Bernoulli’s Equations for the Polarized and Unpolarized Photons and Electrons,
Temperature, Young’s Modulus and Stability of Polarized Photons and Electrons
Daniel, Isaac Mumo, Kirui, M. S. K, Golicha, H. S. A.
Physics Department, Egerton University, P.O. Box 536, Njoro, Kenya.
E-mail:
[email protected]
Key words: Magnetic Charge, Bernoulli’s Equation, Temperature, Isothermal, Adiabatic,
“Big Bang”, Young’s Modulus, Crystal Structure, Radioactive Decay
Abstract
Magnetic charge value has been formulated in the model of a straight wire
carrying an electrical current with the magnetic charge modeled as surrounding the
electrical charge as the magnetic current or magnetic voltage or magnetic field lines when
1
an electrical current flows. All electric charge in the photon has been seen to be the
electric charge, so that polarization of photons is similar to those of electrons. Here, we
consider the flow properties of the photon or electron waves and derive Bernoulli’s
equations for the polarized and unpolarized photon or electron. We proceed to explain
compressibility and incompressibility of magnetic field lines. We give Young’s modulus
for the polarized photon or electron.
1 INTRODUCTION
It was pointed out that magnetic charges (monopoles) could conceivably exist
despite not having been seen so far1. Particle physics theories like the grand unified
theory and superstring theory predict the existence of magnetic charges 2, 3. The quantum
theory of magnetic charge4 showed that if any magnetic charges exist in the universe,
then all electric charge in the universe must be quantized 5. The electric charge is, in fact,
quantized, which suggests that magnetic charges exist5. Experiments have produced
candidate events that were initially interpreted as magnetic charges 6, 7, but are now
regarded as inconclusive8. It has been shown that all electric charge in the photon is
actually the electronic electric charge in wave form, which implies that the photon and
the electron can be handled in a similar manner when we consider perpendicular
polarization of twin photons in the EPR (Einstein-Podolski-Rosen) hidden variable
phenomenon9. The structure of photons (or electrons) based on the particles as consisting
of magnetic waves surrounding the electrical current as the nucleus in the right hand grip
rule has been given9. Magnetic charge in photons (or electrons) has been determined, and
the orientation of magnetic and electric currents given 9. Magnetic charge as the source of
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magnetic field in Maxwell’s equations has been shown9.
Since the formulation of the magnetic charge and the structure of polarized and
unpolarized photons and electrons, no literature on the flow properties of the photon or
electron waves and involved Bernoulli’s equations has been published so far. Here,
vectors are bold while scalars are in italics.
2.1 Bernoulli’s Equation for Unpolarized Photons and Electrons
Consider the three dimensional spherical shape of the unpolarized photon 9. We
regard the circumference as the streamline of a fluid. We consider the flow to be laminar.
The Einsteinian energy9 is ε x = m xυ
2
y
= m x c 2y = hf x . We divide this Einsteinian energy
3
3
by a scalar volume V = 4 3 π r = 4 3 π λ of the sphere, where r ≡ λ is the radius (or
3
wavelength, or amplitude) as has been shown9, and λ = λ x λ y λ z . With the magnitudes
cx = υ
J
m3
x
= constant9,
= Nm
m3
= N
ε x m x c x2
=
V
V
we
m2
ε
have
x
V
=
m x c x2
V
.
This
equation
has
units
. Hence,
(N
m2
)
(1).
The left hand side is equivalent to the instantaneous x -direction relative static pressure
P2x − P1x between two points on a streamline. There is only one energy level for
unpolarized photon9, so, P2x = P1x = Px . It has been shown9 that the mass m x = Uf x ,
3
− 51
h
where U = c 2 = 7.38 × 10 Kgs
x
is constant and h
is Plank’s constant, and
c x = c = 2.998 × 10 8 m / s is the speed of light in vacuum. For unpolarized photon this
mass is a constant since the general frequency f is constant round the circumference of a
circle. Too, the volume V is constant for the sphere. The density of points ρ x = m x
V
is
constant, hence the fluid circumference (photon circumference ) is incompressible since
ρ = constant. The right hand side of equation (1) is the relative dynamic pressure
(
)
ρ υ 12x − υ 22x . Since c x = υ
x
= constant, the right hand side is simply ρ c x2 . Every point on
the circle (or sphere) is characterized9 by c = fλ , which is constant. The pressure
therefore, is a constant and is simply, Px = Fx
A
, where A is a cross sectional area. It
has been shown9 that a point moving along the circumference of a horizontal circle on the
x - z plane can be regarded to perform a to and fro motion along the x -axis. The particle
is regarded to vibrate about a fixed point along the x -axis, so that the force on the
particle changes between centripetal and centrifugal forces. Hence, at one time, the left
hand side of equation (1) is a negative pressure when the force is negative or reverses
direction. It has been shown that along the circumference of a circle (or sphere),
df
dt
= 0 , df
dλ
= 0 , and ∇ ⋅ f = 0 since f is constant and fλ = c is a constant. Along
the x -axis, the pressure gives the Bernoulli’s equation for an unpolarized photon as
Px + ρ x c x2 = constant. For any general direction, we write quantities in scalar form as,
P + ρ c 2 = constant
(2).
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In other words, the photon or boson in its unpolarized ground state is incompressible and
remains so unless otherwise altered by external means.
2.2 Bernoulli’s Equation for Polarized Photons
We choose polarized photon P as an example9 and choose one of the correlation
∆λ

