Previously, we presented several empirical equations using the temperature of the cosmic microwave background (CMB), which were simple and mathematically connected. Next, we proposed an empirical equation for the fine-structure constant. Considering the compatibility among these empirical equations, the values of the CMB temperature ( T c) and the gravitational constant ( G) were calculated to be 2.726312 K and 6.673778 × 10 -11 m 3 ⋅kg -1 ⋅s -2, respectively. Then, for the values of the factors 9/2 and π in our equations, we used 4.48852 and 3.13201, respectively. Using the redefinition of Avogadro’s number and the Faraday constant, we explained that these values can be adjusted back to 9/2 and π. However, our arguments have become quite complex. Thus, we now attempt to simplify these empirical equations. We show that every equation can be explained in terms of the Compton length of an electron ( λ e), the Compton length of a proton ( λ p) and α.
The symbol list is shown in Section 2. We previously discovered Equations (1), (2) and (3) [
G m p 2 h c = 4.5 2 × k T c 1 kg × c 2 (1)
G m p 2 ( e 2 4 π ε 0 ) = 4.5 2 π × m e e × h c × ( C J ⋅ m × 1 kg = 1 V ⋅ m × 1 kg ) (2)
m e c 2 e × ( e 2 4 π ε 0 ) = π × k T c × ( J ⋅ m C = V ⋅ m ) (3)
Next, we discovered an empirical equation for the fine-structure constant [
137.0359991 = 136.0113077 + 1 3 × 13.5 + 1 (4)
13.5 × 136.0113077 = 1836.152654 = m p m e (5)
We believed that Equations (4) and (5) should be related to the transference number [
3.13201 ( V ⋅ m ) = ( m p m e + 4 3 ) m e c 2 e c ( m 2 s × J A ⋅ m = J ⋅ m C = V ⋅ m ) (6)
4.48852 ( 1 A ⋅ m ) = q m c ( m p m e + 4 3 ) m p c 2 ( s m 2 × V ⋅ m J = V J × s m = s C ⋅ m = 1 A ⋅ m ) (7)
Then, ( m p m e + 4 3 ) has units of ( m 2 s ) . Using the redefinition of Avogadro’s number and the Faraday constant, these values can be adjusted back to 9/2 and π [
π ( V ⋅ m ) = ( m p m e + 4 3 ) m e _ n e w c 2 e n e w c ( m 2 s × J A ⋅ m = J ⋅ m C = V ⋅ m ) (8)
4.5 ( 1 A ⋅ m ) = q m _ n e w c ( m p m e + 4 3 ) m p _ n e w c 2 ( s m 2 × V ⋅ m J = V J × s m = s C ⋅ m = 1 A ⋅ m ) (9)
Our first purpose is to simplify these equations, and we have attempted to explain them using thermodynamic principles discovered in the area of solid-state ionics. Unfortunately, the background theory could not be completed. Furthermore, our discussions have become quite complex. Therefore, the purpose of this report is to simplify these equations. The remainder of this paper is organized as follows. In Section 2, we present the list of symbols used in our derivations. In Section 3, we discuss the purpose of this report. In Section 4, we propose six equations that are functions of the Compton length of an electron (λe), the Compton length of a proton (λp) and α. In Section 5, using these six equations, we explain our main equations. The compatibility with the theory of special relativity is discussed. In Section 6, our conclusions are described.
