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**PLANCK CONSTANT DERIVATION:**

On this web page spheromak theory is used to derive the Planck Constant from first principles. It is shown that the Planck constant is actually a geometrical constant that is a function of an electron charge, the speed of light and the permiability of free space. The geometrical constant arises from co-incidence of an integer ratio with an irrational number at the minimum energy state of a spheromak in free space.

It is shown that most methods of experimentally measuring the Planck Constant generate small errors due to the momentum of emitted or absorbed photons. Although the Planck Constant is normally defined in terms of photon properties the photon quantization is actually due to energy quantization within the electromagnetic spheromaks that absorb or emit the photons.

Another issue in high precision experimental measurement of the Planck Constant is suppression of electric and magnetic fields that can slightly distort the spheromak geometry. These issues make the value of the Planck Constant weakly dependent on its precise definition and on the method used for its experimental measurement.

**RADIATION AND MATTER:**

Atomic charged particles have associated electromagnetic spheromaks. Electromagnetic spheromaks can absorb or emit radiation photons. During radiation absorption and emission total energy is conserved. Every packet of potential energy, including both charged particles and photons, has a characteristic frequency. During photon emission the emitting particle's characteristic frequency decreases and the number of photons present increases. During photon absorption the absorbing particle's characteristic frequency increases and the number of photons present decreases.

The energy versus frequency behavior of charged particles and photons is determined by the Planck constant, which lies at the heart of quantum mechanics. Analysis of the Planck constant provides insight into the mechanism by which nature stores photon energy in rest mass and the reasons for quantum mechanical behavior.

In our local universe there is an overall tendency for high energy photons to gradually change into a larger number of lower energy photons. This tendancy determines the direction of evolution of the universe.

**MAXIMUM STABILITY SPHEROMAK:**

The shape of a spheromak can be characterized by its inner radius Rc and its outer radius Rs. The ratio of Rs to Rc is defined by the spheromak parameter So where:

So^2 = (Rs / Rc).

It is shown herein that the total energy of a spheromak has a relative minimum at:

So^2 ~ 4.104474517.

Over time natural spheromaks will absorb or emit photons until they reach this stable minimum energy condition. Hence particles such as electrons and protons that have existed for a long time all have So values that satisfy:

So^2 ~ 4.104474517.

At this value of So the value of (dEtt / dFh) for an electromagmetic spheromak is given by:

(dEtt / dFh) = h = Planck constant,
where:

Fh = the characteristic frequency of the circulating quantum charge that forms an electromagnetic spheromak.

The characteristic frequency is the frequency at which a circulating quantum charge retraces its previous path.

**PLANCK CONSTANT:**

The Planck Constant relates a photon's energy to its characteristic frequency. The origin of the Planck Constant lies in the manner in which energy is stored in an electromagnetic quantum charge spheromak.

The Planck Constant is a fundamental parameter in quantum mechanics. The Planck Constant is a frequently reoccurring composite of other physical and mathematical constants. This web page shows the origin of the Planck Constant.

A spheromak's electric and magnetic field structure allows individual quantized charges to act as stable stores of potential energy. The behavior of these energy stores is governed by electrodynamics. This web page shows the mathematical relationship between spheromaks and quantum mechanics.

It is shown herein that the total field energy Ett of a quantum charge electromagnetic spheromak at steady state in free space is given by:

Ett = h Fh

where Fh is the characteristic frequency of the spheromak and h is a composite constant known as the Planck constant.

When a charged particle gains or loses potential energy by absorption or emission of a photon the spheromak transitions from state "a" with potential energy Etta and natural frequency Fha to state "b" with potential energy Ettb and natural frequency Fhb. The change in spheromak potential energy is:

(Ettb - Etta) = h (Fhb - Fha)

When a charged particle absorbs a photon with energy Ep it also absorbs that photon's linear momentum. From Einstein's famous special relativistic relationship:

E^2 = P^2 C^2 + Mo^2 C^4

the momentum Pp of a photon with no rest mass but with energy Ep is:

Pp = Ep / C

where:

C = speed of light.

