Review of 19th Century Classical Physics
I. Physics Overview
By the late part of the 19th century, physics consisted of two great pillars: a) mechanics including thermodynamics and b) electromagnetism. However, a series
of problems that should have been solvable continued to perplex physicists. Modern physics is the study of the two great revolutions (relativity and quantum mechanics) that solved these problems and our continuing effort to find the ultimate rules of the game.
Note: Classical mechanics is equivalent to the slow speed approximation of the more powerful special theory of relativity. We still use classical mechanics because the math is easier. For large quantum numbers, the more general quantum mechanics reduces to classical mechanics. The present problem is to make a relativistically correct quantum mechanics that also includes gravitation. The general theory of relativity that handles gravity is based upon the math of topology while QM is based upon differential equations.
II. Classical Physics (1800’s)
A. General Ideas of Mechanics
1. A reference frame is a coordinate axis and origin used by an observer to describe the motion of an object.
2. An inertial reference frame is one in which the observer is not accelerating (ie one in which Newton's Laws are valid without adding fake forces to your free- body diagram)
Question: Accelerating with respect to what?
3. Galilean Relativity -
All inertial reference frames are equivalent! Another way of stating this principle is that only relative motion can be detected.
4. Transformation Equations -
If you know what an observer in a particular reference frame observes then you can predict the observations made by an observer in any other reference frame. The equations that enable you to make these calculations are called Transformation Equations.
5. Invariance -
Since the labeling of your coordinate axis and its origin location is arbitrary, the equations of physics should have the same form regardless when you rotate or translate your axis set. Equations that have this property are said to be invariant to the transformation.
Problem 1.10 in your Schaum's Outline Series shows that Maxwell's Equations are not invariant under a Galilean Transformation so E&M and Mechanics are not consistent.
III. Galilean Transformations
A. Time
All observers measure the same time. This was assumed without proof and used to derive our equations in PHYS122.
t = t'
B. Position
Let us consider a ball being measured by two different observers as shown below:
By vector subtraction, we see that the location of the ball according to Sue is given by
where is the position of the ball as seen by Sue
is the position of the ball as seen by Tom
is the position of Sue as seen by Tom
C. Velocity
We now apply the time derivative operator to both sides of our position.
Applying the definition of velocity, we can rewrite the right-hand side of the equation as
where is the velocity of the ball as seen by Tom
is the velocity of Sue as seen by Tom
Since the time measured by both Sue and Tom is the same, we can replace t with t' and apply the definition of velocity to the left-hand side of the equation.
where is the velocity of the ball as seen by Sue.
You should note that the definition of velocity requires that both the time and position be measured by the same observer!
D. Acceleration
We now follow the same procedure as in part C to obtain the relationship between the ball's acceleration as measured by Sue and as measured by Tom.
where is the acceleration of the ball as seen by Tom
is the acceleration of the ball as seen by Sue
is the acceleration of Sue as seen by Tom
Note: If Sue and Tom are not accelerating with respect to each other (ie ), they will agree on the acceleration of the ball and Newton's Laws! The last term on the right-hand side is the reason we add fake forces when using non-inertial reference frames.
Problem: Assuming that , who is accelerating (Tom, Sue, or both)?
E. Special Case of Two Observers in 1-D Uniform Motion
In developing special relativity, we find it convenient to simplify the math by
considering the motion of an object as see by two observers who are in 1-D uniform
motion with respect to each other. Thus, we will assume the following:
1) Tom and Sue are both located at the origin at t = 0 (We can arbitrarily start
measuring time whenever we want so this doesn't limit our results)
2) Sue is traveling at constant speed v in the +x direction as seen by Tom (Thus,
we don't consider acceleration. This is what will be special about special
relativity!)
The position vector of Sue as seen by Tom is given at any instant t by
Inserting our results above into our previous results for the position equation of
the Galilean Transformation, we have
We will see how these equations must be modified for high speed problems
when we study the Lorentz Transformation.
IV. Work and Energy Concepts
A. Work
The work done by a force, , upon a body in displacing the body an amountis defined by the equation
B. Energy
Energy is the ability of a body to perform work. (ie Stored Work!!)
C. Kinetic Energy
Kinetic energy is defined as the energy that an object has due to its motion!
