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In a recent paper, we have studied the nature of the electromagnetic energy radiated over a single period of oscillation by an antenna working in frequency domain under ideal conditions and without losses when the oscillating charge in the antenna is reduced to the elementary charge. Here we extend and expand that study. The energy radiated by an oscillating current in an antenna occurs in bursts of duration
*T/2*
, where
*T*
is the period of oscillation. The results obtained here, based purely on classical electrodynamics, can be summarized by the inequality
U
≥hv
→q
_{0}
≥e
where
*U*
is the energy radiated in a single burst of duration
*T/2*
,
*h*
is the Planck constant,
*ν*
is the frequency of oscillation and
*q*
_{0}
is the magnitude of the oscillating charge associated with the current. The condition *U=hv→q*_{0}*=e*
is obtained when the length of the antenna is equal to the ultimate Hubble radius of the universe (
*i.e.*
the maximum value of the antenna length allowed by nature) and the wavelength is equal to the Bohr radius (resulting from the smallest possible radius of the conductor allowed by nature). The ultimate Hubble radius is directly related to the vacuum energy density. The inequality obtained here is in general agreement with the one obtained in the previous study. One novel feature of this extended analysis is the discovery of an expression, in terms of the elementary charge and other atomic constants, for the vacuum energy density of the universe. This expression predicts the vacuum energy density to be about 4×10^{-10 }
J/m
^{3}
which is in reasonable agreement with the measured value of 6×10^{-10}
J/m
^{3}
.

In several recent publications, Cooray and Cooray [

The starting point of the analysis is a monopole antenna of length L/2 working in frequency domain located in free space over a perfectly conducting ground plane. The antenna is located along the z-axis and the origin of the Cartesian coordinate system to be used in the analysis is located at the point where the antenna meets the perfectly conducting ground plane. The geometry of the problem is shown in

The one and the only assumption that is being made in Paper 1, and also here, is the existence of ideal conditions where all the losses associated with thermal, dispersion and radiation as the current oscillates along the antenna can be neglected. This indeed is far from reality and we will discuss the effects of these losses on the results in the discussion section. Under this assumed ideal conditions the current distribution along the antenna including its image in the perfectly conducting ground plane is given by [

I ( t , z ) = I 0 sin { 2π λ ( L 2 − z ) } e j ω t , 0 ≤ z ≤ L / 2 (1)

I ( t , z ) = I 0 sin { 2π λ ( L 2 + z ) } e j ω t , − L / 2 ≤ z ≤ 0 (2)

With this current distribution, the total power dissipation, P T is given by [

P T = sin 2 ( 2 π t / T ) I 0 2 2 π ε 0 c ∫ 0 π / 2 [ cos ( k L 2 cos θ ) − cos ( k L 2 ) ] 2 d θ (3)

In the above equation k = 2 π / λ where λ is the wavelength, t is the time and T is the period of oscillation. This equation is valid when λ ≫ a , where a is the radius of the conductor [

P a v = I 0 2 4 π ε 0 c ∫ 0 π / 2 [ cos ( k L 2 cos θ ) − cos ( k L 2 ) ] 2 d θ (4)

If the oscillating charge associated with the current is given by q = q 0 e j 2 π ν t (note that j 2 = − 1 ), Equation (4) can be written as

P a v = q 0 2 ( 2 π ν ) 2 4 π ε 0 c ∫ 0 π / 2 [ cos ( k L 2 cos θ ) − cos ( k L 2 ) ] 2 d θ (5)

If L / λ ≪ 1 the integral in Equation (5) can easily be solved resulting in

P a v = q 0 2 ( 2 π ν ) 2 ε 0 c π 6 ( l λ ) 2 (6)

In Paper 1, when L/λ is comparable to or larger than unity, the integral in Equation (5) was solved numerically. As one can see, for very large values of kL (or L/λ) the integrand in Equation (5) oscillates rapidly as a function of θ. This could introduce significant errors in the numerical integration specially when the value of ( k L ) − 1 becomes comparable to the computational accuracy of the computer. Fortunately, an analytical expression in terms of Cosine and Sine Integrals is available for this integral [

P a v = q 0 2 ( 2 π ν ) 2 8 π ε 0 c { γ + ln ( k L ) − C i ( k L ) + 1 2 sin ( k L ) [ S i ( 2 k L ) − 2 S i ( k L ) ] + 1 2 c o s ( k L ) [ γ + ln ( k L / 2 ) + C i ( 2 k L ) − 2 C i ( k L ) ] } (7)

In the above equation, C i is the Cosine Integral, S i is the Sine Integral and γ is the Euler-Mascheroni constant.