**PHOTON ABSORPTION:**

If a charged particle at rest with initial spheromak potential energy Ea absorbs a photon with energy Ep to conserve momentum the charged particle with combined total energy (Ea + Ep) also acquires the photon momentum Pp. Hence:
(Ea + Ep)^2 = Pp^2 C^2 + Eb^2

or

(Ea + Ep)^2 = Ep^2 + Eb^2

where Eb is the charged particle potential energy after absorption of the photon.

Hence:

(Ea + Ep)^2 = Ep^2 + Eb^2

or

Ea^2 + 2 Ea Ep = Eb^2

or

Eb = [Ea^2 (1 + 2 Ep / Ea)]^0.5

= Ea (1 + 2 Ep / Ea)^0.5

Hence:

**(Eb - Ea)** = Ea (1 + 2 Ep / Ea)^0.5 - Ea

= Ea [(1 + (2 Ep / Ea))^0.5 - 1]

~ Ea [1 + (Ep / Ea) - [(2 Ep / Ea)^2 / 8] - 1]

= Ep - (Ep^2 / 2 Ea)

= **Ep [1 - (Ep / 2 Ea)]**

Hence for photon absorption:

Ep = (Eb - Ea) / [1 - (Ep / 2 Ea)]

**PHOTON EMISSION:**

If a charged particle at rest with initial potential energy Ea emits a photon with energy Ep to conserve momentum the charged particle with the new total energy (Ea - Ep) acquires the photon momentum Pp. Hence:
(Ea - Ep)^2 = Pp^2 C^2 + Eb^2

or

(Ea - Ep)^2 = Ep^2 + Eb^2

where Eb is the charged particle potential energy after emission of the photon.

Hence:

(Ea - Ep)^2 = Ep^2 + Eb^2

or

Ea^2 - 2 Ea Ep = Eb^2

or

Eb = [Ea^2 (1 - 2 Ep / Ea)]^0.5

= Ea (1 - 2 Ep / Ea)^0.5

Hence:

**(Ea - Eb)** = Ea - Ea (1 - 2 Ep / Ea)^0.5

= Ea [1 - (1 - 2 Ep / Ea)^0.5]

~ Ea [ 1 - (1 - (Ep / Ea) - (2 Ep / Ea)^2 / 8)]

= Ea [ (Ep / Ea) + (2 Ep / Ea)^2 / 8)]

= Ep + (Ep^2 / 2 Ea)

= Ep [1 + (Ep / 2 Ea)]

Hence for photon emission:

**Ep = (Ea - Eb) / [1 + (Ep / 2 Ea)]**

**THERMAL MOTION:**

Note that if the charged particles are in thermal motion there is broadening of the emission and absorption frequency bands which further complicates precision measurements.

**EXPERIMENTAL MEASUREMENT OF h:**

Ep = h Fp

or

h = Ep / Fp

If the experimental methodology involves measurement of the frequency of photons emitted by charged particles at rest the formula that should be used for determining h is:

**h = (Ea - Eb) / {Fp [1 + (Ep / 2 Ea)]}**

Note that on emission of a photon the change in charged particle potential energy (Ea -Eb) is slightly greater than the photon energy Ep and on photon absorption by a charged particle at rest the change in charged particle potential energy is slightly less than the photon energy Ep. These issues are further complicated by thermal motion of the charged particles.

**EXPERIMENTAL ERROR:**

Some high resolution experimental measurements of h rely on spectroscopic measurement of the frequency of photons emitted by excited electrons. In such experiments lab personnel often incorrectly assume that the term:

[1 +/- (Ep / 2 Ea)] = 1

However, at resolutions in measurement of h with 6 or more significant figures that assumption is wrong and the claimed experimentally measured values of h consistently deviate from the precise theoretically calculated value of Fp = (Eb - Ea) / h. Hence, in high resolution measurements of h it is necessary to account for the charged particle recoil kinetic energy on absorption or emission of a photon.

