The Voltage of the Photon Forrest Bishop, July 25, 2023

The photon is imagined to be a quantized, transverse electromagnetic wave (TEM wave). Proceeding
from that origin, novel and compact expressions for its voltage, current, charge, fields, flux, spin, and
other parameters are developed herein a consilient system with few assumptions. This voltage plays a
crucial role in the quantization criteria for the transverse A, E, and B fields of the photon, the
foundational assumptions of Quantum Electrodynamics, and leads to a further question.

The Voltage of the Photon

A simple question led to this development [Bishop, 2012]. Since the photon has an electric field, E,
measured in volts per meter in the direction of its field lines, what is the voltage of the photon? The
answer turns out to be simple and elegant, with the assumption that the wavelength of the photon is the
length of the photon. How long is a photon? How many times does it wiggle, if it wiggles at all?

From that starting point, many of its expected electromagnetic properties can be written out in closed
form. Its energy is found to be divided evenly between its electric and magnetic fields, as with the
classical Transverse ElectroMagnetic wave (TEM wave). The model is compatible with The Forbidden
Equation, i = qc, [Bishop, 2016], as it has to be to stay within the confines of conventional
electromagnetic theory. A frequency-invariant pair of derived quantum-electromagnetic properties,
which are physical constants, can be associated with its intrinsic spin.

The photon began its conceptual career as a quantized TEM wave, confined with others of its kind in
M. Planck’s cavity oscillator [Planck, 1920a]. In order to derive analytic expressions for its electric and
magnetic fields, another assumption is introduced, an analog of that analytic cavity. From there, the
voltage of the photon is found to be an essential component of the foundational quantization rules.

Derivation of the Electromagnetic Parameters

The below are mostly average values for a photon one wavelength long regardless of the shape of its
envelope. The frame of reference begins from a laboratory rest frame, then changes to a frame in which
the photon is fixed, a frame moving at the speed of light. No regard is given to unobserved, special
relativity effects such as rod-shortening. If the photon shortens up in flight, it would have to shorten to
zero length, and its frequency would go to infinity.

First, a simplification of the constants. Using ϵ0 or μ0 together with c0 in a single equation is

algebraically, as well as geometrically, redundant. Each of these constants is associated with one or two
dimensions in a three-dimensional, (x, y, z), Euclidean space. The speed of light, c0, is in the direction
of propagation of the light: c 0 → c 0 z^ , say, with z^ the unit vector. ϵ0 has two associated, orthogonal
dimensions, one in the direction of propagation, z^ , and the other in the direction of the transverse
electric field, x^ , say. Similarly, μ0 has two associated, orthogonal dimensions, one in the direction of

1
The Voltage of the Photon Forrest Bishop, July 25, 2023

propagation, z^ , and the other in the direction of the transverse magnetic field, y^ , say. So the z^ direction
is being referenced, or ‘overlaid’ three times in a ‘space’ (ϵ0, μ0, c0) that can be described as ((x, z), (y,
z), z), two oriented planes and one oriented line. This adds unnecessary clutter while at the same time
obfuscating the important role of the wave impedance.

With these two equations placed together side by side,

c0 =
1
√ μ0 ϵ 0
and Z0 = √ μ0
ϵ0 (1)

it is a simple matter to cast the electric permittivity, ϵ0, and magnetic permeability, μ0, in terms of the
vacuum wave impedance, Z0, and the vacuum speed of light, c0:

1 Z0
ϵ0 = and μ0 = (2)
Z0 c0 c0

Z0 is associated with the two perpendicular axes transverse to the direction of propagation, the plane of
impedance. The Cartesian space that the set {Z0, c0} refers to can be described as ((x, y), z). For one
example of removing clutter, the fine-structure constant is often written out as α = e 2 / ( 4 π ϵ 0 ℏ c ) ,
which can be condensed to α = Z 0 e 2 / 2 h with this substitution [Bishop, 2007].