First of all observe that from Equation (3) the power generated by the antenna consists of bursts of energy each burst having a duration of T/2, where T is the period of oscillation. This is due to the fact that the radiated power is proportional to sin 2 ( 2 π t / T ) . For example,

kL. This shows that the radiated energy can be separated into individual bursts of duration T/2. Let us consider the radiated energy associated with a single burst of energy. This energy, U, is given by

U = q 0 2 π ν 2 2 ε 0 c T 2 { γ + ln ( k L ) − C i ( k L ) + 1 2 sin ( k L ) [ S i ( 2 k L ) − 2 S i ( k L ) ] + 1 2 c o s ( k L ) [ γ + ln ( k L / 2 ) + C i ( 2 k L ) − 2 C i ( k L ) ] } (8)

This can be written as

U = q 0 2 π ν 4 ε 0 c { γ + ln ( k L ) − C i ( k L ) + 1 2 sin ( k L ) [ S i ( 2 k L ) − 2 S i ( k L ) ] + 1 2 c o s ( k L ) [ γ + ln ( k L / 2 ) + C i ( 2 k L ) − 2 C i ( k L ) ] } (9)

Observe from Equation (9) that for large values of kL the value of U oscillates rapidly with kL. The upper and lower bounds of U occur when k L = n π and k L = m π where n and m are even and odd integers (i.e. when cos ( k L ) = 1 or cos ( k L ) = − 1 ). The median value of U is given by

U m = q 0 2 π ν 4 ε 0 c { γ + ln ( k L ) − C i ( k L ) } (10)

First, note that for large kL the Cosine Integral varies as ∼ cos ( k L ) / ( k L ) 2 and it can be neglected with respect to other terms when kL is very large. Thus for very large kL the function U_{m} reduces to the following

U m = q 0 2 π ν 4 ε 0 c { γ + ln ( k L ) } (11)

Observe that this function goes to infinity as kL or L/λ goes to infinity. However, in reality, L/λ is limited by natural bounds. The magnitude of these natural bounds has been discussed in previous publications [_{m} that can ever be realized in nature for a given oscillating charge is given by

U m m = q 0 2 π ν 4 ε 0 c { γ + ln ( 4 π R ∞ / a 0 ) } (12)

Let us consider the number of energy units, equivalent to the energy of a photon, associated with this radiated energy. Since the energy of a single photon is equal to hν the number of such energy units associated with this energy is given by

N = q 0 2 π 4 ε 0 h c { γ + ln ( 4 π R ∞ / a 0 ) } (13)

This equation shows that the number of such energy units decreases with decreasing charge. The charge associated with a given number of such energy units is given by

q 0 = 4 ε 0 h c N π { γ + ln ( 4 π R ∞ / a 0 ) } (14)

When the number of such energy units (or the number of equivalent photons) is reduced to one, the oscillating charge reduces to

q 0 = 4 ε 0 h c π { γ + ln ( 4 π R ∞ / a 0 ) } (15)

Interestingly, information concerning the maximum possible value of the Hubble radius is ingrained into the fabric of the universe through the magnitude of the vacuum energy density. If the density of the vacuum energy, ρ Λ , of a flat universe is zero then the ultimate size of the Hubble radius is infinity [^{3} [

R ∞ = c 2 3 8 π G ρ Λ (16)

Since this is the maximum ever possible value of the Hubble radius, and its value is determined by the microscopic parameters of the fabric of space, it is the correct parameter to be inserted into Equation (15). With this value for the Hubble radius Equation (15) can be written as

q 0 = 4 ε 0 h c π { γ + ln [ 4 π c 2 a 0 3 8 π G ρ Λ ] } (17)