Using spheromak theory, subject to the accuracy of the spacial energy density assumptions, we can precisely calculate a theoretical values for:

h = (Eb - Ea) / (Fhb - Fha)

= dEtt / dFh

Note that the spheromak spacial energy density assumptions are only truly valid in field free space, which is often not the case during a practical measurement of the Planck Constant. While the particle spheromak magnetic fields are large compared to an applied laboratory magnetic field, the system is not totally distortion free.

**EXPERIMENTAL VALUE FOR PLANCK CONSTANT:**

The published experimentally measured value for the Planck Constant h is:

h = 6.62606957 × 10-34 m^2 kg / s.

The accuracy of measurement of a proton charge has been limited by the experimental error in determination of h. Hence it is highly desirable to find a precise value for h.

**ORIGIN OF PLANCK CONSTANT:**

The parameter h is a function of:

Mu = permiability of free space;

C = speed of light in a vacuum;

Q = proton charge;

Pi = (circumference / diameter) of a circle

= 3.141592653589793

Pi^2 = 9.869604401

To understand the relationship of spheromak parameters to the Planck constant it is necessary to derive a closed form expression for the total field energy of a spheromak and hence an elementary atomic charged particle.

**PARAMETER DEFINITIONS:**

Define for a spheromak in free space:

**Rc** = minimum radius of inner spheromak wall;

**Rs** = maximum radius of outer spheromak wall;

**(K Ro) = (Rs Rc)^0.5** = radius where the potential energy well is deepest;

**So = [Rs / Ro] = [Ro / Rc] ** = spheromak shape parameter;

**So^2 = (Rs / Rc)**;

**Hs** = distance of spheromak wall from the equatoral plane;

**Hf** = maximum value of **|Hs|**

**2 Hf** = spheromak overall length;

**Lh** = charge hose length;

**Np** = number of poloidal charge motion path turns contained in Lh;

**Nt** = number of toroidal charge motion path turns contained in Lh;

**Nr = Np / Nt**;

**Rf** = spheromak wall radius at **H = Hf** and **H = -Hf**;

**Lp = Pi (Rs + Rc)** = wall tangential poloidal turn length;

**Lt = Pi (Rs - Rc)** = wall tangential toroidal turn length;

**Bpo** = poloidal magnetic field strength at the center of the spheromak;

**Upo = (Bpo^2 / 2 Mu)** = maximum field energy density at the center of the spheromak;

**Upo = (Uo)** = maximum field energy density at the center of the spheromak;

As shown on the web page titled ELECTROMAGNETIC SPHEROMAK the peak magnetic field strength Bpo at the center of a spheromak can be expressed as:

**Bpo = [(Mu C Qa) / (4 Pi Ro^2)]**

As shown on the web page titled ELECTROMAGNETIC SPHEROMAK the peak magnetic field strength Bpo at the center of a spheromak can also be expressed as:

**Bpo** = I [(Mu Qs C) / (2 Pi^2 Rc^2)] {Nr / {[Nr (So^2 + 1)]^2 + [So^2 - 1]^2}^0.5}

= I (Mu Qs C) / (2 Pi^2 Ro^2)(Ro / Rc)^2 {Nr / {[Nr (So^2 + 1)]^2 + [So^2 - 1]^2}^0.5}

= **I (Mu Qs C / (2 Pi^2 Ro^2) So^2 {Nr / {[Nr (So^2 + 1)]^2 + [So^2 - 1]^2}^0.5} **

where:

**I = Integral from Z = 1 to Z = So^2 of:
Z^3 dZ Nr / ([(So^2 - Z)(Z - 1)]^0.5 {[Nr Z]^2 + [(So^2 - 1) / 2]^2}^0.5 [So^2 Z - So^2 + Z]^1.5)**

where:

and

where:

Np = integer number of poloidal magnetic field generation turns

and

Nt = integer number of toroidal magnetic field generation turns.