Figure 1. The fields of a classical TEM wave in free space

The set of independent physical constants {c0, Z0} is used below to the exclusion of the set {ϵ0, μ0} or
the set {ϵ0, μ0, c0 }. Any occurrences of ϵ0 and μ0 in any imported equations are replaced using
Equations (2). Z0 plays an important role in all of this development.

2
The Voltage of the Photon Forrest Bishop, July 25, 2023

With h as M. Planck’s constant, c0 the speed of light in vacuum, and f the frequency, the energy, EP, of
the quantum-mechanical photon is

E P = hf , or equivalently, E P = hc 0 / λ , with f = c 0 / λ and λ = c 0 /f (3)

For this one-full-wavelength photon, its time-of-passage, Tp, through a fixed, transverse plane, is

T p = 1/f , (4)

which is the minimum amount of time required to recognize or register the frequency. The average
power, PP, of this photon is then

PP = EP /T p → PP = ( hc 0 / λ )∗( c 0 / λ ) → PP = hc 20 / λ 2 , or PP = hf
2
(5)

Equation (5) can only hold if the wavelength of the photon is the length of the photon. For the TEM
wave in free space, or guided by two conductors arranged to have an impedance of Z0,

i =
√ P
Z0
(6)

Therefore the electric current, iP, of this photon is

iP =
√ PP
Z0
(7)

Substituting Equation (5) in for PP

iP = f
√ h
Z0
(8)

and, with V = i/Z, the voltage of this photon is

or
V P =i P Z 0=f
√ h
Z
Z0 0
(9)

V P = f √ h Z0 (10)

Putting in numbers, the root term is 4.99624007*10-16 J-sec/coulomb. At 1 gHz, the voltage is then
5*10-7 V. The voltage of the 121.6nm H first ground transition photon (Lyman 1->2) is 1.23V. These
values are not new; they, or perhaps some arithmetic multiplier of them, are inherent in the

3
The Voltage of the Photon Forrest Bishop, July 25, 2023

conventional description of both the classical light field and of the quantum-electrodynamical photon
field, as shown below.

From Equation (8) the electric charge on one side of the photon is

±QP =
√ h
Z0
and so
2
QP =
h
Z0
(11)

The electric charge is the same for all photons. The value of ±QP is 1.32621132*10-18 coulomb. This is
8.27755100 times larger than the charge of the electron and 1.414… times larger than the Planck
charge, qP. commonly used in theoretical physics. q P = 4 π ϵ 0 ℏ c 0 , is q P = √ 2 QP .

This Equation (11) establishes a link between the electric-charge concept, Planck’s constant, and the
wave impedance of the vacuum. The square of the electric charge of this photon is Planck’s constant
divided by the wave impedance. The plus/minus signifies that this QP is an analog of the charge on the
plates of a capacitor/transmission line. The total electric charge is then QP + -QP = 0. The divergence is
still ∇⋅E = 0 outside of this photon, a condition required in the derivation of Maxwell’s wave
equations. In one view, “electric charge” is a name for the sides of the TEM wave where it stops,
always with a positive side and a negative side as depicted in Figure 1., always moving at the speed of
light according to the governing equation of electric current, i = qc0.

The mysterious and celebrated fine-structure constant, α, can be expressed as the ratio of the squares
of the invariant electronic and photonic charges:

2 2 2
e Z e e
α = ↔ α = 0 ↔ α = , (12)
4 π ϵ 0 ℏ c0 2h 2Q P
2

a very different way of looking at it. Electric charge, Q, is a “square-root of reality” type of physics
notion, as is voltage and many others. Bare Q is has never been measured all by itself, only its product
with another such Q, such as the Q2 in Coulomb’s law. The squares in the last of Equations (12) are
closer to a description of reality than the collection of terms in the conventional expression for α.