Substituting values for all the known parameters in Equation (17) we obtain q 0 = 1.603 × 10 − 19 C, which is within 0.1% of the elementary charge. It is important to point out here that, had we used the current Hubble radius corresponding to cosmological times spanning from less than 0.1 billion years until the time at which the Hubble radius had reached a steady value, the values of q_{0} we would have obtained would still remain within the order of magnitude of the elementary charge. This shows that when the length of the antenna is stretched and the radius of the antenna is compressed to its natural limits, and oscillating charge in the antenna is equal to the elementary charge, the median value of the energy it radiates during each half period of oscillation is equal to that of a photon corresponding to the frequency of oscillation. The Bohr radius is given by a 0 = ε 0 h 2 / π m e e 2 and substituting this expression for the Bohr radius in Equation (17) we obtain

q 0 = 4 ε 0 h c π { γ + ln [ 4 π 2 c 2 m e e 2 ε 0 h 2 3 8 π G ρ Λ ] } (18)

In the above equation m_{e} is the rest mass of the electron. Recall that Equation (17) and (18) are based purely on classical electrodynamics. However, these results show us that we could have arrived at the same equations if we had made the following two assumptions at the beginning of the analysis. (a) The minimum oscillating charge that can radiate in an antenna is the elementary charge and (b) The minimum energy associated with a single burst of radiation has to be larger than or equal to the energy of a photon. Of course here we are departing from classical electrodynamics because the quantization of the charge or the concept of a photon is not part of the classical electrodynamics. With these two reasonable assumptions, we end up with the following relationship between the elementary charge and the density of the vacuum energy.

e = 4 ε 0 h c π { γ + ln [ 4 π c 2 a 0 3 8 π G ρ Λ ] } (19)

Note that Equation (17) is identical to Equation (19) except that q_{0} is replaced by e. Equation (19) can be written as

ρ Λ = 3 8 π G { a 0 4 π c 2 exp [ 4 ε 0 h c e 2 π − γ ] } 2 (20)

Note that the basis of the derivation of Equations (17) and (19) is different. Equation (17), which is based purely on classical electrodynamics, shows that the smallest oscillating charge that can generate a burst of energy equal to that of a single photon by an antenna stretched to its limiting dimensions is on the order of the elementary charge. On the other hand, Equation (19) results when we assume that the smallest oscillating charge possible is the elementary charge and the smallest energy that could be produced by a single burst of radiation by an antenna stretched to its limiting dimensions is equal to the energy of a photon.

One can interpret Equation (20) as an alternative derivation of the vacuum energy density of the universe. In fact, this is the first time that a connection between the vacuum energy density and the other atomic constants, including the elementary charge, is implied. If we insert the values of the known parameters into Equation (20), we will obtain ρ Λ = 4.3 × 10 − 10 J/m^{3}. This is slightly less but a good estimation of the vacuum energy. Indeed, this is the value we have to substitute in Equation (17) to make q_{0} exactly equal to the elementary charge. It is important to point out that this result is based on the median value of the radiated energy. In order to distinguish between upper and lower limits of the energy, it is necessary to determine the Hubble radius to an accuracy better than about the Bohr radius. Quantum uncertainties as dictated by uncertainty principle would prohibit a measurement of the Hubble radius to such an accuracy. Thus, the use of median value has its merits. However, this point needs further investigations.

The connection between the vacuum energy density and the elementary charge is not surprising given the fact that both the observable value of the elementary charge and the vacuum energy are controlled by vacuum fluctuations [

As mentioned in the introduction, a study similar to the one reported here was conducted previously in Paper 1. There are several differences both in the methodology and the results obtained between that study and the present one. First, in Paper 1 it was observed that the energy dissipation over a given period of time reaches more or less a constant value with increasing L/λ. However, the results presented above show that instead of reaching a constant value the energy increases slowly (i.e. logarithmically) with increasing L/λ. This slow increase was not captured in that paper due to rounding off errors caused by the numerical integration. Second, in Paper 1 the energy is calculated over a single period rather than over half a period as was done in the present study. We believe that the energy dissipation over half a period is more appropriate than over a full period because the energy is dissipated in bursts of half period durations. Third, in Paper 1 the radiation generated by a dipole antenna fed by two currents at the center was investigated. Here, the radiating system is reduced to its basic element by placing the antenna over a perfectly conducting ground plane and exciting it by a single current. Notwithstanding all these differences the oscillating charge corresponding to a single photon of energy estimated in Paper 1 does not differ more than a factor of two, approximately, and thus the conclusions made in that paper are confirmed in the present study.

