Interference Between Source-Free Radiation and Radiation from Sources:   Particle-Like Behavior for Classical Radiation

Timothy H. Boyer

arXiv:1703.06309·physics.class-ph·November 15, 2017

Interference Between Source-Free Radiation and Radiation from Sources: Particle-Like Behavior for Classical Radiation

Timothy H. Boyer

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TL;DR

This paper demonstrates how classical electromagnetic radiation can exhibit particle-like behavior through interference effects between source-free standing waves and radiation from a current pulse, illustrating concepts related to photon-like phenomena.

Contribution

It introduces a simple electrodynamics problem showing classical interference effects that mimic particle-like behavior, connecting classical waves to photon concepts.

Findings

01

Interference patterns depend on phase and magnitude relations.

02

Symmetric radiation occurs when source-free waves are negligible.

03

Asymmetric, particle-like radiation emerges when source-free waves are significant.

Abstract

A simple junior-level electrodynamics problem is used to illustrate the interference between a source-free standing plane wave and a wave generated by a pulse in a current sheet. Depending upon the relative phases between the standing wave and the current pulse and also upon the relative magnitudes, we can find quite different patterns of emitted energy and momentum. If the source gives a large radiation pulse so that the source-free plane wave can be neglected, then the radiation spreads out symmetrically on either side of the current sheet. However, if the radiation sheet gives a pulse with fields comparable to those of the standing wave, then we can find a single radiation pulse moving to the right while the current sheet recoils to the left, or the situation with the directions reversed. The example is a crude illustration of particle-like behavior arising from conventional…

Equations92

V (r, t) = V^{in} (r, t) + \int d^{3} r^{'} \frac{ρ ( r , t - ∣ r - r ^{'} ∣/ c )}{∣ r - r ^{'} ∣}

V (r, t) = V^{in} (r, t) + \int d^{3} r^{'} \frac{ρ ( r , t - ∣ r - r ^{'} ∣/ c )}{∣ r - r ^{'} ∣}

A (r, t) = A^{in} (r, t) + \int d^{3} r^{'} \frac{J ( r , t - ∣ r - r ^{'} ∣/ c )}{c ∣ r - r ^{'} ∣}

A (r, t) = A^{in} (r, t) + \int d^{3} r^{'} \frac{J ( r , t - ∣ r - r ^{'} ∣/ c )}{c ∣ r - r ^{'} ∣}

K (t) = x K_{0} cos (ω t)

K (t) = x K_{0} cos (ω t)

E_{r} (r, t) = x E_{0} cos (k z - ω t), B_{r} (r, t) = y E_{0} cos (k z - ω t) for z > 0,

E_{r} (r, t) = x E_{0} cos (k z - ω t), B_{r} (r, t) = y E_{0} cos (k z - ω t) for z > 0,

E_{l} (r, t) = x E_{0} cos (k z + ω t), B_{l} (r, t) = - y E_{0} cos (k z + ω t) for z < 0,

E_{l} (r, t) = x E_{0} cos (k z + ω t), B_{l} (r, t) = - y E_{0} cos (k z + ω t) for z < 0,

\frac{4 π}{c} K (t) = z \times [B_{r} (0, t) - B_{l} (0, t)] = - x 2 E_{0} cos (ω t),

\frac{4 π}{c} K (t) = z \times [B_{r} (0, t) - B_{l} (0, t)] = - x 2 E_{0} cos (ω t),

E_{0} = - \frac{2 π}{c} K_{0} .

E_{0} = - \frac{2 π}{c} K_{0} .

P / A = - K (t) \cdot [E_{r} (0, t) + E_{l} (0, t)] /2 = - K_{0} E_{0} cos^{2} (ω t) = \frac{c}{2 π} E_{0}^{2} cos^{2} (ω t),

P / A = - K (t) \cdot [E_{r} (0, t) + E_{l} (0, t)] /2 = - K_{0} E_{0} cos^{2} (ω t) = \frac{c}{2 π} E_{0}^{2} cos^{2} (ω t),

S_{r} (0, t) = \frac{c}{4 π} E_{r} (0, t) \times B_{r} (0, t) = z \frac{c}{4 π} E_{0}^{2} cos^{2} (ω t),

S_{r} (0, t) = \frac{c}{4 π} E_{r} (0, t) \times B_{r} (0, t) = z \frac{c}{4 π} E_{0}^{2} cos^{2} (ω t),

S_{l} (0, t) = \frac{c}{4 π} E_{l} (0, t) \times B_{l} (0, t) = - z \frac{c}{4 π} E_{0}^{2} cos^{2} (ω t),

S_{l} (0, t) = \frac{c}{4 π} E_{l} (0, t) \times B_{l} (0, t) = - z \frac{c}{4 π} E_{0}^{2} cos^{2} (ω t),

P / A = \frac{c}{2 π} E_{0}^{2} cos^{2} (ω t) = z \cdot S_{r} (0, t) + (- z) \cdot S_{l} (0, t) .