D. Work-Energy Theorem
The work done by the net external force upon an object (or equivalently the net work done by all forces upon the object) is equal to the change in the objects kinetic energy!!
The work-energy theorem is the heart of all energy concepts as it relates the connection between Newton II, work and energy!!
We used this theorem to derive the conservation of mechanical energy in PHYS1224 and to develop the classical formula for computing the kinetic energy of a body.
E. Classical Formula For Finding The Kinetic Energy of an Object
The kinetic energy of an object traveling at speeds much less than the speed of light (ie classical physics) can be obtained using the formula
Proof:
Inserting Newton II into the work energy theorem, we have that
We now apply the definition of velocity and linear momentum to our equation
In classical mechanics, the mass of a particle is constant (an assumption we will have to re-examine in special relativity) so
We can simplify our equation by using the following Calculus
Thus, our equation is simplified to
By comparing the individual terms, we obtain our classical formula for kinetic energy.
Q.E.D.
F. Temperature and Average Energy
1. In our study of thermodynamics, we developed a relationship between average
energy of a monatomic gas molecule with three degrees of motion and the
temperature of the gas. The constant of proportionality is called the Boltzman
constant and is related to the ideal gas constant R.
The calculation of the average energy of the particle involves two separate steps: 1) determining the number of degrees of freedom for the particle and 2) determining the average energy for each degree of freedom.
We can rewrite our previous result for a gas particle with three degrees of freedom as
Thus, we see that each degree of freedom (translation in the x, y, and z directions) have on average of energy. This fact can be generalized as the equipartition theorem.
2. Equipartion Theorem
The average energy of any degree of freedom involving square of a generalized co-ordinate is .
The equipartition theorem is very useful in classical calculations of systems containing many particles like gases. We will use this concept when considering the classical computation of the thermal radiation of black-bodies.
The ability to correctly calculate the number of degrees of freedom for more complicated systems is considered in Mechanics (PHYS 3313). The application of this material in determining macroscopic properties of systems based upon their atomic nature is considered in your advanced thermodynamics class (PHYS 3333 Statistical Mechanics) and in Solid State Physics (PHYS 4363).
V. General Ideas of E&M in 19th Century
1. Maxwell’s Equations in Integral Form
“Gauss’ Law for Electric Fields”
“Gauss’ Law for Magnetic Fields”
“Ampere-Maxwell Law”
“Faraday’s Law of Magnetic Induction”
These global equations are best for verifying Maxwell’s equations experimentally. The differential forms of Maxwell’s equations which are obtained from the integral forms using vector calculus are usually more powerful from a theoretical standpoint.
2. Maxwell’s Equations in Differential Form
3. Light is an electromagnetic wave with a wavelength between 400-700 nm. If you see a scratch on a table then it must be larger than the wavelength of light. Thus, a highly polished surface in optics or semiconductor manufacturing which appears smooth to the eye is said to have a sub-micron finish.
4. Light is a transverse wave.
5. The speed of light depends only on the medium through which it travels and not upon the observer.
Proof:
Since there are no current or charge densities in outer space ( and ), our equation after substituting in Maxwell’s equations becomes:
This is the vector wave equation. It is a generalization of the scalar wave equation
where v is the speed of the wave. By comparison, we see that the electric field propagates with a speed dependent only upon the product of and. I will leave derivation of the wave equation for the magnetic field for you.
6. The values for the permittivity and permeability constants of free space are already specified in our work with static electric and magnetic fields (Coulomb’s Law and Ampere’s Law). Thus, the speed in free space for any E&M disturbance
is independent of the observers reference frame.
“Coulomb’s Law”
“Ampere’s Law”
7. According to 19th century physicists, light propagated from the sun to the earth through the luminiferous ether.
8. Properties of the Ether Fluid
i) non-viscous - Earth doesn't slow down while traveling through the ether.
ii) incompressible - speed of light is very fast
iii) massless
VI. Important Physics Problems of Late 19th Century
Modern Physics was developed as the solution to some extremely important
problems in the late 19th century that stumped physicists. We will study these
important problems and how they have caused us to change our notions of time,
space, and matter. Some of these important problems include a) the ether
problem, b) stability of the atom, c) blackbody radiation, d) photoelectric effect,
and e) atomic spectra.