12 The Photo-Dynamic Vector

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1 1 The Photo-Dynamic Vector Photo-dynamic force may be represented as a 3D vector. The components of force are; inertial force, electric force, and magnetic force. The binding energy of the Hydrogen electron is well documented. This energy is associated with a special state of photo-dynamic force. Another special state of photodynamic force will give emissive force, which is associated with emissive power and the Stephan- Boltzmann constant. Reciprocal forces will give the Faraday law. The sum of force components will give the Ampere law. The sum of vector components will also give the Klein-Gordon equation. The Quantization rule; The quantization rule states that only an integer number (n) of wavelength (λ) may be imposed upon the circumference (πr) of a circular orbit, partial wavelengths are not permitted. The quantization rule is; nλ = πr Where; n is the principle quantum number (integer) λ is wavelength r is the radius of a circular orbit Angular Momentum; Angular momentum (L m ) is; De Broglie angular momentum (h) is; Electric angular momentum is; L m = mvr h = mv m λ m h = qv q λ q Uncertainty; Angular momentum be written as; Force may be written as; L = ½Nђ = xp = Et F = Nђ/xt = p/t = E/x Where; ђ is the reduced Plank constant (ђ = h/π) N is the uncertainty ratio; N = L/ђ = 4πL/h (N > 1) x is position and p is momentum E is energy and t is time

2 The Force Vector; Photo-dynamic force may be represented as a 3D vector (F); F = F 1 i + F j + F 3 k Where; i, j, k are unit vectors of force (directional vectors in 3D) F 1, F, F 3 are components of force The force vector has magnitude; F = F 4 The components are related to the magnitude; F 1 + F + F 3 = F 4 Sub-components of force (F 5, F 6, F 7, F 8, F 9 ) are defined as; F 5 = F 8 + F 9 The scalar force matrix (F M ) is; F M = F 1 F F 3 F 4 F 5 F 6 F 7 F 8 F 9 Component Geometry; F 5 = F 1 + F = F 4 - F 3 F 6 = F + F 3 = F 4 - F 1 F 7 = F - F 9 = F 1 - F 8 F 8 = F 1 F 7 F 9 = F F 7 The force vector (F) has scalar parts (F n ) arranged as geometry; F 1 = F 5 Cos(A 1 ) and F = F 5 Sin(A 1 ) F 8 = F 1 Cos(A 1 ) and F 7 = F 1 Sin(A 1 ) F 7 = F Cos(A 1 ) and F 9 = F Sin(A 1 ) F 5 = F 4 Cos(A ) and F 3 = F 4 Sin(A ) F = F 6 Cos(A 3 ) and F 3 = F 6 Sin(A 3 ) F 1 = F 4 Cos(A 4 ) and F 6 = F 4 Sin(A 4 ) October 9, 018 Page

3 Force Properties; The scalars of force are associated with natural properties; F 1 is inertial force (also dynamic or thermal force) F is electric force F 3 is magnetic force F 4 is photo-dynamic force F 5 is electro-dynamic force F 6 is electro-magnetic force F 7 is radiant force (also photonic or wave force) (also a photonic force) F 8 is thermo-emissive force F 9 is electro-emissive force States of Force; Angular conditions represent a state. If a state is defined, then the force vector may represent unique states of force. Two special states are; the binding state (for hydrogen) and the emissive state. The Binding State; It is reasonable to assume that reciprocal forces (F 4, F 10 ) are required to balance the electron in a steady state. The Faraday law may be written as; F 10 + F 4 = 0 (reciprocal forces) Where; F 10 is the magnitude of binding force F 4 is the magnitude of photo-dynamic force Geometry gives; Cos(A 1 ) = F 1 /F 5 = F 7 /F F 1 F = F 5 F 7 F 1 F = F 5 F 7 F 1 F = F 4 F 7 Cos(A )/Cos(A 1 ) The binding state is; A 1 = A Giving; F 1 F = F 4 F 7 October 9, 018 Page 3