In order to determine the spheromak operating point for each value of So^2 find the corresponding value of Nr^2 using the common boundary condition formula:

**Nr^2 = {(8 / Pi^2) - [(So^2 - 1) / (So^2 + 1)]^2} / {1 - (16 / [Pi (So^2 - 1)]^2)}**

which formula is derived on the web page titled: ELECTROMAGNETIC SPHEROMAK and then do a numerical integration to determine I.

**RELATIONSHIPS BETWEEN Efs, Ett and Fh:**

An upper limit on the possible spheromak field energy is Efs. As shown on the web page titled: SPHEROMAK ENERGY this upper limit is given by:

**Efs** = **(Bpo^2 / 2 Mu) (Ro)^3 Pi^2**

As shown on the web page titled: SPHEROMAK ENERGY the actual field energy trapped by a spheromak in free space at steady state is given by:

**Ett = Efs {1 - [(So -1)^2 / (So^2 + 1)]^2}**

where:

**So ~ 2.026**

Thus:

**(Ett / Efs) = {1 - [(So -1)^2 / (So^2 + 1)]^2}**

The characteristic frequency Fh of an atomic particle spheromak is given by:

**Fh** = (C / Lh)

= C / {[2 Pi Np (Rs + Rc) / 2]^2 + [2 Pi Nt (Rs - Rc) / 2]^2}^0.5

= C / [Pi {[Np (Rs + Rc)]^2 + [Nt (Rs - Rc)]^2}^0.5]

= C / [Pi Nt {[Nr (Rs + Rc)]^2 + [(Rs - Rc)]^2}^0.5]

= C / [Pi Nt Rc {[Nr (So^2 + 1)]^2 + [(So^2 - 1)]^2}^0.5]

= C Ro / [Pi Nt Rc Ro {[Nr (So^2 + 1)]^2 + [(So^2 - 1)]^2}^0.5]

= [C / (Pi Nt Ro)] [So / {[Nr^2 (So^2 + 1)^2] + [(So^2 - 1)]^2}^0.5]

Hence:

[1 / Ro] = Fh Pi Nt {[Nr^2 (So^2 + 1)^2] + [(So^2 - 1)]^2}^0.5 / [C So]

Thus:

**Efs = (Bpo^2 / 2 Mu) (Ro)^3 Pi^2**

= (Bpo^2 / 2 Mu) (Ro)^4 Pi^2 [1 / Ro]

= (Bpo^2 / 2 Mu) (Ro)^4 Pi^2 Fh Pi Nt {[Nr^2 (So^2 + 1)^2] + [(So^2 - 1)]^2}^0.5 / [C So]

As shown on the web page ELECTROMAGNETIC SPHEROMAK from far field spheromak energy density matching:

**Bpo^2** = **[Mu^2 C^2][Qa / (4 Pi Ro^2)]^2**

Thus:

**Efs** = (Bpo^2 / 2 Mu) (Ro)^4 Pi^2 Fh Pi Nt {[Nr^2 (So^2 + 1)^2] + [(So^2 - 1)]^2}^0.5 / [C So]

= ([Mu^2 C^2][Qa / (4 Pi Ro^2)]^2 / 2 Mu) (Ro)^4 Pi^2 Fh Pi Nt {[Nr^2 (So^2 + 1)^2] + [(So^2 - 1)]^2}^0.5 / [C So]

= Fh ([Mu C Qa^2] / 32) Pi Nt {[Nr^2 (So^2 + 1)^2] + [(So^2 - 1)]^2}^0.5 / [So]

= Fh ([Mu C Qa^2] / 4 Pi) [Pi^2 / 8] Nt {[Nr^2 (So^2 + 1)^2] + [(So^2 - 1)]^2}^0.5 / [So]

The spheromak field energy Ett is given by:

