Physics 2 (Modern Physics)

Objectives

• To become familiar with the different
branches of Modern Physics
• To state the postulates of the special
theory of relativity
• To differentiate between inertial and
non-inertial reference frames

Modern physics

⇒ started around the beginning of the
20th century

Relativity ⇒ physics of the very, very fast
(speeds approaching c)

Branches of Modern Physics

Atomic and Nuclear Physics
– study of the composition, structure and
behavior of the nucleus of the atom


Quantum Physics
– study of the discrete nature of phenomena
at the atomic and subatomic levels
⇒ its focus is on the indivisible units of
energy called quanta as described by the
Quantum Theory
⇒ physics of the very, very small (protons,
electrons, …)


Relativistic Physics
– study of phenomena that take place
in frame of reference that is in
motion with respect to an observer

⇒ physics of the very, very fast
(speeds approaching c)


Solid State Physics
– study of all the properties of solid
materials, including electrical
conduction in crystals of semi-
conductors and metals,
superconductivity and photo-
conductivity


Condensed Matter Physics
– study of the properties of condensed
materials (solids, liquids and those
intermediate between them, and
dense gas) with the ultimate goal of
developing new materials with
better properties


Plasma Physics

– study of the fourth state of matter


Low-Temperature Physics
– study of the production and
maintenance of temperatures
down to almost absolute zero,
and the various phenomena that
occur only at such temperature

Modern Physics showed that Newton’s
laws were incomplete.

Newton’s laws only apply to objects of
macroscopic size (bigger than protons
and electrons) and relatively small
speeds (much less than the speed of
light)

Albert Einstein (1879 – 1955)

- published three papers of
extraordinary importance
1. an analysis of Brownian motion
2. photoelectric effect (Nobel Prize)
3. special theory of relativity

The special theory of relativity has
made wide-ranging changes in the
understanding of nature.

Special
Relativity
In 1905, Albert Einstein
described in his theory of
Special Relativity how
measurements of time
and space are affected by
the motion between the
observer and what is
being observed.

Special
Relativity

The Theory of Special
Relativity revolutionized
the world of physics by
connecting space and time,
matter and energy, and
electricity and magnetism

The Special Theory of Relativity
defies common sense!

But, the results of the Special Theory of
Relativity have been extensively tested
numerous times and are in fact true!

Postulates of relativity

1. The laws of Physics are the same in
every inertial frame of reference.
(No experiment can be done in an inertial
reference frame to detect its state of motion.)

2. The speed of light (3 x 108 m/s) is the same
in all inertial frames of reference and is
independent of the motion of the source
(The speed of light in vacuum has the same
value when measured by any observer,
regardless of the observer’s state of motion.)

Inertial Reference Frame

 An inertial reference frame is one
 in which no accelerations are observed in
the absence of external forces (acceleration
is the result of a force)
 that is not accelerating
 Newton’s laws hold in all inertial reference
frames.

Inertial Reference Frame
examples
1. This room
Experiment: Drop a ball.
It accelerates downward at 9.8 m/s2 due to
the force of gravity.
2. Inside of a car moving at constant speed along a
straight road.
Repeat the experiment:
Results are the same as in #1.
3. Inside of an elevator that is moving either upward
or downward at constant speed.
Repeat the experiment:
Results are the same as in #1 and #2.

Noninertial Reference Frame

 A noninertial reference frame is one
that is accelerating with respect to an
inertial reference frame.
 In a noninertial reference frame, bodies
have accelerations in the absence of
applied forces.

Noninertial Reference Frame
examples
1. The interior of a car that is either speeding up,
slowing down, or going around a curve.
Experiment: Drop a ball.
If the car is slowing down, the ball accelerates downward and
towards the front of the car. The acceleration toward the front of
the car is not due to a force on the ball.
2. The inside of an elevator that is accelerating
either upward or downward.
Repeat the experiment.
If the elevator is accelerating upward, the ball accelerates downward
faster than 9.8 m/s2. The additional downward acceleration is not due
to a force on the ball.

Reference Frames
Platform at rest, tree moving—ball is Platform moving. Observer on
seen by observers on platform as being the ground (inertial frame) sees
deflected, but no force acts on it. ball move in a straight line, but
Violation of Newton’s second law. sees the catcher move away.

Platform is accelerating Ground is the
noninertial frame inertial frame

Special Relativity:
Consequences
• time dilates
Time to moving objects appear to slow down

• length shrinks
Moving objects appear shorter

• mass increases
Moving objects appear to be massive

Time dilation

If there is relative motion between two observers
(if they are moving at different velocities), they will
not agree in their measurements of space and time.

However, the two observers will agree on their
measurement of the speed of light.

Since speed equals distance divided by time,
both observers will measure the same ratio of
space (distance) and time

Space
= Space = c
time
time

Time dilation

⇒ An observer in the rocket moving with the clock
sees the light traveling straight up and down.