In a previous publication related to a similar topic but with transient currents (i.e. [

The results presented in the previous section shows that under conditions where the dissipated energy by the antenna for a given charge is a maximum (i.e. L/λ reaches its upper limit), at least one elementary charge has to be associated with the oscillating current so that the energy dissipated over half a period to be larger than or equal to that of a photon. However, we have only considered the median value of the energy corresponding to a given value L/λ even though the number of photons (or the equivalent energy units) associated with a given charge oscillates between two limits in the vicinity of any given L/λ. Note that the energy oscillates between upper and lower bounds when the ratio L/λ changes by 0.5. In the vicinity of L / λ ∼ 10 36 (corresponding to 2 R ∞ / a 0 ) the relative change necessary in this ratio to make the radiated energy jump between the two bounds is infinitesimally small and we are not in a position to determine exactly the value of the charge corresponding to this ratio because neither of the parameters R ∞ and a 0 are known to such an accuracy. Under these circumstances the best one can do is to estimate the median value of the energy and the corresponding charge.

current for N = 1 when one estimates it using either the upper bound, lower bound or the median value as given by Equation (9). This shows that in the vicinity of L / λ = R ∞ / a 0 even the upper and lower bounds of the charge necessary to generate a single photon within T/2 are still in the order of magnitude of the elementary charge. Thus the results obtained in this paper can be summarized by the order of magnitude relationship U ≥ h ν → q ≥ e where U is the energy dissipated over half a period, q is the charge associated with the oscillating current and e is the electronic charge.

Observe that the results presented above are based on absolutely ideal and lossless conditions. In reality, the presence of losses will modify the assumed ideal current distribution and this in turn will give rise to a reduction in the energy radiated. Interestingly, in the presence of losses, a larger charge is necessary to radiate a given amount of energy over a given amount of time than in the ideal case. Thus, the relationship U ≥ h ν → q ≥ e still remains valid.

Finally, it is important to stress here that the Equations (17) to (19) given in this paper are based purely on classical electrodynamics. Given the fact that the elementary charge or the concept of photon is not a part of the classical electrodynamics (note that the electron was discovered nearly 30 years after the development of classical electrodynamics), it is remarkable that its predictions lead to the above relationship. Of course, in order to derive an expression for the vacuum energy density we had to utilize both the experimental fact that the electric charge is quantized and the concept of photons from quantum mechanics. However, this semi-classical analysis led to a rather accurate expression for the vacuum energy density as a function of the well-known atomic constants including the elementary charge. However, as mentioned before more investigations are needed on this point.

The energy radiated by an oscillating current in an antenna occurs in bursts of duration T/2, where T is the period of oscillation. The results obtained here, based purely on classical electrodynamics, can be summarized by the inequality U ≥ h ν → q 0 ≥ e where U is the energy radiated in a single burst of duration T/2, h is the Planck constant, ν is the frequency of oscillation and q_{0} is the magnitude of the oscillating charge associated with the current. The condition U = h ν → q 0 = e is obtained when the length of the antenna is equal to the ultimate radius of the universe and the wavelength is equal to the Bohr radius. The inequality obtained here is in general agreement with the one obtained in the previous study.

One novel feature of the analysis is the discovery of an expression in terms of atomic constants including the elementary charge for the vacuum energy density of the universe. This expression predicts the vacuum energy density to be about 4 × 10 − 10 J/m^{3} which is in reasonable agreement with the measured value of about 6 × 10 − 10 J/m^{3}.

Authors appreciate the suggestions, support and encouragement given by Prof. Farhad Rachidi, Prof. Marcos Rubinstein, Prof. Carlo Mazzetti and Prof. Yoshihiro Baba during the development of ideas presented in this paper.

Cooray, V. and Cooray, G. (2018) Remarkable Predictions of Classical Electrodynamics on Elementary Charge and the Energy Density of Vacuum. Journal of Electromagnetic Analysis and Applications, 10, 77-87. https://doi.org/10.4236/jemaa.2018.105006