**Ett** = (Ett / Efs) Efs

= (Ett / Efs) Fh ([Mu C Qa^2] / 4 Pi) [Pi^2 / 8] Nt {[Nr^2 (So^2 + 1)^2] + [(So^2 - 1)]^2}^0.5 / [So]

= **h Fh**

where the Planck constant h is given by:

**h** = ([Ett / Efs] [Mu C Qa^2] / 4 Pi) [Pi^2 / 8] Nt {[Nr^2 (So^2 + 1)^2] + [(So^2 - 1)]^2}^0.5 / [So]

= **({1 - [(So -1)^2 / (So^2 + 1)]^2} [(Mu C Qa^2) / (4 Pi)] [Pi^2 / 8] Nt {[Nr^2 (So^2 + 1)^2] + [(So^2 - 1)]^2}^0.5 / [So]**

In this formula at steady state So spontaneously adopts the value that minimizes Ett. As shown by the following graph of the So dependent term of Ett vs So the operating value of So is:

**So ~ 2.026**

Note that in plotting this graph Nr^2 is itself a complex function of So.

In the expression for the Planck constant:

**So ~ 2.026**

**Pi = 3.141592653589793**

and

**Pi^2 = 9.869604401**

As shown on the web page titled: SPHEROMAK ENERGY at the spheromak minimum energy operating point the approximate values are:

**{1 - [(So -1)^2 / (So^2 + 1)]^2} {[Nr^2 (So^2 + 1)^2] + [(So^2 - 1)]^2}^0.5 / [So] ~ [2.2882]**

**So ~ 2.026**

**So^2 ~ 4.104676**

**Nr^2 ~ 0.5297583716**

**Nr ~ 0.7278450189**

**Ett = [Mu C Qa^2 / 4 Pi] [Pi^2 / 8] Fh Nt [2.2882]**

**Ett = h Fh**

The published experimentally measured value of h is:

h = 6.62606957 × 10-34 m^2 kg / s

= 6.62606957 × 10-34 J-s

C = 2.99792458 X 10^8 m / s

Mu = 4 Pi X 10^-7 T^2 m^3 / J

[Pi^2 / 8] = 1.23370055

Hence:

**Nt** = h / {[Mu C Qa^2 / 4 Pi] [Pi^2 / 8] [2.2882]

= 6.62606957 × 10-34 J-s / {10^-7 X 2.99792458 X 10^8 m / s X (1.60217662 X 10^-19 coulombs)^2 x 1.23370055 X 2.2882}

= 6.62606957 × 10-34 J-s / {(7.695582224 X 10^-37) X 1.23370055 X 2.2882}

= **305.0076759 turns**

Then:

Np = Nr Nt

= 0.0.7278450189 (305.0076759)

= **221.9983176 turns**

Since Np and Nt must be exact integers with no common factors the only possible exact solution for Np, Nt is:

Nt = 305

Np = 222

= 222 / 305

=

**EXACT VALUES:**

Since Np and Nt are integers we know that the exact value of Nr is given by:

**Nr** = (Np / Nt)

= 222 / 305

= **0.7278688525**

Hence the exact value of Nr^2 is given by:

**Nr^2** = (0.7278688525)^2

= **0.5297930664**

Since we now have an accurate value for Nr^2 we can accurately determine So^2 using the common boundary condition. With an accurate value of So^2 we can calculate an accurate value of h in terms of Mu, C and Q.

Recall that from the common boundary condition:

Nr^2 = ([8 - {[Pi]^2 [(So^2 - 1) / (So^2 + 1)]^2}] / {[Pi]^2 - [16 / (So^2 - 1)^2]})

This boundary condition is only precisely valid in field free space.

Using the precise value of Nr^2 we can find the corresponding precise value of So^2. In order to do so use the approximate value of So^2 in combination with the slope of the Nr^2 versus So^2 curve at the approximate solution to find the exact solution.