G: gravitational constant: 6.6743 × 10−11 (m3∙kg−1∙s−2)
(we use the compensated value 6.673778 × 10−11 in this report)
Tc: temperature of the CMB: 2.72548 (K)
(we use the compensated value 2.726312 K in this report)
k: Boltzmann constant: 1.380649 × 10−23 (J·K−1)
c: speed of light: 299,792,458 (m/s)
h: Planck constant: 6.62607015 × 10−34 (J∙s)
ε0: electric constant: 8.8541878128 × 10−12 (N·m2·C−2)
μ0: magnetic constant: 1.25663706212 × 10−6 (N·A−2)
e: electric charge of one electron: −1.602176634 × 10−19 (C)
qm: magnetic charge of one magnetic monopole: 4.13566770 × 10−15 (Wb)
(this value is only a theoretical value, qm = h/e)
mp: rest mass of a proton:1.6726219059 × 10−27 (kg)
(we use the compensated value 1.672621923 × 10−27 kg in this report)
me: rest mass of an electron: 9.1093837 × 10−31 (kg)
Rk: von Klitzing constant: 25812.80745 (Ω)
Z0: wave impedance in free space: 376.730313668 (Ω)
α: fine-structure constant: 1/137.035999081
λp: Compton wavelength of a proton: 1.32141 × 10−15 (m)
λe: Compton wavelength of an electron: 2.4263102367 × 10−12 (m)
e n e w = e × 4.48852 4.5 = 1.59809 E − 19 ( C ) (10)
q m _ n e w = q m × π 3.13201 = 4.14832 E − 15 ( Wb ) (11)
h n e w = e n e w × q m _ n e w = h × 4.48852 4.5 × π 3.13201 = 6.62938 E − 34 ( J ⋅ s ) (12)
R k _ n e w = q m _ n e w e _ n e w = R k × 4.5 4.48852 × π 3.13201 = 25958.0 ( Ω ) (13)
We observe that Equation (13) can be rewritten as follows.
R k n e w = 4.5 ( 1 A ⋅ m ) × π ( V ⋅ m ) × m p m e = 25957.9966027 ( Ω ) (14)
Z 0 _ n e w = α × 2 h n e w e n e w 2 = 2 α × R k n e w = Z 0 × 4.5 4.48852 × π 3.13201 = 378.849 ( Ω ) (15)
We observe that Equation (15) can be rewritten as follows.
Z 0 _ n e w = 4.5 ( 1 A ⋅ m ) × π ( V ⋅ m ) × 2 α × m p m e = 378.8493064 ( Ω ) (16)
μ 0 _ n e w = Z 0 _ n e w c = μ 0 × 4.5 4.48852 × π 3.13201 = 1.26371 E − 06 ( N ⋅ A − 2 ) (17)
ε 0 _ n e w = 1 Z 0 _ n e w × c = ε 0 × 4.48852 4.5 × 3.13201 π = 8.80466 E − 12 ( F ⋅ m − 1 ) (18)
c _ n e w = 1 ε 0 _ n e w μ 0 _ n e w = 1 ε 0 μ 0 = c = 299792458 ( m ⋅ s − 1 ) (19)
In Equation (19), the value of the speed of light should not be changed because the units for 1 m and 1 s are unchanged. The Compton wavelength (λ) is as follows.
λ = h m c (20)
This value (λ) should be unchanged since the unit for 1 m is unchanged. However, in Equation (12), the Planck constant is changed. Therefore, the unit for the masses of one electron and one proton should be redefined.
m e _ n e w = 4.48852 4.5 × π 3.13201 × m e = 9.11394 E − 31 ( kg ) (21)
m p _ n e w = 4.48852 4.5 × π 3.13201 × m p = 1.67346 E − 27 ( kg ) (22)
From the dimensional analysis in the previous report [
k T c _ n e w = 4.48852 4.5 × π 3.13201 × k T c = 3.7659625 E − 23 ( J ) (23)
Next, to simplify the calculation, GN is defined as follows.
G N = G × 1 kg ( m 3 ⋅ s − 2 ) (24)
Now, we hope that the value of GN should remain unchanged. However, according to the dimensional analysis in the previous report [
G N _ n e w = G N × 4.5 4.48852 ( m 3 ⋅ s − 2 ) = 6.69084770 E − 11 ( m 3 ⋅ s − 2 ) (25)
As a result of our previously proposed redefinition method [
Procedure 1: The MKSA units should be redefined.
Procedure 2: The equations should be recalculated using the redefined values.
Procedure 3: The calculated values should be converted back to the MKSA units.
For convenience, Equations (8) and (9) are rewritten as follows.