4 The binding state forces are defined as; Inertial force; F 1 = ½mc /r Electric force; F = Zq /4πε 0 r Photonic force; F 7 = hf/r Where; f is frequency (fλ = c) Substituting force definitions; (½mc /r)(zq /4πε 0 r ) = F 4 (hf/r) (½mc /r)(z q 4 /16π ε 0 r 4 ) = F 4 (h c /λ r ) The quantization rule gives; (½mc /r)(z q 4 /16π ε 0 r 4 ) = F 4 (h c n /4π r 4 ) (½m/r)(Z q 4 /4ε 0 ) = F 4 (h n ) mz q 4 /8ε 0 r = F 4 (h n ) Giving photo-dynamic force; mz q 4 /8ε 0 h n r = F 4 The binding force is; -mz q 4 /8ε 0 h n r = F 10 Binding energy (E 10 ) is; -mz q 4 /8ε 0 h n = E 10 (Bohr energy) The Radiant State; Four conditions are required to define the radiant state; Condition 1; Sin(A 1 ) = ¼ giving; F 5 = 4F Condition A 1 = A 3 giving; Cos(A 1 ) = Cos(A 3 ) and; F 7 /F = F /F 6 Condition 3 Dynamic energy (E 1 ) is thermal energy; E 1 = ½k B T Condition 4 Wave energy (E 6 ) is quantized; E 6 = ђv/nλ (v is velocity) Vector components give; F 5 = F 1 + F Condition 1 gives; 16F = F 1 + F 15F = F 1 Condition gives; 15F 7 F 6 = F 1 The radiant state forces (F 1, F 6 ) are defined as; F 1 = E 1 /ђc = (½k B T) /ђc (condition 3) F 6 = E 6 /ђv = (ђv/nλ) /ђv = ђv/n λ (condition 4) October 9, 018 Page 4

5 Substitute forces (15F 7 F 6 = F 1 ); 15F 7 (ђv/n λ ) = (½k B T) 4 /ђ c 15(vF 7 )(ђ/n λ ) = (½k B T) 4 /ђ c Power (P 7 ) is; P 7 = vf 7 15P 7 (ђ/n λ ) = (½k B T) 4 /ђ c The quantization rule gives; 15P 7 (ђ/4π r ) = (½k B T) 4 /ђ c 15(P 7 /r )(ђ/4π ) = k 4 B T 4 /16ђ c Brightness (B 7 ) is; B 7 = P 7 /r = power/area 15B 7 (ђ/4π ) = k 4 B T 4 /16ђ c 15B 7 = π k 4 B T 4 /4ђ 3 c 15B 7 = π 5 k 4 B T 4 /h 3 c B 7 = (π 5 k 4 B /15h 3 c )T 4 = σt 4 The Stefan-Boltzmann constant (σ) is; σ = π 5 k 4 B /15h 3 c The Ampere Law; The sum of force components gives the Ampere law. From vector components; F 9 + F 8 = F 5 Vector geometry gives; F 9 + F 7 /Tan(A 1 ) = F 5 Definitions are; F 9 = k e Zq /r F 7 = k e Zqꝺq/4r F 5 = k e Zqꝺq/rꝺr ꝺt = ttan(a 1 ) The sum of components may be written as; k e Zq /r k e Zq /tr + (k e Zqꝺq/4r )(t/ꝺt) = k e Zqꝺq/rꝺr + k e Zqꝺq/4ꝺtr = k e Zqꝺq/trꝺr q/r t + ꝺq/4r ꝺt = ꝺq/rꝺrt October 9, 018 Page 5

6 q/πr t + ꝺq/4πr ꝺt = ꝺq/πrꝺrt (q/t)(1/πr ) + (ꝺ/ꝺt)(q/4πr ) = (ꝺ/ꝺr)(q/t)(1/πr) Electro-magnetic characteristics are; I = q/t J = I/πr D = q/4πr H = I/πr Giving; J + ꝺd/ꝺt = ꝺh/ꝺr Transformation rules are; ꝺ/ꝺt transforms to a time operator (ꝺ/ꝺt f) (frequency) ꝺ/ꝺr transforms to a spatial operator (ꝺ/ꝺr delx) Transformations give; Giving the Ampere law; J + fd = delxh J + ꝺd/ꝺt = delxh The Faraday Law; Reciprocal force components give the Faraday law. From vector components; F 10 + F 4 = 0 Definitions are; F 10 = k e Zqꝺq/xꝺλ F 4 = ½k e Zqꝺq/rꝺr ꝺλ = cꝺt x = ct The sum of components may be written as; k e Zqꝺq/xꝺλ + ½k e Zqꝺq/rꝺr = 0 k e Zqꝺq/c Tꝺt + ½k e Zqꝺq/rꝺr = 0 ꝺq/c Tꝺt + ½ꝺq/rꝺr = 0 October 9, 018 Page 6