Let X = So^2 - 1

Then:

Nr^2 = {8 - [Pi]^2 [X / (X + 2)]^2} / {[Pi]^2 - [16 / (X^2)]}

Recall that:

Pi^2 = 9.869604401

**Try So = 2.0260000**

Then:

X = So^2 - 1

= 3.104676

and

X^2 = 9.639013065

giving a trial value of Nr^2 as:

Trial Nr^2 = {8 - [Pi^2] [X / (X + 2)]^2} / {[Pi]^2 - [16 / (X^2)]}

= {8 - [9.869604401] [3.104676 / (5.104676)]^2} / {[9.869604401] - [16 / (9.639013065)]}

= 4.34913352 / {8.209683422}

= 0.5297565444

Recall that the target value of Nr^2 is:

(222/ 305)^2 = 0.5297930664

Thus for a trial value of So = 2.02600000 the target value of Nr^2 exceeds the trial value of Nr^2 by: 0.5297930664 - 0.5297565444 = 3.652198 X 10^-5

We need to make the trial value of Nr^2 slightly larger which implies making the trial value of So slightly smaller.

**Try So = 2.02590000**

Then:

X = So^2 - 1

= 3.10427081

and

X^2 = 9.636497262

giving a trial value of Nr^2 as:

Trial Nr^2 = {8 - [Pi^2] [X / (X + 2)]^2} / {[Pi]^2 - [16 / (X^2)]}

= {8 - [9.869604401] [3.10427081 / (5.10427081)]^2} / {[9.869604401] - [16 / ( 9.636497262)]}

= 4.349506903 / 8.209250066

= 0.5298299928

Recall that the target value of Nr^2 is:

(222 / 305)^2 = 0.5297930664

Thus for a trial value of So = 2.02590000 the trial value of Nr^2 exceeds the target value of Nr^2 by:
0.5298299928 - 0.5297930664 = 3.692635 X 10^-5

**FINDING EXACT VALUES:**

Thus:

d(Trial value of Nr^2) / dSo = (0.5298299928 - 0.5297565444) / (2.02590000 - 2.0260000)

= 7.34484 X 10^-5 / (- 10^-4)

= **- 0.734484**

Hence:
[So|target] = [So|trial] + [dSo / dNr^2] {[Nr^2|target] - [Nr^2|trial]}

= 2.02600000 + [- 1 / 0.734484] [0.5297930664 - 0.5297565444]

= 2.02600000 - [4.972470469 X 10^-5]

= 2.02600000 - .00004972470469

= **2.025950275**

The corresponding value of X is:

X = So^2 - 1

= 3.104474517

X^2 = 9.637762026

We can check this calculation by recalculating Nr^2.

Nr^2 = {8 - [Pi^2] [X / (X + 2)]^2} / {[Pi^2] - [16 / (X^2)]}

= {8 - [9.869604401] [3.104474517 / (5.104474517)]^2} / {[9.869604401] - [16 / (9.637762026)]}

= 4.349319182 / 8.209467955

= 0.529793064

Hence:

Nr = 0.7278688508
as compared to target value of:

(222 / 305) = 0.7278688525

Thus the calculated precise So value gives a precise Nr value that is accurate to eight significant figures.

Thus at the spheromak operating point So is:

**So = 2.025950275**

and

So^2 = 4.104474517

Recall that the theoretical value of the Planck constant h is given by:

**h = [(Mu C Qs^2) / (4 Pi)] [Pi^2 / 8] Nt
{1 - [(So -1)^2 / (So^2 + 1)]^2}
{[Nr^2 (So^2 + 1)^2] + [(So^2 - 1)]^2}^0.5 / [So]**

Evaluation of the terms of h gives:

[(Mu C Qs^2) / (4 Pi)] [Pi^2 / 8] Nt

= {(7.695582224 X 10^-37) X 1.23370055 X 305 J-s

= **2.895683427 X 10^-34 J-s**

In evaluating this term it is important to be aware that the published value of the proton charge Qs may not be accurate due to being derived from a value for h which is inaccurate.