∆λ

equations ∆ f x =  2 Px λ  f 2 Px =  2 Px λ  f1Px as has been shown9, where ∆ f x is
1Px 
2 Px 


the polarization frequency, ∆ λ 2 Px is the polarization wavelength, λ 1Px is the final
wavelength after photon P is polarized from original wavelength λ 2 Px , f 1Px is the final
frequency after photon P is polarized from original frequency f 2 Px , and f 2 Px is the
original frequency of photon P before polarization such that f 2 Px λ 2 Px = c x = c , and
f 1Px λ 1Px = c x = c . We have, Uc x = h c . The instantaneous force in the x -direction9 is,
x
(
)
2
∆ Fx = xˆ Uc x ( ∆ f x ) . Squaring
∆ fx
and substituting into the force we have,
2

∆λ
 
∆ Fx = xˆ Uc x f 22Px  2 Px

λ 1Px   . The elliptical shapes of polarized photons are regarded



as streamlines. The area of an ellipse is π × (amplitude in x -direction) × ( amplitude in y direction). In the polarized photon P, it has been shown 9 that the final wavelengths (or
amplitudes or radii) are: the original λ 2 Py since there is no polarization along y -axis, and
λ 1Px ≡ λ 1Pz since polarization occurs along the x and z -axes with x ≡ z -axis in two
5
dimensions. Therefore, the area of an elliptical photon in two dimensions is and is
π rx ry = π λ 1Px λ 2 Py . With λ 1Px = λ 1Pz , the area is also π λ 1Pz λ 2 Py . The force divided by the
area is the pressure. We have seen that at one time, the force is negative due to a to and
fro motion of a vibration about a fixed point on the circumference. Choosing that moment
of a negative force, we have,
 Uc x ( ∆ f x ) 2
ˆ
− x
 π λ 1Pz λ 2 Py

2
2 


 = xˆ Uc x f 2 Px ( ∆ λ 2 Px ) 
 π λ 1Pz λ 2 Py λ 1Px ( λ 1Px ) 




(3).
Considering direction along the x -axis, the left hand side of equation (3) gives the
relative change in static pressure − ( ∆ P2x − ∆ P1x ) between two different streamlines or
energy levels along the x -axis. The quantity π λ 1Pz λ 2 Py λ 1Px in the right hand side is a
volume of an ellipsoid without the factor 4 3 so the right hand side has units
(
)
(
)
 Kg
 m2
= ∆ ρ x ∆ υ 12x − ∆ υ 22x , where ∆ υ 1x and ∆ υ
3
m 
s2

