The Basic Laws Of Physics In A Nutshell (Part 2)

 Basic laws of Heat – thermal behavior:

Definition of Energy:

Operational : "The ability to do work" - e.g. W = F x d (force acting in a direction x displacement in that direction)

Ideal definition: (Noether's): “Energy is that quantity that’s conserved because of time-displacement symmetry”

The last segment, “time-displacement symmetry” refers to the constancy of physical laws in time. Time goes on, but the laws of physics retain a constancy of their properties within it.

Law of conservation of energy-mass:

“The total amount of mass-energy in the universe in a system remains constant.”

This includes as a generalization the first law of thermodynamics, e.g. that internal (heat) energy is conserved, i.e. if two bodies are in thermal contact – and at different temperatures- the cooler body will have heat energy transferred to it, while the hotter will LOSE heat energy.

The Second law of thermodynamics (entropy law) is simply stated as:

“The entropy (degree of disorder of a system, usually denoted by the symbol S) increases in all natural processes”

There are two subsidiary statements of it:

I) The Kelvin -Planck statement:

It is impossible to construct a heat engine, operating in a cycle, which produces no other effect than absorption of thermal energy from a hot reservoir and the performance of an equal amount of work.

II): The Clausius statement:

It is impossible to construct a cyclical machine that produces no other effect than to transfer heat continuously from one body to another at higher temperature.

Thus, for example, gasoline once burnt in your car engine cannot be captured from the exhaust gases and used over again. Also, any energy process will also have a large part of any energy produced coming off as unusable waste energy. There is no way, or any process that can deliver 100% usable energy.

Basic laws of Electricity and Magnetism:

Coulomb’s Law: Any 2 charged particles (e.g. electrons, ions) attract (or repel) each other with a force inversely proportional the square of the distance between them and directly proportional to the product of the charges (Q, q)

F = k Qq/ r²

where k is a constant, Q, q the charges, and r the separation. Note the similar mathematical form to the universal law of gravitation. The difference is that the latter is ONLY attractive, while the Coulomb interaction may also be repulsive. (E.g. two like charges will always repel)

Gauss’ Law: The net number of electric field lines passing through a surface that encloses a net electric charge is proportional to the charge enclosed within the surface.

Other Electric –magnetic laws:

“There are NO magnetic monopoles. All magnetic field lines must end on one or other of two poles.”

Changes in magnetic flux always produce an induced electric current (Faraday’s law)

Moving electric charges in a closed circuit or loop give rise to a magnetic field (Ampere’s law)


From Maxwell’s E-M equations:

“At every instant, the ratio of the electric field magnitude to the magnetic field magnitude equals the speed of light, c” viz.

(ú Eú /ú Bú  = c )

Generalized law for E-M waves arising from Maxwell’s equations, laws:

“Electro-magnetic waves are generated by accelerating charges and consist of oscillating electric and magnetic fields which are at right angles to each other and also at right angles to the direction of wave propagation.”

In general, Maxwell’s equations will be expressed:

i)  Ñ X H  =  J    + ¶D / ¶ t     (A current density J arises from a magnetic field)

 

ii)               Ñ X E  =  - ¶B / ¶ t       (A magnetic field can arise from an electric field)

iii)             Ñ · B  =  0                        (There are no magnetic monopoles)

iv)             Ñ · D  =  r                       (Charges are conserved)

In addition, there are three “constitutive relations” that allow any of the above vectors to be re-cast in slightly different forms:

v)  D  =   e E

vi)  B =  m H 

vii) J  = s E

 

In the equations above, H represents the magnetic field intensity, B is the magnetic induction, E the electric field intensity, D the displacement current, and J is the current density. The constants, e  and m, denote the permittivity and the magnetic permeability – each for media. In vacuo, the constants used are: e ₀  and m ₀ and the speed of light can be expressed:  c =  1/ Öe ₀  Ö m ₀  .

Basic laws of quantum mechanics:


Whenever electrons change position (energy levels) in an atom, energy is given off in a discrete packet such that:

E = h n

where h is the Planck constant, and f is the frequency of the emitted light (photon) corresponding to the difference of energies in the levels.


This may also be written: E = hc/ l

where l is the wavelength

Every material particle has associated with it a de Broglie wave with a wavelength

l = h/ mv

where h is Planck’s constant, and m the mass, v the velocity.

All atoms represent systems that can be described in terms of probability waves, such that these waves disclose the probability of where the constituent electrons are at any given time.  The core equation foundational to describe these waves is the Schrodinger Equation:

Basic principles of quantum mechanics:

Whenever electrons change position (energy levels) in an atom, energy is given off in a discrete packet such that:

E = h n

where h is the Planck constant, and f is the frequency of the emitted light (photon) corresponding to the difference of energies in the levels.