{1 - [(So -1)^2 / (So^2 + 1)]^2}

= {1 - [(1.025950275)^2 / (5.104474517)]^2}

= {1 - 0.0425209706}

= **0.9574790294**

{[Nr^2 (So^2 + 1)^2] + [(So^2 - 1)]^2}^0.5 / [So]

= {[0.5297930664 (5.104474517)^2] + [(3.104474517)]^2}^0.5 / [2.025950275]

= {[13.80410806] + [9.637762027]}^0.5 / [2.025950275]

= **2.389831854**

Hence the theoretical value of h is:

**h** = [2.895683427 X 10^-34 J-s] [0.9574790294] [2.389831854]

= **6.625943023 X 10^-34 J-s**

By comparison the published experimental value for h is:

h = 6.62606957 × 10-34 m^2 kg / s.

These two values for h are in agreement to an accuracy of 2 parts in 100,000. The discrepency may be due to error in Qs or due to conservation of linear momentum on the emission of a photon by an excited charged particle or may be due to spheromak distortion in a high magnetic field. It is necessary to examine exactly how h is experimentally measured to understand these error sources. The fractional discrepency is:

[(6.62606957 - 6.625943023) / 6.62606957] = **1.9098 X 10^-5**

The error will depend on the energy Ep of the photons used and the particle mass which is often an electron.

Consider an electron with rest mass Me. The rest potential energy of the electron is:

Etta = Me C^2.

The ionization energy of a gas is typically of the order of:

Ep = 20 eV.

For electrons:
Etta = Me C^2
= 9.1 X 10^-31 kg X (3 X 10^8 m / s)^2 X 1 eV / 1.602 X 10^-19 J

= 51.12 X 10^4 eV

Hence if h is measured via photon emission from an ionized gas and if the term:

[1 +(Ep / 2 Etta)]

is assumed to be unity we can expect an error in the experimentally determined value of h of about:

(Ep / 2 Er) = 20 eV / [2 (51.12 X 10^4 eV)]

= 2 / (102.24 X 10^3)

~ **1.95 X 10^-5**

Thus the small discrepency between the theoretical and experimental values of h may be caused by failure to properly take into account recoil kinetic energy when an electron emits a photon. When an electron emits a photon about 2 / 100,000 of the change in potential energy becomes electron linear kinetic energy rather than photon energy. If this error is uncorrected the experimentally observed value of h will be slightly larger than the theoretical value calculated herein. Note that the h value calculated herein is actually the change in charged particle potential energy with respect to a change in charged particle natural frequency. The energy carried away by the photon will be slightly less than the decrease in charged particle potential energy due to the small increase in charged particle kinetic energy on emission of a photon. This increase in charged particle kinetic energy must occur to satisfy the law of conservation of linear momentum.

For precise measurement of h it is important that the participating atoms be cold to minimize error due to thermal molecular motion.

Note that an experimental measurement of h may be more precise if the charged particle emitting or absorbing the photon is a proton instead of an electron. Then the recoil momentum results in less recoil energy. This issue improves the resolution of medical Magnetic Resonance Imaging (MRI) equipment that relies on photon absorption and emission by protons in water. However, in that application there is still some disturbance by the magnetic fields caused by the electrons in the hydrogen.

**CONCLUSION:**

The spheromak model of a charged particle provides a means of accurately calculating the Planck constant h in terms of Pi, Mu, Q and C. In highly accurate experimental measurements of h it is necessary to take into account the charged particle recoil kinetic energy. When recoil kinetic energy is properly taken into account there is agreement between the theoretical and experimental determinations of the Planck constant h to at least 7 significant figures.

Note that for a spheromak at steady state conditions the Planck constant is independent of the charged particle spheromak nominal radius Ro and hence is also independent of the charged particle total field energy Ett.

This web page last updated February 19, 2018.

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