The units of electric permittivity, ϵ0, are farad/meter, or capacitance per unit length. The length in
question is in the direction of propagation of the TEM wave in vacuum, so the permittivity can be
thought of as the scale-invariant “capacitance per unit length of space”. The length in question here is
the wavelength, λ, so the total capacitance of the photon is

1
C P = ϵ 0 λ , or, with Equations (2) and (3), C P = . (13)
fZ 0

Crosschecking against the definition of capacitance,

4
The Voltage of the Photon Forrest Bishop, July 25, 2023

√
QP
C =
Q
V
→ CP =
VP
=
h 1
Z0 f √ h Z0
which reduces to C P =
1
fZ 0
. (14)

The units of magnetic permeability, μ0, are henry/meter, or inductance per unit length. The length in
question is in the direction of propagation of the TEM wave in vacuum, so the permeability can be
thought of as the scale-invariant “inductance per unit length of free space”. The length in question here
is the wavelength, λ, so the total inductance of the photon is

( ) f ) , which reduces to L
(
Z0 c 0 Z0
LP = μ 0 λ or, using Equ. (2) and (3), LP = P = . (15)
c0 f

The magnetic flux, φP, can be found from the definition of inductance, with subscript L for “per unit
length”, a geometric form factor in the plane of impedance, GF, of 1 for free space, and multiplying
by the wavelength:

√
φL φL λ Z0 h c0
LL = = GF μ 0 → LP = = μ0 → φ P = μ0 i P λ = f = √h Z 0. (16)
i i c0 Z0 f

For Maxwell et al’s transverse electromagnetic wave (TEM wave), its energy content is split evenly
between its electric field and its magnetic field. This criterion should continue to hold in the instant
case, such that each has, from Equation (3), E = hf, hf/2 worth of the total energy. With WE the energy
of the electric field and WM the energy of the magnetic field,

WE =
1
2
1
2
1
VQ = V P QP . With Equs. (10) and (11), W E = f √ hZ 0
2
h
Z0 √ (17)

which reduces to
1
WE = hf (18)
2

For the magnetic field energy,

1 2 1 2 1 Z0 2 h
WM = Li = LP i P . Substituting in Equations (8) and (15), W M = f (19)
2 2 2 f Z0

which reduces to
1
WM = hf (20)
2

The electric charge, Q, when invoked as an electric current, i, has been shown to only move at the
speed of light according to its defining equation of electric current, The Forbidden Equation, i = qc0

5
The Voltage of the Photon Forrest Bishop, July 25, 2023

with q the charge per unit length. The common definition, i = Q/t, masks this equation. As a crosscheck
on the development so far, with λ the length in question,

√ √
QP c 0 h c0 h
i = qc 0 → iP = . With Equ. (10) and (3), i P = , or i P = f (21)
λ Z0 λ Z0

in agreement with Equation (8). The directions of c0 and λ are the same, so their directions in the ratio
cancel. The speed of light, the magnetic flux, the capacitance, the voltage, the electric current, and of
course any length, are always in reference to one or two directions in space when used in a descriptive
equation, unlike scalars such as temperature. This issue arises again below.

The expression for the voltage of the photon, Equation (10), can be crosschecked against the requisite
Companion Forbidden Equation of voltage using Equation (16) divided by the wavelength,

V = ϕ L co → V p =
√ h Z0 c =
√ h Z0 f λ = f √h Z 0 (22)
λ 0
λ

in agreement with Equation (10).

Intrinsic Spin

The photon has a spin of either zero for linearly polarized light,
or +-h/(2π) for circularly polarized, an invariant property of all
photons regardless of the frequency.

Both Equation (11) for the positive and negative electric charge
QP, and Equation (16) for the total magnetic flux φP, show
invariant properties, which do not depend on the frequency or
wavelength of the photon:

±QP =
√ h
Z0
and φP = √ h Z0 (23)
Figure 2. Circular polarization

Each of these quantities is a physical constant, being composed of other physical constants. The SI unit
of QP is coulomb, and the units of the magnetic flux φP are kg*m*/amp, which can be thought of as
angular momentum per coulomb, or as action per coulomb.