⇒ The observer and the clock are in the same
frame of reference.

Time dilation

⇒ An observer on the ground (who is not in the
same reference frame as the clock) sees the
light traveling in a diagonal path.

⇒ In the frame of reference of the observer on
Earth, the light travels a longer distance.

Time dilation

⇒ Since the speed of light is the same in all
reference frames the light must travel for a
longer time in the Earth than in a reference
frame of the rocket.

⇒ The stretching out of time is called time dilation.

Time dilation

⇒ Moving clocks run slow.

⇒ Time dilation has nothing to do with the mechanics
of clocks but with the nature of time itself.

⇒ Time passes more slowly in a reference frame
that is moving than in a reference frame that is
at rest.

Time dilation

⇒ is given as

Δt’
Δt =
√ 1 – v2/c2

Δt time interval in the moving frame
Δt’ time interval in the frame at rest
v speed of relative motion
c speed of light

Length shrinks

⇒ The lengths of objects appear to be contracted
(shortened) when they move at relativistic speeds.

⇒ This length contraction is really a contraction
of space.

⇒ As the speed increases, length in the direction
of motion decreases. Lengths in the perpendicular
direction do not change.

Length shrinks

⇒ Length contraction is given as

l = l’√1 – v2/c2

where l length measurement of the moving frame
l’ length measurement of the frame at rest
v speed of the moving frame
c speed of light

Mass increases

⇒ The mass of an object moving at a speed v relative to
the observer is larger than its mass when at rest
relative to the observer
⇒ the relativistic mass is given as

m = m’
√1 – v2/c2

where m relativistic mass
m’ mass of the object at rest
v speed of the moving frame
c speed of light

General Theory of Relativity

Relativity refers to the observation of the motion
of a body by two different observers in relative
motion to each other

General Theory of Relativity is a geometrical
theory of gravitation published by Albert
Einstein in 1915


unifies special relativity and Sir Isaac
Newton's law of universal gravitation with
the insight that gravitation is not due to a
force but rather is a manifestation of curved
space and time, with this curvature being
produced by the mass-energy and
momentum content of the spacetime.


has three parts:

– equivalence of inertial and gravitational
mass (Galileo’s principle)

– laws of physics same in freely falling lab
as in lab at rest far from any mass

– physical laws in accelerating lab same
as in stationary lab in gravitational field

The Equivalence Principle
New ton Einstein

This compartment is This compartment is
at rest in the Earth’s moving in a gravity-free
gravitational field. environment
The apple hits the floor of The apple hits the floor
the compartment because of the compartment
the Earth’s gravity because the compart-
accelerates the apple ment accelerates.
downward


“Equivalence Principle”:

Observers cannot distinguish between
inertial forces due to acceleration and
uniform gravitational forces due to a
massive body.

• Consequence: Gravity, inertia, and
acceleration are related to the
curvature of space-time


“Mass tells space how to curve.
Curvature tells mass how to accelerate”.

In the context of the Theory of General Relativity,
gravitation was redefined as a property of the
space-time continuum. The force was replaced by
the strength of the curvature of the space which
is depending on the mass and the size of an
object, i.e., its ability to bend space.


Representation of the warping of
space and time due to large mass


currently the most successful gravitational
theory, being almost universally accepted
and well confirmed by observations such as:
• gravitational redshift

• deflection of light by mass

• bending of light by gravitation

• perihelion precession of Mercury

Perihelion is the point in the path of a celestial body
(as a planet) that is nearest to the sun.

 Gravitational redshift
- the effect when light or other forms of
electromagnetic radiation of a certain
wavelength originating from a source
placed in a region of stronger gravitational
field (and which could be said to have
climbed "uphill" out of a gravity well) will
be found to be of longer wavelength when
received by an observer in a region of
weaker gravitational field.
The gravitational redshift of a
light wave as it moves upwards
If applied to optical wave-lengths this manifests against a gravitational field
(caused by the yellow star below).
itself as a change in the color of the light as the http://en.wikipedia.org/wiki/Gra
vitational_redshift
wavelength is shifted toward the red (making it
less energetic, longer in wavelength, and lower in
frequency) part of the spectrum

 Gravitational redshift

Light leaving a region where
the gravitational force is large
will be shifted towards the red
(its wavelength increases;
similarly, light falling into a
region where the gravitational
pull is larger will be shifted The gravitational redshift

towards the blue. http://physics.ucr.edu/~wudka
/Physics7/Notes_www/node89.h
tml#fig:redshift

 Deflection of light by mass
One immediate consequence of the curvature of the space-
time is that light must also be subject to gravity

http://library.thinkquest.org/C0116043/generaltheory.htm
Figure above shows a beam of light from a star passing by the Sun
and continuing on to the Earth. Because the light ray is bent, the
star appears to be shifted from its actual location.
This prediction was first tested in 1919 during a total solar eclipse.

