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Now that we know of what constitutes c, k, and h, we can study Max Planck's equation for cavity radiancy, which calculates the amount of energy, Eλ, of a particular wavelength, λ, that is radiated from a cavity at a particular temperature, K. Planck's first version of the equation, a modification of Wilhelm Wien's original one, is, as follows:
Eλ = c1 λ-5 [exp(c2 K-1 λ-1) - 1]-1 (155
where:
c1 = Wien's first radiation constant,
c2 = Wien's second radiation constant,
K = the temperature of the cavity,
λ = the wavelength of the radiancy, and
exp(x) = e, which is 2.71828 ..., to the power of x where,
x = c2 K-1 λ-1, the exponent of base e.
Originally, in order to make the equation work, the magnitudes of the two radiation constants, c1 and c2, were determined empirically because, up to that time, no one knew their formulas.
The SI values of their magnitudes are:
c1 = 3.7 × 10-16 kg·m4·s-3 or J·s-1·m2 (156
where J = joule, the SI unit of energy;
and
c2 = 1.4 × 10-2 K·m (157
where K = kelvin, the SI unit of temperature.
Eventually, Planck determined that the two constants consist of factorial combinations of π, c, k, and of a hitherto unknown factor, which, amazingly (for that time), occurs in both c1 and c2. It is Planck's constant, h. Planck's findings heralded the discovery of the Quantum World and earned him a Nobel Prize in physics. His equations for these constants are:
c1 = 2 π c2 h (158
and
c2 = h c k-1 (159
Planck's final version of the cavity-radiancy equation is:
Eλ = 2 π c2 h λ-5 [exp(h c k-1 K-1 λ-1) - 1]-1 (160
Because of the presence of many FUPCONs, this equation is difficult to analyze. However, at the time, Planck could not have refined it further--most of the quantum attributes of the electron had yet to be discovered.
Replacing the FUPCONs in the two radiation constants with the quantum attributes of the electron makes the cavity-radiancy equation easier to understand. This replacement process is, as follows:
c1 = 2 π (λe·te-1)2 (me·λe2·te-1) =
(Ee·te-1) (2 π λe2) (161
which we can read as: The rest-mass energy of an electron, Ee, radiates during each time period, te, from each unit area, λe2, of the surface of a hemisphere whose area is (2 π λe2).
c2 = (me·λe2·te-1) (λe·te-1)(me·λe2·te-2·ke-1)-1 = ke·λe (162
which ensures that the factor, (K λ), in the exponent of e is expressed in SE units of measure.
When the cavity-radiancy equation uses unitary electronic attributes rather than FUPCONs, its form is, as follows:
Eλ = (Ee·te-1) (2 π λe2) λ-5 [exp(ke K-1 λe λ-1) - 1]-1 (163
Radiation Constants in SE Units
When using the SE of unit measures, the magnitudes of the two radiation constants possess the following values:
c1 = 2 π (164
and
c2 = 1 (165
Therefore, the constants, c1 and c2, do not exist as such.
When the cavity-radiancy equation uses SE units of measure, its form is:
Eλ = 2 π λ-5 [exp(K-1 λ-1) - 1]-1 (166
where the magnitudes of the temperature, K, and wavelength, λ, variables are expressed in SE units of measure.
To test the new SE cavity-radiancy equation, we solve an example problem, first, by using Planck's equation, which uses the SI units, then by using the SE version. If the two answers are the same, the SE version is valid.
The problem is to calculate the magnitude of the radiancy at wavelength, λ, from a cavity at temperature, K, where the SI and SE values for the magnitudes of λ and K are, respectively, as follows:
λ = 1.5 × 10-6 m = (1.5 × 10-6 m)(2.4 × 10-12 m·λe-1)-1 =
6.2 × 105 λe (167
K = 2000 K (kelvin) = (2000 K)(5.9 × 109 K·ke-1)-1 =
3.4 × 10-7 ke (168
Using Planck's equation with the empirical values for c1 and c2, the problem is formulated, as follows:
Eλ = c1 λ-5 [exp(c2 K-1 λ-1) - 1]-1 (169
Eλ = (3.7 × 10-16 Js-1m2) (1.5 × 10-6 m)-5 ×
{exp[(1.4 × 10-2 Km) (2000 K)-1 ×
(1.5 × 10-6 m)-1] - 1}-1 (170
By reducing the number of factors to one, the solution is:
Eλ = 4.1 × 1011 Js-1m-3 (171
Using the SE equation, the problem is formulated, as follows:
Eλ = 2 π λ-5 [exp(K-1 λ-1) - 1]-1 (172
Eλ = 2 π (6.2 × 105 λe)-5 ×
{exp[(3.4 × 10-7 ke)-1 (6.2 × 105 λe)-1] - 1}-1 (173
By reducing the number of factors to one, the solution is:
Eλ = 5.8 × 10-31 Ee·te-1·λe-3 (174
Comparing the SI and SE Solutions
To compare the SI and SE solutions, we convert the SE solution into SI units of measure, as follows:
Eλ = (5.8 × 10-31 Ee·te-1·λe-3)(8.2 × 10-14 J·Ee-1) ×
(8.1 × 10-21)-1 (2.4 × 10-12)-3 =
4.1 × 1011 J·s-1·m-3 (175
By comparing Equations 171 and 175, we see that the two solutions equal each other; therefore, the SE version of the cavity-radiancy equation is valid.
By juxtaposing the two versions of the cavity-radiancy equation, we see the simplicity and improvement of the SE equation over the one that uses FUPCONs:
Eλ = 2 π c2 h λ-5 [exp(h c k-1 K-1 λ-1) - 1]-1 (176
compared to
Eλ = 2 π λ-5 [exp(K-1 λ-1) - 1]-1 (177
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