P / A = \frac{c}{2 π} E_{0}^{2} cos^{2} (ω t) = z \cdot S_{r} (0, t) + (- z) \cdot S_{l} (0, t) .

E_{S} (z, t) = - x 2 E_{0} cos (k z) cos (ω t), B_{S} (z, t) = - y 2 E_{0} sin (k z) sin (ω t)

E_{S} (z, t) = - x 2 E_{0} cos (k z) cos (ω t), B_{S} (z, t) = - y 2 E_{0} sin (k z) sin (ω t)

E_{S} (z, t)

E_{S} (z, t)

B_{S} (z, t)

E_{r} (r, t) = x E_{0} cos (k z - ω t), B_{r} (r, t) = y E_{0} cos (k z - ω t) for - π /2 < k z - ω t < π /2,

E_{r} (r, t) = x E_{0} cos (k z - ω t), B_{r} (r, t) = y E_{0} cos (k z - ω t) for - π /2 < k z - ω t < π /2,

E_{l} (r, t) = x E_{0} cos (k z + ω t), B_{l} (r, t) = - y E_{0} cos (k z + ω t) for - π /2 < k z + ω t < π /2.

E_{l} (r, t) = x E_{0} cos (k z + ω t), B_{l} (r, t) = - y E_{0} cos (k z + ω t) for - π /2 < k z + ω t < π /2.

\frac{U}{A} = \int_{t = - π /2 ω}^{t = π /2 ω} d t \frac{P}{A} = \int_{t = - π /2 ω}^{t = π /2 ω} d t \frac{c}{2 π} E_{0}^{2} cos^{2} (ω t) = \frac{1}{4 k} E_{0}^{2} .

\frac{U}{A} = \int_{t = - π /2 ω}^{t = π /2 ω} d t \frac{P}{A} = \int_{t = - π /2 ω}^{t = π /2 ω} d t \frac{c}{2 π} E_{0}^{2} cos^{2} (ω t) = \frac{1}{4 k} E_{0}^{2} .

\frac{U _{r}}{A} = \int_{z = c t - π /2 k}^{z = c t + π /2 k} d z \frac{1}{8 π} (E_{r}^{2} + B_{r}^{2}) = \int_{z = c t - π /2 k}^{z = c t + π /2 k} \frac{d z}{8 π} 2 E_{0}^{2} cos^{2} (k z - ω t) = \frac{1}{8 k} E_{0}^{2},

\frac{U _{r}}{A} = \int_{z = c t - π /2 k}^{z = c t + π /2 k} d z \frac{1}{8 π} (E_{r}^{2} + B_{r}^{2}) = \int_{z = c t - π /2 k}^{z = c t + π /2 k} \frac{d z}{8 π} 2 E_{0}^{2} cos^{2} (k z - ω t) = \frac{1}{8 k} E_{0}^{2},

P_{r} / A = \int_{z = c t - π /2 k}^{z = c t + π /2 k} d z \frac{1}{4 π c} E_{r} \times B_{r} = z \int_{z = c t - π /2 k}^{z = c t + π / k} d z \frac{1}{4 π c} E_{0}^{2} cos^{2} (k z - ω t) = z \frac{1}{8 ω} E_{0}^{2} .

P_{r} / A = \int_{z = c t - π /2 k}^{z = c t + π /2 k} d z \frac{1}{4 π c} E_{r} \times B_{r} = z \int_{z = c t - π /2 k}^{z = c t + π / k} d z \frac{1}{4 π c} E_{0}^{2} cos^{2} (k z - ω t) = z \frac{1}{8 ω} E_{0}^{2} .