The value of +-QP is 1.32621132*10-18 coulomb ( or amp-second). The value of φP is 4.99624007*10-

16
kg*m2/amp (or weber). Their ratio is the wave impedance:

6
The Voltage of the Photon Forrest Bishop, July 25, 2023

ϕP
QP
= √ h Z0 √ Z0
h
= Z0 . (24)

Their product is Planck’s constant:

QP ϕ P = (√ Zh )( √h Z ) = h .
0
0 (25)

If a circularly-polarized, one-wavelength long photon makes one full rotation about its long axis, the
associated electrical parameters above would also rotate through that angle of 2π. The product of
charge and magnetic flux would rotate one Planck’s constant worth of action per 2π radians, or h/(2π).
The intrinsic spin of the photon can then be expressed as

J z = ±ℏ = ±Q P ϕ P /2 π (26)

Quantization of A, E, and B, the width of the photon

For the above Equations (4) through (26) only one spatial dimension was assumed, the wavelength in
the propagation direction (z axis) of the photon. To find the expressions for the electric and magnetic
fields and fluxes, an additional assumption about the size of its transverse dimensions (x and y axes)
has to be introduced. How wide is a photon?

The cavity oscillator is a cubical box of arbitrary size, lined with perfectly-reflecting mirrors on its
inside faces, and sitting on the laboratory bench. For the instant analytic volume, its length is the
wavelength, λ and it moves with the photon. One choice for the transverse dimensions is the
wavelength, to form a cubical box of volume λ3. With that, the electric field strength, EP, becomes

VP f √ h Z0 √ h Z0 ,
EP = = = c0 (27)
λ λ λ2

and the magnetic flux density, BP, is

EP f √h Z 0
BP = =
λ c0
√ h Z0 = . (28)
c0 λ2

As a crosscheck of Equations (27) and (28), with B = μ 0 H = H Z 0 /c 0 , the magnetic field, HP, is
H P = B P / μ 0 = c B P /Z 0 = E P /Z 0 , and

7
The Voltage of the Photon Forrest Bishop, July 25, 2023

Z0 =
E
H
=
EP
HP
=
EP Z0
c0 B P
=
Z0 f
c0 λ
√ h Z0 ( )( f √λhcZ ) =
0

0
Z0 (29)

The classical Hamiltonian, H, for an analytic volume, a box on the theoretical lab bench filled with
electromagnetic radiation, is

1
H = ϵ 0∭|E ( r , t )|2 +c 02|B ( r ,t )|2 d 3 r (30)
2 Vol

In this single-photon case, with the box attached to and moving with the photon, there are no additional
components, modes, or photons to integrate over. As is done for the quantization of this Hamiltonian in
quantum electrodynamics, substitute the squares of EP and BP into Equation (30). Dropping the
unnecessary notation, and using Equation (2), the volume integral resolves to

1 1
H = λ 3 ( E 2P +c 20 B 2P ) (31)
2 Z0 c 0

As above, Equations (17) and (19), the electric and magnetic energies should each be equal to 1/2hf.
For the electric energy, WE, and using Equations (2) and the square of Equation (27),
2
1 1 1 1 3f 1 1 1
WE = λ 3 E2 = λ 2 h Z0 = λ f2h = h f (32)
2 Z0 c0 2 Z0 c0 λ 2 c0 2

For the magnetic energy, WM, using Equations (2) and the square of (28),

1 1 1 1 hZ 1
WM = λ 3 B2 = λ 3 40 c20 = h f , so (33)
2 Z0 c0 2 Z0 c0 λ 2

1
WE = WM = hf (34)
2

as before. Equations (32) and (33) appear to support the choice of the analytic volume, λ3, and the
expressions for EP and BP. It is consistent with the rest of this model. But it isn’t necessary. The
transverse dimensions needed for this derivation cancel out of the energy equations, so they are still
indeterminate, with one constraint.