2x
are the relative polarization
velocities9 characterizing two different energy level, different from original c x . This
gives the change in relative dynamic pressure. Like the force, the pressure depends on the
frequency and wavelength at that particular point on the streamline, so, between
streamlines (energy levels) along the x -axis, the frequencies are different for polarized
photons. Even though the magnitudes of the speeds points c = fλ are constant at any
point along the circumference of a photon energy level 9, whether polarized or
unpolarized, different energy levels or streamlines have different relative pressures for the
polarized photons as frequency f is different between two different streamlines on the x
6
-axis and after polarization ∆ υ ix ( i = 1 , 2 , 3 ,…) are different. Thus a relative dynamic
pressure gradient exists between two different streamlines. Polarization velocities for
polarized photon P are shown to be given by the equation
∆ f x λ 1Px − f 1Px ∆ λ 2 Px − ∆ f x ∆ λ 2 Px = 0
(4).
∆ υ 1Px − ∆ υ
2 Px
− ∆ υ 3 Px = 0
Replacing c x with ∆ υ ix in equation (3), we write,
U ( ∆ υ 1Px − ∆ υ 2 Px ) f 22Px ( ∆ λ 2 Px ) 2 U ( ∆ f x λ 1Px − f1Px ∆ λ 2 Px ) f 22Px ( ∆ λ 2 Px ) 2
−
=
π λ 1Pz λ 2 Py λ 1Px ( λ 1Px )
π λ 1Pz λ 2 Py λ 1Px ( λ 1Px )
(5).
The quantities Uf1 and Uf 2 are the masses on two different energy levels. Part of the
denominator of the right hand side of equations (3) and (5) is the volume of one ellipse
and is constant only for that ellipse (or ellipsoid in three dimensions), but is different for
different ellipses (ellipsoids) of different energy levels in a polarized photon. Thus, the
density is not constant and varies between two consecutive energy levels. It has been
shown9 that even though fλ = c = constant along the circumference of any polarized or
unpolarized structure, for polarized structures along the circumference, df dt ≠ 0 ,
df
dλ
≠ 0 , and ∇ ⋅ f ≠ 0 . Since pressure depends on frequency and wavelength at that
particular point on a polarized structure, then, along the circumference of a polarized
structure, a pressure gradient exists. Polarized photons are thus compressible as density is
not constant. Using scalar density, the right hand side of equation (5) becomes
7
( ∆ ρ 1x λ 1Px −
∆ρ
2x
=
∆ρ
Uf1Px
2x
∆ λ 2 Px ) f 22Px ( ∆ λ 2 Px )
π λ 1Pz λ 2 Py λ
∆ ρ 1x = ∆ ρ 1 , ∆ ρ
2x
where
∆ ρ 1x =
U∆ f x
π λ 1Pz λ 2 Py λ
and
1Px
. The Bernoulli’s equation for a polarized structure is thus,
λ 1Px
generally
( λ 1Px
),
1Px
− ( ∆ P2x − ∆ P1x ) =
Let
2
( ∆ ρ 1x λ 1Px −
∆ρ
2x
∆ λ 2 Px )
λ 1Px
λ 1Px
f 22Px ( ∆ λ 2 Px )
f 22Px ( ∆ λ 2 Px ) = ( ∆ υ 1 ) ,
2
2
2
∆ λ 2 Px
(6).
λ 1Px
f 22Px ( ∆ λ 2 Px ) = ( ∆ υ
2
2
)2,
= ∆ ρ 2 , and replacing vectors with scalars in pressures, generally,
∆ P1x = ∆ P1 , and ∆ P2x = ∆ P2 . Bernoulli’s equation (6) for any general direction becomes,
( ∆ P1 −
∆ P2 ) = ∆ ρ 1 ( ∆ υ 1 ) 2 − ∆ ρ
2
(∆ υ 2 )2
(7),
or,
∆ P1 − ∆ ρ 1 ( ∆ υ 1 ) 2 = ∆ P2 − ∆ ρ
2
(∆ υ 2 )2
(8).
The density is not constant and a pressure gradient exists, hence, polarized photons are
compressible back to their original unpolarized state by regaining 9 ∆ f x for polarized
photon P, and by the loss of ∆ f x by polarized photon Q.
2.3 Temperature of Polarized Photons and Electrons
We
have
∆ f x = 2.985 × 10 8 Hz ,
seen
that9
f 1Px = 1.3 × 10 6 Hz ,
f 3Qx = f 2Qx + ∆ f x = 5.983 × 10 8 Hz ,
f 2 Px = f 2Qx = 2.998 × 10 8 Hz ,
λ 1Px = λ 1Pz = 230.615m ,
λ 2 Px = λ 2 Py = λ 2 Pz = λ 2Qx = λ 2Qy = λ 2Qz = 1m , and λ 3Qx = 0.501m . Upon polarization,
8
unpolarized photon Q is compressed to polarized photon Q with reduction in volume
3
2
(from linear dimensions λ 2Qx λ 2Qy λ 2Qz = λ 2Qx to λ 3Qx λ 2Qy λ 3Qz = λ 3Qx λ 2Qy ). We have
seen that the temperature in the unpolarized and polarized states in a photon is given as9,
T=
2hf 2 Px
3K B
= Mf 2 Px
(K )
∆T =
2h∆ f x
3K B
(9).
= M∆ f x
− 13
where, M = 2h 229.615( 3K ) = 1.395 × 10 Ks is a constant, K B is the Boltzmann
B
constant. Since temperature is directly proportional to the frequency, polarized photon Q
reduces in volume but its temperature along the x -axis increases from T = Mf 2Qx to
T = Mf 3Qx at constant pressure boundary fλ = c , meaning the process of polarization is
isothermal and not adiabatic Unpolarized photon P expands to polarized photon P (from
3
2
linear dimensions λ 2 Px λ 2 Py λ 2 Pz = λ 2 Px to λ 1Px λ 2 Py λ 1Pz = λ 1Px λ 2 Py ) at constant pressure
boundary fλ = c , and cools from T = Mf 2 Px to T = Mf1Px .
Since it has been shown that h = q m q e = ∆ q m ∆ q e , where, q m , q e are the
unpolarized magnetic and electric charges and ∆ q m , ∆ q e are their polarized counterparts,
then, Bernoulli’s equations involve compressions or expansions in magnetic and electric
charge waves.
2.4 State of Polarized Photons Universes
Let compressibility mean compression or expansion. Polarized photon P can be
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compressed back to the unpolarized photon at constant pressure boundary fλ = c with
reduction
in
volume
(from
linear
dimensions
λ 1Px λ 2 Py λ 1Pz = λ 12Px λ 2 Py
to
λ 2 Px λ 2 Py λ 2 Pz = λ 32 Px ) and increase in temperature (from T = Mf1Px to T = Mf 2 Px ), while
polarized photon Q can expand back to unpolarized photon Q with increase in volume
2
3
(from linear dimensions λ 3Qx λ 2Qy λ 3Qz = λ 3Qx λ 2Qy to λ 2Qx λ 2Qy λ 2Qz = λ 2Qx ) at constant
pressure boundary fλ = c , and reduction in temperature (from T = Mf 3Qx to T = Mf 2Qx ).
Since polarization involved momentary interaction of photons and then
separation9, then, this situation can be compared to the “Big Bang”, moment of creation
of our universe. Thus polarized photon P universe is contracting and heating up while
polarized photon Q universe is expanding and cooling. We have seen that two
perpendicularly polarized twin photons in the EPR phenomenon communicate with each
other10. Therefore, after the “Big Bang” that created the universe we live in, there is a
possibility there is another universe polarized perpendicularly to our own and
communicates with our own universe. By this theory, one universe is heating up while the
other is cooling.
2.5 Young’s Modulus
We deal with polarized photon P. Polarization occurs on the x - z plane and we have
seen that ∆ f x ≡ ∆ f z , f1Px ≡ f1Pz , λ 1Px ≡ λ 1Pz . An area in the x - z plane is given as the
product, π λ 1Px λ 1Pz = π λ 12Px . The force per unit area is the pressure or a stress. We have
10
∆ f x as the extension, and f 2 Px as the original frequency. We write the strain in scalar
quantities as
∆ fx
f 2 Px . We thus have
∆ Fx π λ 12Px
Y=
= 1.19 × 10 − 30 N / m 2
∆ f x f 2 Px
(10).
Values for polarized photon Q can be obtained by replacing quantities as in equation (10).
(
)
Uc x = h
2
∆ Fx = xˆ Uc x ( ∆ f x ) ,
Since
h = q m qe = ∆ qm ∆ qe ,
c x and
then,
 h( ∆ f x ) 2 
 ∆ q ∆ q (∆ f )2 
∆ Fx = xˆ
 = xˆ  m e x
 . We have seen9 that for a current flowing in
c
c
x
x