π ( V ⋅ m ) = ( m p m e + 4 3 ) m e _ n e w c 2 e n e w c ( V ⋅ m ) (26)
4.5 ( 1 A ⋅ m ) = q m _ n e w c ( m p m e + 4 3 ) m p _ n e w c 2 ( 1 A ⋅ m ) (27)
We observe that the coefficient ( m p m e + 4 3 ) cannot be uniquely determined. For example, ( m p m e + 3.78 3 ) is allowed. In this case, the value of G should be 6.67431E−11, which may be a more suitable value. However, the calculated value of Tc is then 2.72642 K and becomes larger than the observed value. The purpose of this report is to simplify every equation to elucidate the redefinition method. To simplify very complex calculations from the unexpected original aspects is useful to combine two different theories to give a single unified theory [
We propose the following 6 equations. After redefinition, the Compton wavelength (λ) is unchanged. Therefore, the right side of each of these equations should be constant.
m e _ n e w c 2 × ( m p m e + 4 3 ) 2 ( J ⋅ m 4 s 2 ) = π 4.5 ( V ⋅ m ⋅ A ⋅ m = J ⋅ m 2 s ) × λ p c ( m 2 s ) = 2.76564 E − 07 ( J ⋅ m 4 s 2 ) = constant (28)
e n e w c × ( m p m e + 4 3 ) ( A ⋅ m 3 s ) = 1 4.5 ( A ⋅ m ) × λ p c ( m 2 s ) = 8.80330 E − 08 ( A ⋅ m 3 s ) = constant (29)
m p _ n e w c 2 × ( m p m e + 4 3 ) 2 ( J ⋅ m 4 s 2 ) = π 4.5 ( J ⋅ m 2 s ) × λ e c ( m 2 s ) = 5.07814 E − 04 ( J ⋅ m 4 s 2 ) = constant (30)
q m _ n e w c × ( m p m e + 4 3 ) ( V ⋅ m 3 s ) = π ( V ⋅ m ) × λ e c ( m 2 s ) = 2.28516 E − 03 ( V ⋅ m 3 s ) = constant (31)
k T c _ n e w × 2 π α × ( m p m e + 4 3 ) 3 ( J ⋅ m 6 s 3 ) = π 4.5 ( J ⋅ m 2 s ) × λ p c × λ e c = 2.011697 E − 10 ( J ⋅ m 6 s 3 ) = constant (32)
G N _ n e w ( m 3 s 2 ) × ( m p m e + 4 3 ) ( m 2 s ) = ( λ p c ) 2 ( m 4 s 2 ) × c ( m s ) × 9 α 8 π = 1.22943 E − 07 ( m 5 s 3 ) = constant (33)
Every equation is written using the values after redefinition. However, the calculated values should be converted back to the MKSA units. The main problem is Equation (25). For convenience, Equation (25) is rewritten as follows.
G N _ n e w = G N × 4.5 4.48852 ( m 3 ⋅ s − 2 ) = 6.69084770 E − 11 ( m 3 ⋅ s − 2 ) (34)
To explain Equation (34), we have discovered the following equations.
m e _ n e w e n e w = m e e × π 3.13201 (35)
It means that the mass-to-charge ratio should be changed. But the ratio between the number of electrons in 1C and the number of electrons in 1kg should not be changed. Therefore,
1 C n e w 1 C = e e n e w = 4.5 4.48852 (36a)
1 k g n e w 1 k g = e e n e w = m e m e _ n e w × π 3.13201 = 4.5 4.48852 × 3.13201 π × π 3.13201 = 4.5 4.48852 (36b)
Therefore,
G N 1 k g = G N _ n e w 1 k g n e w × 4.48852 4.5 ( m 3 s − 2 k g ) = 6.6737778667 E − 11 ( m 3 s − 2 k g ) (37)
Therefore, the associated problem can be solved.
From this section onward, the values used are those obtained after redefinition. Strictly speaking, therefore, me should be written as me_new. However, we omit the subscript “new” to avoid unnecessarily notational complexity.
For convenience, Equation (1) is rewritten as follows.