Let the transverse dimensions of the analytic box be some unknown X and Y, so the box volume is XYλ.
For free space the geometric factor, GF, is the ratio between the height, X, and the width, Y, in the
plane of impedance of the area the TEM wave moves through. It is a square with GF = 1, therefore Y =
X.

8
The Voltage of the Photon Forrest Bishop, July 25, 2023

With Y = X, the electric field strength, Equation (27), becomes an unknown function of X, the width
of the photon:

VP f √ h Z0 .
EP =
X
=
X
√ h Z0 → X = f
EP
(35)

And similarly for the magnetic flux density, Equation (28):

EP f √h Z 0 .
BP = =
c 0 X c0
√ h Z0 =
λX
(36)

So no information about the width of the photon, its waveform, or its electric-field strength can be
gleaned from the above analysis. These conclusions can be applied to the presumptions and equations
of Classical Electromagnetism and of Quantum Electrodynamics to find the voltage and to search for X,
the width of the photon, as follows.

With wave number |k| = 2 π f k /c 0 = ω k /c 0 , t for †, eμ the transverse polarization basis vectors, and
VOL = L3 the cubical, macroscopic analytic volume, the foundational quantized operator fields are

√ ℏ
{ (μ)
}
( μ) (μ) t (μ ) −i k⋅r
∑
i k⋅r
A (r ) = e a ( k ) e + ē a ( k ) e (37)
k, μ 2 ω k V OL ϵ 0

√
ℏωk
E ( r) = i ∑
k, μ 2V OL ϵ 0
{ (μ ) (μ ) i k⋅r (μ) t (μ )
e a ( k ) e − ē a ( k ) e
−i k⋅r
} (38)

√ ℏ
{( k × e (μ) (μ ) (μ)
}
t( μ ) −i k⋅r
B (r ) = i ∑
i k⋅r

2 ω k V OL ϵ 0
) a ( k) e − ( k × ē ) a (k ) e (39)
k, μ

For the root terms in Equations (37) for A(r) and (39) for B(r), the use of Equations (2) reveal, again,
something that is obscured by mixing together redundant physical constants. Substituting in
ϵ 0 = 1/ Z 0 c 0 , ω = 2 π f , and resolving ℏ as h/2 π , yields

√ ℏ
2 ω k V OL ϵ 0
=
√ h Z0 c0
2
8 π f k V OL
⇒
1
2π √ h Z0 c 0
2
2 f k L X LZ
=
1 ϕP
2 π LX √
λk
2 LZ
(40)

using the invariant magnetic flux density, φP, from Equation (15) for the latter expression, Two of the
macroscopic lengths, the transverse LX , have been labeled to emphasize their transverse directions and
then brought out from under the root using the arguments leading to Equation (34). With |Lx| = |Ly| = |Lz|
the root term of Equation (38) for E(r) becomes

9
The Voltage of the Photon Forrest Bishop, July 25, 2023

√
ℏωk
2 V OL ϵ 0
=
√ h ( Z 0 c 0)( 2 π f k )
2π 2L
3
=
√ h f k Z0 c0
2 L x L y Lz
⇒ fk
ϕP
Lx √
λk
2 Lz
= c0
ϕP
√
λk
λ k L x 2 Lz
(41)

With subscript j for summing over the three perpendicular axes in this quantum electrodynamic
construct, k is commonly restricted to the countable

2π nj
|k j| = L
, n j = ±1,±2 ,±3. .., for j = 1 ,2 , 3 (42)

For unit vectors ( x^ , y^ , z^ ) along the (x, y, z) axes, k is in the direction |k| z^ , for example, and the
solutions to Equations (37)-(39) are time-independent standing TEM waves propagating perpendicular
to the faces of the mirrored box, still with an undetermined analytic width, X, of the quantized photon
of wavenumber k. Considering propagation in the z^ direction only, and so dropping the unneeded
subscript j of Equation (43),