the y -direction in the right hand grip rule for a straight wire, with ∆ f x = ∆ f y = ∆ f z (even
though there is no polarization on the y -axis, the change in magnetic and electric
currents
∆ Fx =
are
∆ i ey × ∆ i mz
Y=
∆ i mz = ∆ q m ∆ f z ,
and
∆ i ey = ∆ q e ∆ f y
respectively.
Therefore,
c x , hence,
∆ i ey × ∆ i mz π c x λ 12Px
(11).
∆ f x f 2 Px
We have seen that the space-time equations ∆ f x =
polarized photon P and ∆ f x =
∆ λ 2Qx f 2Qx
λ 3Qx
=
∆ λ 2 Px f 1Px
∆ λ 2Qx f 3Qx
λ 2 Px
=
∆ λ 2 Px f 2 Px
λ 1Px for
λ 2Qx for polarized photon Q.
Therefore, many values of Young’s moduli are available for a polarized photon since also,
11
c x can be replaced by ∆ υ ix in equation (11).
2.6 Stability of Polarized Photons and Electrons and Radioactive Decay
It has been shown10 that photons and electrons have a nucleus. Since
perpendicularly polarized photons and electrons are compressible, this can be interpreted
that polarized photon and electrons are unstable or constitute an unstable nucleus. On the
other hand, unpolarized photons and electrons are incompressible and are regarded as
stable, or, constitute a stable nucleus.
There are several scenario here. (a) since a calcite crystal reunites perpendicularly
polarized light into the original beam and the process is reversible, then, perpendicularly
polarized photons and electrons can reunite in a reversible process to the original
unpolarized states (b) polarized photons and electrons can polarize other neighbouring
photons (or electrons) in their path and form a crystal structure (c) polarized photons (or
electrons) can decay radioactively to other forms of photons (or electrons).
CONCLUSION
Derivation of Bernoulli’s equations for unpolarized and polarized photons has
enabled us understand the possible existence of another universe polarized
perpendicularly to our own and communication with us, and that one universe is heating
up while the other is cooling down.
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12
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