in the momentum, then the following is true:
From this, if 𝑚 is the mass of the particle, then the uncertainty in the particle’s velocity is must be…
RFQFQ
The position of an electron (with mass 𝑚 = 9. 11 × 10
− 31
kg) in an atom is measured to an accuracy of
0.018ˍnm. What is the uncertainty in the velocity of the electron?
- Velocity Uncertainty
A. 3.768E+06ˍm/s D. 3.090E+06ˍm/s
B. 3.217E+06ˍm/s E. 3.624E+06ˍm/s
C. 3.478E+06ˍm/s F. 3.330E+06ˍm/s
Radioactivity and Half Life
In the context of atomic nuclei, radioactivity (or radioactive decay) is the release of energy by a nucleus
by the emission of a particle.
Gamma (𝛾) Decay: emission of a photon (particle of light)
Alpha (𝛼) Decay: emission of a helium nucleus (two protons and two neutrons)
Beta (𝛽) Decay: emission of an electron
In a process such as beta decay, a neutron in a nucleus emits an electron and in the process changes into a
proton. (Note that the net charge of zero is conserved.) During this process, the atomic number of the
nucleus changes, and so also the type of nucleus changes.
While a radioactive nucleus will eventually decay, the time it takes to do so is unpredictable.
Nevertheless, an “average” value can be found. Usually, it is the half-life (rather than true average) that is
given.
The half-life of a type of radioactive material is the time it takes for one-half of radioactive nuclei to decay.
If 𝑁 0
is the number of radioactive nuclei initially present and 𝜏 is the half-life, then the number of nuclei 𝑁
that are still radioactive after time 𝑡 is given by…
0
𝑡 𝜏
⁄
RFQRY
Suppose a radioactive material has a half-life of 0.0083ˍs. If 6.56E+14 nuclei are initially present, then
approximately how many nuclei are expected to still be radioactive after 0.13ˍs?
- Radioactive nuclei remaining after 0.13ˍs
A. 1.331E+10 D. 1.508E+
B. 1.446E+10 E. 1.554E+
C. 1.399E+10 F. 1.265E+
Quantized Orbitals
According to the classic theory of electrodynamics, an electron in orbit about a nucleus would radiate
energy and spiral into the nucleus. However, this spiral is not observed experimentally. If it did occur,
probably our universe would not exist.
The physicist Neils Bohr suggested that an electron could only be in certain discrete orbits (and don’t
radiate while in such an orbit). These orbits would derive from standing waves. And just as only certain
Orbital Jumps
The principle quantum number 𝑛 (which must be a natural number 1 , 2 , 3 …) is an index for the various
possible orbits of an orbital electron in an atom. The electric potential energy associated with the 𝑛
th
orbit is 𝐸 𝑛
. As seen in the diagram below, for instance, the electric potential energy for the third possible
orbit is 𝐸 3
= − 1. 51 eV. (The negative value comes form the way the reference level is defined for electric
potential energy.)
Orbital Energy Levels for Hydrogen Atom
An electron can transition (or jump) from one energy level to another. If it falls from a higher level 𝑚 to a
lower level 𝑛, then the atom releases a photon having energy Δ𝐸 = −(𝐸
𝑛
𝑚
). But if it transitions from
a lower level 𝑚 to a higher level 𝑛, then the atom absorbs a photon with energy Δ𝐸 = 𝐸 𝑛
𝑚
RGAMM
Suppose an electron jumps from the 𝑚 = 5 orbital level to the 𝑛 = 2 orbital level. Then the atom emits a
photon with what energy?
- Photon Energy Δ𝐸
A. 2.486ˍeV D. 2.603ˍeV
B. 2.86ˍeV E. 2.967ˍeV
C. 3.091ˍeV F. 2.735ˍeV