Lz
|k| = 2λπ = 2 π n , n = ±1,±2 ,±3. .. and so λ k = (43)
k Lz n

where Lz is the length of the box in the z^ and k direction. With Equation (43), Equation (40) reduces to

√
1 ϕP
√ √
ℏ λk 1 ϕP 1
⇒ = [ weber/meter ] (44)
2 ω k V OL ϵ 0 2π Lx 2 Lz 2π Lx 2n

and similarly for Equation (41), for the magnitude of the electric field, E(r), and for an amplitude of 1
of any component λk:

√ √ √
ℏω k ϕP λk ϕP 1
|E ( r )| =
k
2V OL ϵ 0
⇒ fk
Lx 2 Lz
= c0
λ k Lx 2 n
[ volt/meter ]. (45)

For B(r), Equation (39), the dimensions of k in the cross product, 2π/λk, are brought out and multiplied
with Equation (45). For an amplitude of 1 of any component,

√ 2π 1 ϕ P
√ √
2π ℏ 1 ϕP 1
|B ( r )| =
k λk 2 ω k V OLϵ 0
⇒
λ k 2 π Lx 2n
=
λ k Lx 2n
[ weber/meter2 ] (46)

which crosschecks against E = c0 B .

The voltage, Vk, of any component λk of the expansion of Ek(r), Equations (45), (41), and (38), is seen
to be

10
The Voltage of the Photon Forrest Bishop, July 25, 2023

V k = f k ϕ P = f k √ h Z0 (47)

The voltage of the photon, and the TEM wave, was there in the conventional models ab initio.

The cubical cavity oscillator used for the foundational Quantum Electrodynamic Equations (37)-(39)
has unique, geometric properties about it. Any cross-section of a cube, taken parallel to any face, is a
square. Orienting this cube parallel to the (x, y, z) axes of a Cartesian coordinate system, the k vectors
are taken as parallel to x, y, or z. The plane TEM waves are restricted to propagating through a square
in the plane of their impedance. The geometric factor, GF, of a square is GF = 1, therefore the
impedance, Z, of this confined space, in the specified x^ , y^ , z^ directions, is Z = GF * Z0 = Z0, the
impedance of free space, a fact not evident in the literature.

The implied impedance can also be found by equating the electric and magnetic fields as follows:

Zk =
Ek
=
|
Ek ( r ) Z 0 |
=
Z 0 c 0 ϕ P / λ k Lx √ 1/2n
= Z0 (48)
Hk |
Bk ( r ) c 0 | c0 ϕ P / λ k L x √ 1/2 n

With the wavelength, λK, in the direction of propagation of the photon, and k selected perpendicular to
any pair of the walls of the box, the remainder dimensions of VOL are in the transverse directions of k
when applied to any particular photon that participates in the summation, the plane or subspace
spanned by the eμ basis vectors. As shown above, these are undetermined dimensions X and Y, with the
constraint that Y = X when Z = Z0.

With the quantized photon’s demonstrated impedance of Z k = Z 0 , it is possible to equate any

component of |Ek| with the independently-derived EP in an unknown X, identified above as the width of
the photon, from Equation (35), with λk = λ:

√
ϕP 1 ϕP
|E ( r )| =
k
c0
λ k LX 2n
= c0
λX
(49)

and solve for X:

X = L √2 n (50)

If this were true, then the width of the photon would be larger than the box it is supposed to be confined
in, for all n, an incredible proposition. Another interpretation is X = L, if the root term is ignored for
some reason.

There are five different values for the width of the photon, X, considered so far: zero, infinity, λ, L, and
L √ 2 n . The X = 0 width of the point particle implies an infinite field strength. For X→ infinity, E→0,
also an untenable result. So X is still an undetermined, finite length, greater than zero and less than
infinity.

11
The Voltage of the Photon Forrest Bishop, July 25, 2023

Summary and Findings

The novel equations above were initially derived and crosschecked using the dimensional-analytic set
{F, L, T, Q} (force, length, time, charge), then translated back into derived SI units.