This may also be written: E = hc/ l

where l is the wavelength

Every material particle has associated with it a de Broglie wave with a wavelength

l = h/ mv  =   h/ p

where h is Planck’s constant, and m the mass, v the velocity.

Postulates of quantum mechanics:

Postulate 1. The state of a quantum mechanical system is completely specified by a function  that depends on the coordinates of the particle(s) and on time. This function, called the wave function or state function, has the important property that:

y (r,t) y *(r,t) dV

is the probability that the particle lies in the volume element dV.

The wavefunction must satisfy certain mathematical conditions because of this probabilistic interpretation. For the case of a single particle, the probability of finding it somewhere is 1, so that we have the normalization condition:

P _aa’  =     ò ^a’_a   ‖y‖²  dx  =   1

All atoms represent systems that can be described in terms of probability waves, such that these waves disclose the probability of where the constituent electrons are at any given time.  The Schrodinger equation embodies the foundational description and principles to encompass this subatomic behavior. 

An analogous approach can be used to obtain the probability density (e,g, for electron in 1s state) for the hydrogen electron (in 3 dimensions):

P = ½y (1s) y (1s) *½

 yielding:

                                       The n = 1 electron orbital for hydrogen

As well as the energy quantization equation for the spectral lines. e.g.

E = - ħ² / 2m [a²  - 2a/ r] - e² / r   

Postulate 2. To every observable in classical mechanics there corresponds a linear, Hermitian operator in quantum mechanics.

For example there are 3 angular momentum operators for each of three Cartesian dimensions, x, y, z, e.g.

L _{x op}   =  -i h   [y  (¶/¶x ) – x (¶/¶y)]

L _{y op}   =  -i h   [z  (¶/¶x ) – x (¶/¶z)]

L _{z op}   =  -i h   [x  (¶/¶y ) – y (¶/¶x)]

Postulate 3. In any measurement of the observable associated with operator  A^, the only values that will ever be observed are the eigenvalues a, which satisfy the eigenvalue equation:

 A^y   =   a y

This postulate captures the central point of quantum mechanics--the values of dynamical variables can be quantized (although it is still possible to have a continuum of eigenvalues in the case of unbound states). If the system is in an eigenstate of  A^  with an eigenvalue a  then any measurement of the quantity A will yield a.

Heisenberg Uncertainty  Principle:

Given two canonically conjugate variables (e.g. energy and time, position and momentum), they cannot both be measured to the same precision at the same time.

Pauli Exclusion Principle:

“No 2 electrons in an atom can ever be in the same quantum state, that is – no two electrons in the same atom can have the same exact set of quantum numbers (n, l, m(s) or ml)”

A summary of  the simpler Bohr model for atomic applications is found in the flow chart below:

                          ATOMIC SUMMARY FLOW CHART: To find(a) Total energy of hydrogen atom, b) radii of allowed orbits, c) Allowed energies of atom for different stationary electron states

Einstein’s mass-energy equation: E = mc²

“In any fusion or fission reaction, the total rest mass of the products is less than the rest mass of the reactants – the change (decrease) in rest mass appearing as energy released in the reaction.”

Thus: D E (change in energy) = [m_R - m_p] c²

where the bracketed quantity on the right side is the difference in rest masses between reactants and products.

Special relativity:

Postulates: 

1) The laws of physics are the same in all inertial reference frames.

I.e. "All inertial observers are equivalent."

2) The velocity of light, c, is independent of the state of motion of the source and the observer.

This implies:

 “the speed of light has the same value in all inertial reference frames”

Thus, in no inertial reference frame can any material object exceed the speed of light.


Time dilation: 

All moving clocks run slower relative to an identical clock in a stationary frame.

Finally, the “fourth dimension” is NOT a basic law. Rather it arises out of the Principle of Relativity  by virtue of referencing all physical actions, laws in terms of time as well as the three dimensions of space (x, y, z).

In fact, in principle, given the four dimensions x, y, z and t -  ANY ONE could be “the fourth dimension”! (The order of choice is not important, what’s important is that four dimensions are required to specify and follow physical laws between difference reference frames).

 The Interval:

 Consider the diagram shown below showing two reference frames (x,y,z, t and x', y' z' and t') in relative motion with respect to an even P.

Then the space-time intervals (r and r') will be such that:

 r² = x² + y² + z² - c² t²

And:

r’² = x’² + y’² + z’² = c² t’²

And also:

x² + y² + z² -  c² t²  = x’² + y’² + z’² - c² t’²

The Inertia of Energy:

This is embodied in Einstein's famous equation:

E = m c²

which is more accurately posed as:

E = (D m) c²

where D m is the "mass defect" or difference, say in a nuclear reaction, and c is the velocity of light.