G m p 2 h c = 4.5 2 × k T c 1 kg × c 2 (38)
G N m p 2 h c = 4.5 2 × k T c c 2 (39)
Using Equations (28)-(33), the left side is rewritten as
G N m p 2 h c = ( λ p c ) 2 × c × 9 α 8 π × ( m p m e + 4 3 ) − 1 × { π 4.5 × λ e c − 1 × ( m p m e + 4 3 ) − 2 } 2 1 4.5 × λ p × ( m p m e + 4 3 ) − 1 × π × λ e × ( m p m e + 4 3 ) − 1 × c (40)
Therefore,
G N m p 2 h c = λ p × λ e × α 4 × ( m p m e + 4 3 ) − 3 (41)
The right side is
4.5 2 × k T c c 2 = 4.5 2 c 2 × λ p c 9 × λ e c × ( m p m e + 4 3 ) − 3 × α = λ p × λ e × α 4 × ( m p m e + 4 3 ) − 3 (42)
From Equations (41) and (42), we obtain
G m p 2 h c = 4.5 2 × k T c 1 kg × c 2 (43)
For convenience, Equation (2) is rewritten as follows.
G m p 2 ( e 2 4 π ε 0 ) = 4.5 2 π × m e e × h c × ( C J ⋅ m × 1 kg = 1 V ⋅ m × 1 kg ) (44)
Therefore,
G N m p 2 h c = 4.5 2 π × m e e × ( e 2 4 π ε 0 ) (45)
According to Equation (41), the left side is
G N m p 2 h c = λ p × λ e × α 4 × ( m p m e + 4 3 ) − 3 (46)
Regarding the right side,
4.5 2 π × m e e × ( e 2 4 π ε 0 ) = 4.5 2 π × m e × e c 4 π ε 0 c = 4.5 2 π × m e × e c 4 π × Z 0 (47)
For convenience, Equation (16) is rewritten as follows.
Z 0 = 9 π × α × m p m e (48)
Therefore,
4.5 2 π × m e e × ( e 2 4 π ε 0 ) = 4.5 2 π × m e × e c 4 π × 9 π × α × m p m e = 4.5 8 π × 9 m p × e c × α (49)
Hence,
4.5 8 π × 9 α × e c × m p = α 4 × λ e × λ p × ( m p m e + 4 3 ) − 3 (50)
From Equations (46) and (50), we obtain
G N m p 2 h c = 4.5 2 π × m e e × ( e 2 4 π ε 0 ) (51)
Therefore,
G m p 2 ( e 2 4 π ε 0 ) = 4.5 2 π × m e e × h c (52)
For convenience, Equation (3) is rewritten as follows.
m e c 2 e × ( e 2 4 π ε 0 ) = π × k T c × ( J ⋅ m C = V ⋅ m ) (53)
The left side is
m e c 2 × e 4 π ε 0 = m e c 2 × e c 4 π ε 0 c = m e c 2 × e c 4 π × Z 0 (54)
Therefore, using Equation (48), we obtain
m e c 2 × e c 4 π × Z 0 = m e c 2 × e c 4 π × 9 π × α × m p m e = m p c 2 × e c × 9 4 α (55)
Using Equations (29) and (30), we obtain
m p c 2 × e c × 9 4 α = π 4.5 × λ e c × ( m p m e + 4 3 ) − 2 × 1 4.5 × λ p c × ( m p m e + 4 3 ) − 1 × 9 4 α (56)
Therefore,
m p c 2 × e c × 9 4 α = π α 9 × λ e c × λ p c × ( m p m e + 4 3 ) − 3 (57)
The right side is
π × k T = π α 9 × λ e c × λ p c × ( m p m e + 4 3 ) − 3 (58)
From Equations (57) and (58), we obtain
m e c 2 × e 4 π ε 0 = π × k T c (59)
We attempt to prove the following Equation (60).
k T c e 2 c 4 π ε 0 = 1 1837.485988 ( s 2 m ) = 1 ( m p m e + 4 3 ) ( s 2 m ) (60)
e 2 c 4 π ε 0 = e 2 c 2 4 π × Z 0 = 1 4 π × { 1 4.5 × λ p c × ( m p m e + 4 3 ) − 1 } 2 × 9 π α × λ e λ p = α 9 λ p λ e c 2 ( m p m e + 4 3 ) − 2 (61)
According to Equation (32),
k T c = α 9 × λ e c × λ p c × ( m p m e + 4 3 ) − 3 (62)
From Equations (61) and (62), we obtain
k T c e 2 c 4 π ε 0 = ( m p m e + 4 3 ) − 1 (63)
We attempt to prove the following Equation (64).