Photon Property Equation SI Derived Units {F, L, T, Q} Units

Electric permittivity ϵ 0 = 1/ ( Z 0 c 0 ) farad/meter Q2 L-2 F-1

Magnetic permeability μ 0 = Z0 /c0 henry/meter F T2 Q-2

Electric current i P = f √ h/Z 0 ampere QT-1

Voltage V P = f √ h Z0 volt F L Q-1

Component voltage V k = f k √h Z 0 volt F L Q-1

Electric charge ±QP = √ h/Z 0 coulomb Q

Capacitance C P = 1/ fZ 0 farad Q2 F-1 L-1

Inductance LP = Z 0 /f henry F L T2 Q-2

Magnetic flux φP = √ h Z0 weber F L T Q-1

Electric energy W E = 1/2 h f joule FL

Magnetic energy W M = 1/2h f joule FL

Intrinsic spin J z = ±ℏ = ±Q P ϕ P /2 π action quantum FLT

Electric field strength E P =f ϕ P / X volt/meter F Q-1

Electric field strength |E ( r )| =

k
f k ϕ P / Lx √ 1 /2 n volt/meter F Q-1

Magnetic flux density BP = ϕ P / ( λ X ) tesla F T Q-1 L-1

Magnetic flux density |B ( r )| =

k
ϕ P / ( λ k L x ) √ 1/2 n tesla F T Q-1 L-1

Magnetic field strength H P = ( f / X ) √ h/Z 0 ampere/meter Q L-1 T-1

Geometric factor, GF varies dimensionless -

Fine-structure constant α = e 2 / ( 2Q P2 ) dimensionless -

12
The Voltage of the Photon Forrest Bishop, July 25, 2023

Findings:

• The discovery of the mainstream voltage of the photon, which has apparently not been addressed.
Since the photon has an unsung voltage in their model, along with electric and magnetic fields, it also
has a finite width, by their definition.
• A mathematical and logical unification of the TEM wave on the line with the TEM wave on the
aether, which Maxwell and his followers had failed to accomplish. This may lead to entirely new lines
of inquiry and possible application.
• Derivations of the apparently previously-unknown, closed expressions for the charge, capacitance,
flux, etc. of the photon.
• An intriguing interpretation of the fine-structure constant, in which it is seen as the ratio of
electronic and photonic charge. This constant only appears when photons and massive particles
interact, and so its interpretation as their ratio of intrinsic electric charges is most direct.
• The identification of the occult wave impedance used in the QED analytic cavity, a figure not
generally emphasized.
• The identification of the variable, X, which represents the width of the photon, as well as the TEM
wave on the line or in free space, in conventional electromagnetics.
• An expression for the intrinsic spin in terms of the photon’s charge and magnetic flux density.

Several useful mathematical relations and techniques were employed in the above, not all of which
were called out. There are the crucial substitutions of Equations (2), which condense the number of
physical constants employed. There is The Forbidden Equation: i = qc, the governing equation of
electric current, which is found to be compatible with this model. The equal partition of the electric and
magnetic field energies of the electromagnetic wave was used twice to verify the derivations. The wave
impedance of free space, or the aether, Z0, along with its geometric form factor, GF, [Catt, et al, 1978]
is shown to be a crucial concept for any photon model, being as fundamental as the speed of light.

The initial assumption, that the wavelength of the photon is the length of the photon, is supported by
the derived quantities. Any other length assumption, from a point-like zero to an infinite extent, either
complicates or renders absurd the expressions for electric field, current, voltage, and the other
equations that involve frequency or wavelength.

A second assumption, famously proposed by Maxwell and the others, is that light is an electromagnetic
wave, similar to the transverse electromagnetic waves of an electric circuit/transmission line. That
hypothesis was used directly to apply the electrical parameters of the TEM wave on the transmission
line to this photon model with and including the quantization conditions.