7 (ꝺq/c Tꝺt)(1/πε 0 r) + (½ꝺq/rꝺr)(1/πε 0 r) = 0 ꝺq/πε 0 rc Tꝺt + ꝺq/4πε 0 r ꝺr = 0 ꝺqμ 0 /πrtꝺt + ꝺq/4πε 0 r ꝺr = 0 (where; μ 0 ε 0 = 1/c ) (ꝺ/ꝺt)μ 0 (q/πrt) + (ꝺ/ꝺr)(q/4πε 0 r ) = 0 (ꝺ/ꝺt)μ 0 (I/πr) + (ꝺ/ꝺr)e = 0 (where; E = E ) (ꝺ/ꝺt)μ 0 H + (ꝺ/ꝺr)e = 0 (ꝺ/ꝺt)β + (ꝺ/ꝺr)e = 0 ꝺβ/ꝺt + ꝺe/ꝺr = 0 Transforming to the Faraday law; ꝺβ/ꝺt + delxe = 0 (where; ꝺe/ꝺr delxe) The Klein-Gordon Equation; The sum of force components will transform to the Klein-Gordon equation of a wave function (ψ). Definitions are; F 1 = ђf/r F = (ђf/ψ)(v/c) F 7 = ђf/λ F 8 = (mc ) /ђc F 5 = ђf/ψ = ђc/ψλ = ђc(ꝺ /ꝺr ) Giving; 1 = ꝺ (ψλ)/ꝺr (spatial operator) ꝺψꝺλ rꝺr Cos(A 1 ) = F 1 /F 5 = ψ/r Sin(A 1 ) = F /F 5 = v/c The Lorentz factor (γ) is; γ = 1/Cos(A 1 ) = F 5 /F 1 = r/ψ γ = 1/[1 Sin (A 1 )] ½ = 1/[1 v /c ] ½ From vector components; F 9 + F 8 = F 5 Vector geometry gives; F 7 Tan(A 1 ) + F 8 = F 5 F 7 Sin(A 1 )/Cos(A 1 ) + F 8 = F 5 October 9, 018 Page 7

8 (ђf/λ)(v/c)(r/ψ) + (mc ) /ђc = ђc(ꝺ /ꝺr ) (ђf/λ)(v/c)r + ψ(m c 4 )/ђc = ђc(ꝺ ψ/ꝺr ) (1/c )(f/λ)vr + ψ(m c )/ђ = (ꝺ ψ/ꝺr ) (1/c )(r/λ)fv + ψ(m c )/ђ = (ꝺ ψ/ꝺr ) Assume; f = ꝺ/ꝺt (time operator, frequency) v = ꝺw/ꝺt (velocity) giving; (1/c )(r/λ)(ꝺ/ꝺt)(ꝺw/ꝺt) + ψ(m c )/ђ = (ꝺ ψ/ꝺr ) Assume; rw = λψ Giving; (1/c )(ꝺ/ꝺt)(ꝺψ/ꝺt) + ψ(m c )/ђ = (ꝺ ψ/ꝺr ) Transformation rules; (ꝺ/ꝺt) ꝺ /ꝺt Giving the Klein-Gordon equation; Conclusion; ꝺ/ꝺr del (1/c )ꝺ ψ/ꝺt + ψ(m c )/ђ = del ψ Components of a force vector may be associated with; magnetism, electric charge, and inertia. If the scalar forces are suitably defined and a state is represented by an angular relationship, then Bohr energy or radiant energy may be easily obtained. Forces may also transform to the Faraday and Ampere laws and the Klein-Gordon equation. Nomenclature; m is the mass of the electron Z is the atomic number (proton number) q 1 is the charge of a proton q is the charge of an electron k e is the electric constant (4πε 0 k e = 1) ε 0 is the electric spatial flow constant (permittivity) μ 0 is the magnetic spatial flow constant (permeability) h is the Plank constant October 9, 018 Page 8

9 ђ is the reduced Plank constant (ђ = h/π) r is distance from nucleus to electron f is frequency October 9, 018 Page 9