9 × m e × m p × c 2 2 π h = 9 × m e × m p × c 2 2 π ( e × q m ) = 1 ( m p m e + 4 3 ) 2 = 2.96177 E − 07 (64)
From Equations (28)-(31), we obtain
9 × m e × m p × c 2 2 π h = 9 2 π × π 4.5 × λ p c × π 4.5 × λ e c × ( 1 4.5 × λ p c × π × λ e c ) − 1 × ( m p m e + 4 3 ) − 2 (65)
Therefore,
9 × m e × m p × c 2 2 π h = ( m p m e + 4 3 ) − 2 (66)
For convenience, Equation (8) is rewritten as follows.
π ( V ⋅ m ) = ( m p m e + 4 3 ) m e _ n e w c 2 e n e w c ( V ⋅ m ) (67)
According to the theory of special relativity, the value of the electric charge should not be changed.
m e _ n e w c 2 × ( m p m e + 4 3 ) = π × e n e w c (68)
However, according to special relativity, the mass should be increased.
( m p m e + 4 3 ) m e c 2 1 − ( v c ) 2 = e c × π (69)
where v is the velocity. Then, ( m p m e + 4 3 ) has units of ( m 2 s ) . Therefore,
{ ( m p m e + 4 3 ) × 1 − ( v c ) 2 } × m e c 2 1 − ( v c ) 2 = e c × π = unchanged (70)
Hence, when ( m p m e + 4 3 ) is unchanged, compatibility with special relativity is maintained.
Our first purpose is to simplify our equations, and we have attempted to explain them using thermodynamic principles discovered in the area of solid-state ionics. Unfortunately, the background theory could not be completed. Furthermore, our discussions have become complex. Therefore, the purpose of the report is to simplify these equations. In this report, we attempt to simplify our empirical equations by proposing the following six equations.
m e _ n e w c 2 × ( m p m e + 4 3 ) 2 = π 4.5 × λ p c = 2.7656397 E − 07 = constant (71)
e n e w c × ( m p m e + 4 3 ) = 1 4.5 × λ p c = 8.80330473 E − 08 = constant (72)
m p _ n e w c 2 × ( m p m e + 4 3 ) 2 = π 4.5 × λ e c = 5.07814 E − 04 = constant (73)
q m _ n e w c × ( m p m e + 4 3 ) = π × λ e c = 2.28516154 E − 03 = constant (74)
k T c _ n e w × 2 π α × ( m p m e + 4 3 ) 3 = π 4.5 × λ p c × λ e c = 2.011697 E − 10 = constant (75)
G N _ n e w × ( m p m e + 4 3 ) = ( λ p c ) 2 × c × 9 α 8 π = 1.22943389 E − 07 = constant (76)
Regarding Equation (76), to solve the associated problem, the following expression is also proposed.
G N 1 k g = G N _ n e w 1 k g n e w × 4.48852 4.5 ( m 3 s − 2 k g ) = 6.6737778667 E − 11 ( m 3 s − 2 k g ) (77)
Using these six equations, we have proven our three main equations and two other important equations. Furthermore, the compatibility with the theory of special relativity is discussed.
The author declares no conflicts of interest regarding the publication of this paper.
Miyashita, T. (2023) Simplification of Various Empirical Equations for the Electromagnetic Force in Terms of the Cosmic Microwave Background Temperature. Journal of Modern Physics, 14, 1217-1227. https://doi.org/10.4236/jmp.2023.148068