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 motion2. 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 byconnecting space and time, matter and energy, and electricity and magnetism

The Special Theory of Relativitydefies 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.

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

(No experiment can be done in an inertial reference frame to detect its state of motion.)

(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.

1. This room Experiment: Drop a ball. It accelerates downward at 9.8 m/s2 due to

the force of gravity.

Inertial Reference Frame

examples

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.

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

Noninertial Reference Frame

examples1. 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 is accelerating noninertial frame

Ground is the inertial frame

Platform at rest, tree moving—ball is seen by observers on platform as being deflected, but no force acts on it. Violation of Newton’s second law.

Platform moving. Observer on the ground (inertial frame) sees ball move in a straight line, but sees the catcher move away.

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

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.

Time dilation

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 time

Space time

= = c

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

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

Time dilation

is given as

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

Δt’

√ 1 – v2/c2Δt =

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

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

the relativistic mass is given as

m =

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

This compartment is at rest in the Earth’s gravitational field.

This compartment is moving in a gravity-

free environment

The apple hits the floor of the compartment because the compart-ment accelerates.

The apple hits the floor of the compartment because the Earth’s gravity accelerates the apple downward

Newton Einstein


“Equivalence Principle”:

Observers cannot distinguish betweeninertial forces due to acceleration anduniform gravitational forces due to amassive 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 thespace-time continuum. The force was replaced bythe strength of the curvature of the space whichis 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

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

• perihelion precession of Mercury

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

Gravitational redshift


The gravitational redshift of a light wave as it moves upwards against a gravitational field (caused by the yellow star below). http://en.wikipedia.org/wiki/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. If applied to optical wave-lengths this manifests itself as a change in the color of the light as the 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 towards the blue.

The gravitational redshift http://physics.ucr.edu/~wudka/Physics7/Notes_www/node89.html#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

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.

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

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



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.

deflection angle df tells how far away from a straight line the path of the light pulse in question was deflected by the Sun.

turning point R0 is the closest distance that the light pulse gets to the Sun. f=0 corresponds to R = R0,

Bending of light by gravitationGeneral Theory of Relativity

Newtonian gravitydoesn't work well for describing theproperties of light, which can be modeledlike 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))1/2where 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.

Bending of light by gravitationGeneral Theory of Relativity

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)/R0

2, - 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.




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 arc per century.




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.

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

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)

Education

Transcript of Physics 2 (Modern Physics)