 Deflection of light by mass

A light ray arriving from the left would be bent
inwards such that its apparent direction of origin,
when viewed from the right, would differ by an angle
(α, the deflection angle, see diagram) whose size is
inversely proportional to the distance (d) of the closest
approach of the ray path to the center of mass.


 Bending of light by gravitation

Light travels always the
shortest distance in a
curved space-time.


The figure above shows three different possible (mathematical)
paths for a pulse of light travelling around the Sun: the path with
no gravity, the path as predicted by Newtonian gravity, and
the path as predicted by Einstein's General Theory of Relativity


deflection angle df turning point R0 is
tells how far away the closest distance
from a straight line that the light pulse
the path of the light gets to the Sun. f=0
pulse in question was corresponds to R = R0,
deflected by the Sun.

No gravity, the path is a straight line. The path of a straight line in polar
coordinates centered at the center of the Sun would be: 1/r = (1/R0) cos(f)
To find df, look at the figure to the left and imagine the straight line path extending
infinitely far to the right and left of your screen.
When r = infinity, by symmetry of the coordinate system 0 = (1/R0) cos(df/2).
Therefore Df = p is the total difference in angle swept out by the light pulse as it
comes in from infinitely far away and travels back out infinitely far away.
The deflection angle here is df = df - p = 0, as it should be for a straight line.


Newtonian gravity
doesn't work well for describing the
properties of light, which can be modeled
like the propagation of a massless
particle. But it is possible using the
equation for a Newtonian hyperbolic orbit:
1/r = (G M(m/L)2)(1 + e cos(f)), e = (1 + (2E/m)(L/GMm)2))1/2
where the eccentricity e is a function of the incoming particle's energy E, mass m and
angular momentum L. The turning point R0 = (L/m)2/(G M (1 + e)).
To fake the propagation of light in Newtonian gravity,
the energy E = m v2/2 = m c2/2 so that (2 E/m) = c2. The angular momentum per unit
incoming mass (L/m) becomes L/m = R0 c.
The total angular sweep df = p + df is given by 0 = (1/R0) cos(df/2) + (G M/c2)/R02,
- cos(p/2 + df/2) = sin(df/2) ~ df/2 = (G M/c2)/R0
Finally, dfN = 2 (G M/c2)/R0 is the deflection angle for light found by naively using
the Newtonian model for a particle with velocity c.


Einstein's General Relativity

In General Relativity, the path of a
light pulse is described as a null geodesic
satisfying the geodesic equation for the
Schwarzschild metric, the distance
function that solves the Einstein equations
around a massive object in outer space such
as the Sun. An approximate equation for the
trajectory is 1/r = (1/R0) cos(f) + ((G M/c2)/R02) (2 - cos2(f)).
The term cos2(f) can be neglected if the deflection angle df is very small and
df/2 is close to p/2.
Therefore, to lowest order in df the 0 = (1/R0) cos(df/2) + 2 (G M/c2)/R02,
- cos(p/2 + df/2) = sin(df/2) ~ df/2 = 2 (G M/c2)/R0. Therefore dfE = 4 (G M/c2)/R0 =
2 dfN is the deflection angle for light found by using null geodesics in the
Schwarzschild metric according to General Relativity.


 Perihelion Precession of Mercury
The orbit of Mercury did not behave as required by Newton's
equations.(a long-standing problem in the study of the Solar System)

As Mercury orbits the Sun, it
follows an ellipse...but only
approximately: it is found that
the point of closest approach of
Mercury to the sun does not
always occur at the same place
but that it slowly moves around
the sun. This rotation of the
orbit is called a precession. Artist’s version of the precession of mercury’s
orbit around the sun
http://physics.ucr.edu/~wudka/Physics7/Notes_
www/node98.html


All the planetary orbits
precess and Newton's theory
predicts these effects, as
being produced by the pull of
the planets on one another. .
The precession of the orbits
of all planets except for
Mercury's can, in fact, be
understood using Newton;s
equations. But Mercury
seemed to be an exception. Artist’s version of the precession of mercury’s
orbit around the sun
www/node98.html


As seen from Earth the
precession of Mercury's orbit is
measured to be 5600 seconds
of arc per century (one second
of arc=1/3600 degrees).
Newton's equations, predicts a
precession of 5557 seconds of
arc per century. There is a
discrepancy of 43 seconds of Artist’s version of the precession of mercury’s
arc per century. orbit around the sun
www/node98.html


http://library.thinkquest.org/C0116043/generaltheory.htm

Most of the effect is due to the pull from the other
planets but there is a measurable effect due to the
corrections to Newton's theory predicted by the
General Theory of Relativity.

References

 Serway, Raymond A., Vuille, Chris and
Faugnn, Jerry S.(2009). College Physics
(Volume 2) 8th ed. Brooks/Cole Cengage
Learning
 http://www.youtube.com/watch?v=ev9zrt__lec

Physics 2 (Modern Physics)

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