M. Planck, in his June, 1920 Nobel Prize acceptance speech, titled in translation “The Genesis and
Present State of Development of the Quantum Theory” [Planck, 1920b] addresses the problem of the
volume of the photon in his closing remarks:

“There is one particular question the answer to which will, in my opinion, lead to an extensive
elucidation of the entire problem. What happens to the energy of a light quantum after its emission?
Does it pass outwards in all directions, according to Huygens’s wave theory, continually increasing in

13
The Voltage of the Photon Forrest Bishop, July 25, 2023

volume and tending towards infinite dilution? Or does it, as in Newton’s emanation theory, fly like a
projectile in one direction only?

“In the former case the quantum would never again be in a position to concentrate its energy at a
spot strongly enough to detach an electron from its atom; while in the latter case it would be necessary
to sacrifice the chief triumph of Maxwell’s theory- the continuity between the static and the dynamic
fields- and with it the classical theory of the interference phenomena which accounted for all their
details, both alternatives leading to consequences very disagreeable to the modern theoretical physicist.
Whatever the answer to this question, there can be no doubt that science will some day master the
dilemma, and what may now appear to us unsatisfactory will appear from a higher standpoint as
endowed with a particular harmony and simplicity. But until this goal is reached the problem of the
quantum of action will not cease to stimulate research, and the greater the difficulties encountered in its
solution the greater will be its significance for the broadening and deepening of all our physical
knowledge.”

Science has not yet mastered this dilemma, as the famous metaphysics of the wave-particle duality and
the results of the double-slit experiment highlight. Nor has Maxwell triumphed in establishing “the
continuity between the static and the dynamic fields”, for it has not be proven that the static field exists
in the first instance, nor that it can be instantly accelerated and transformed into a field that moves at
the speed of light. The math works out to some degree, but that cannot make it govern anything, as the
case history of Quantum Electrodynamics shows [Consa, 2021].

There are several different conceptual models for the photon in addition to the quantum electrodynamic
“packet of energy” idea. But all of them have to have an electric field, E, and its attendant voltage, VP,
and electric charge, QP, or whatever may come to replace those ideas, in order to conform to
observation. No such model can be taken seriously without either an explicit accounting of all of the
electromagnetic parameters of the TEM wave in free space or, perhaps, by replacing large swaths of
electrical engineering and physics with something better. Short of the latter, and following M. Planck,
they must address this question about its electric field strength, E, directly and without subterfuge:

How wide is a photon?

References

Bishop, F., “The Fine-structure Constant and Some Relationships Between the Electromagnetic Wave
Constants”, [2007], http://redshift.vif.com/JournalFiles/V14NO4PDF/V14N4BIS.pdf

Bishop, F., "The Forbidden Equation: i = qc”, [2016],

https://www.scribd.com/document/320890002/The-Forbidden-Equation-i-q
http://www.naturalphilosophy.org/pdf/abstracts/abstracts_paperlink_7395.pdf

14
The Voltage of the Photon Forrest Bishop, July 25, 2023

Bishop, F., [2012], “Reforming Electromagnetic Units, Equations, and Concepts: An Extension of Ivor
Catt’s Theory”, http://www.naturalphilosophy.org//pdf//abstracts/abstracts_6554.pdf

Catt, I., M. Davidson and D. Walton, [1978], Digital Electronic Design, Vol 1, CAM Publishing,
http://forrestbishop.mysite.com/DEDV1/Digital%20Electronic%20Design%20Vol1%20searchable.pdf

Consa, O., [2021], “Something is wrong in the state of QED”, https://arxiv.org/abs/2110.02078

Planck, M., [1920a], The Theory of Heat Radiation, https://www.gutenberg.org/ebooks/40030

Planck, M., [1920b], “The Genesis and Present State of Development of the Quantum Theory”
https://www.nobelprize.org/prizes/physics/1918/planck/lecture

15