Download Classical Physics Review - Tarleton State University

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.
Physics Problem
Fast Object
No
Large
Object
Yes
Large
Object
Quantum
Mechanics
(Hard Math)
Yes
Yes
Relativity
(Hard Math)
No
Classical Physics
(Easier Math)
Or
No
Extremely Hard
Problem
(Need Relativity & QM)
Quantum Mechanics
(Hard Math)
Or
Relativity
(Hard Math)
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 freebody diagram)


dp
 Fext 
dt
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:
y
y'
Ball

r'

R

r
x'
Sue
x
Tom
By vector subtraction, we see that the location of the ball according to Sue is
given by
  
r ' r  R

where r ' is the position of the ball as seen by Sue
r is the position of the ball as seen by Tom

R is the position of Sue as seen by Tom
C.
Velocity
We now apply the time derivative operator to both sides of our position.

d r '  d r   d  R

 
dt
dt
dt  
Applying the definition of velocity, we can rewrite the right-hand side of the
equation as
 
d 

r '  u  v
dt

where u is the velocity of the ball as seen by Tom

v 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.
 
d 

r '  u  v
dt '
  
u'  uv

where u' 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.
d  d 
d 

u '  u   v
dt
dt
dt
 
d 

u '  a  A
dt
 
d 

u '  a  A
dt '
  
a '  a A

where a is the acceleration of the ball as seen by Tom

a ' is the acceleration of the ball as seen by Sue

A is the acceleration of Sue as seen by Tom

Note: If Sue and Tom are not accelerating with respect to each other (ie A  0 ), they
will agree on the acceleration of the ball and Newton's Laws! The last term on the righthand side is the reason we add fake forces when using non-inertial reference frames.

Problem: Assuming that A  0 , 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

R  v t î  0 ĵ  0 k̂
Inserting our results above into our previous results for the position equation of
the Galilean Transformation, we have
t'  t
x'  x  v t
y'  y
z'  z
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, F , upon a body in displacing the body an amount ds is
defined by the equation


W   F  ds
final
initial
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!!
final
Wnet  
 F   ds  K
initial
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
K
1
Mv
2
2
Proof:
Inserting Newton II into the work energy theorem, we have that
final
Wnet  
initial

 dp   ds  K
 dt 

 ds
ΔK   dp 
dt
initial
final
We now apply the definition of velocity and linear momentum to our equation
   v
final

ΔK   d mv
initial
In classical mechanics, the mass of a particle is constant (an assumption we will
have to re-examine in special relativity) so
final
 
ΔK   m dv  v
initial
We can simplify our equation by using the following Calculus
  dv  v   dv  v  v  dv  2 dv  v
d v
2
Thus, our equation is simplified to
final
ΔK  
initial
ΔK 
1
2
K final  K initial 
1

m d v2
2

final
m  d v2
initial
1
2
2
m v final 
1
2
2
mv initial
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.
ε average 
3
kT
2
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
1 
ε average  3 k T 
2 
Thus, we see that each degree of freedom (translation in the x, y, and z directions)
1
have on average
k T of energy. This fact can be generalized as the
2
equipartition theorem.
2.
Equipartion Theorem
The average energy of any degree of freedom involving square of a generalized
1
co-ordinate is
k T.
2
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

  Q enclosed
E  dA 
ε
“Gauss’ Law for Electric Fields”

 
B  dA  0
“Gauss’ Law for Magnetic Fields”


 
dΦ E
B  d l  μ I  με
dt
 
dΦ B
E  dl  
dt
“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
 ρ
E
ε

  B0


B
E
t



E
  B  μ J  με
t
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.
1
v

Proof:







 E 
B
t




   E     E  
B
t





Since there are no current or charge densities in outer space ( J  0 and ρ  0 ),
our equation after substituting in Maxwell’s equations becomes:






E

0    2 E    με
 t   t 

2


E
 2 E  με
 t2
This is the vector wave equation. It is a generalization of the scalar wave equation
 2f
1  2f

 x 2 v2  t 2
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 μ and ε . 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.
 1 qQ
F
r̂
4π r 2

 
B dl μ I
“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.