E_{S} (r, t) = x E_{S} cos (k z + ϕ) cos (ω t + θ), B_{S} (r, t) = y E_{S} sin (k z + ϕ) sin (ω t + θ)

E_{S} (r, t) = x E_{S} cos (k z + ϕ) cos (ω t + θ), B_{S} (r, t) = y E_{S} sin (k z + ϕ) sin (ω t + θ)

u_{S} (z, t)

u_{S} (z, t)

= \frac{E _{S}^{2}}{8 π} [cos^{2} (k z + ϕ) cos^{2} (ω t + θ) + sin^{2} (k z + ϕ) sin^{2} (ω t + θ)] .

S_{S} (z, t) = \frac{c}{4 π} E_{S} (r . t) \times B_{S} (r, t) = z \frac{c E _{S}^{2}}{16 π} sin [2 (k z + ϕ)] sin [2 (ω t + θ)],

S_{S} (z, t) = \frac{c}{4 π} E_{S} (r . t) \times B_{S} (r, t) = z \frac{c E _{S}^{2}}{16 π} sin [2 (k z + ϕ)] sin [2 (ω t + θ)],

P / A

P / A

= - K_{0} cos (ω t) E_{S} cos (ϕ) cos (ω t + θ) - K_{0} E_{0} cos^{2} (ω t)

= - K_{0} E_{S} cos^{2} (ω t) cos ϕ cos θ + K_{0} E_{S} cos (ω t) sin (ω t) cos ϕ sin θ - K_{0} E_{0} cos^{2} (ω t) .

U / A = \int_{t = - π /2 ω}^{t = π /2 ω} d t \frac{P}{A} = \frac{E _{S} E _{0}}{4 k} cos ϕ cos θ + \frac{E _{0}^{2}}{4 k},

U / A = \int_{t = - π /2 ω}^{t = π /2 ω} d t \frac{P}{A} = \frac{E _{S} E _{0}}{4 k} cos ϕ cos θ + \frac{E _{0}^{2}}{4 k},

U_{R} (z, t) / A

U_{R} (z, t) / A

= \int_{z = c t - π /2 k}^{z = c t + π /2 k} d z \frac{1}{8 π} {[E_{S} (z, t) + E_{r} (z, t)]^{2} + [B_{S} (z, t) + B_{r} (z, t)]^{2} - [E_{S}^{2} (z, t) + B_{S}^{2} (z, t)]}

= \int_{z = c t - π /2 k}^{z = c t + π /2 k} d z \frac{1}{8 π} {2 E_{S} (z, t) \cdot E_{r} (z, t) + 2 B_{S} (z, t) \cdot B_{r} (z, t) + E_{r}^{2} (z, t) + B_{r}^{2} (z, t)}

\int_{z = c t - π /2 k}^{z = c t + π /2 k} d z \frac{1}{8 π} [2 E_{S} (z, t) \cdot E_{r} (z, t)]

\int_{z = c t - π /2 k}^{z = c t + π /2 k} d z \frac{1}{8 π} [2 E_{S} (z, t) \cdot E_{r} (z, t)]

= \int_{z = c t - π /2 k}^{z = c t + π /2 k} d z \frac{1}{4 π} [E_{S} cos (k z + ϕ) cos (ω t + θ) E_{0} cos (k z - ω t)]

= \frac{E _{S} E _{0}}{8 k} cos (ω t + θ) cos (ω t + ϕ),

U_{R} / A = \frac{E _{S} E _{0}}{8 k} cos (ϕ - θ) + \frac{E _{0}^{2}}{8 k} .

U_{R} / A = \frac{E _{S} E _{0}}{8 k} cos (ϕ - θ) + \frac{E _{0}^{2}}{8 k} .

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Full text

Interference Between Source-Free Radiation and Radiation from Sources:

Particle-Like Behavior for Classical Radiation

Timothy H. Boyer

Department of Physics, City College of the City University of New York, New York, New York 10031

Abstract

A simple junior-level electrodynamics problem is used to illustrate the interference between a source-free standing plane wave and a wave generated by a pulse in a current sheet. Depending upon the relative phases between the standing wave and the current pulse and also upon the relative magnitudes, we can find quite different patterns of emitted energy and momentum. If the source gives a large radiation pulse so that the source-free plane wave can be neglected, then the radiation spreads out symmetrically on either side of the current sheet. However, if the radiation sheet gives a pulse with fields comparable to those of the standing wave, then we can find a single radiation pulse moving to the right while the current sheet recoils to the left, or the situation with the directions reversed. The example is a crude illustration of particle-like behavior arising from conventional classical electromagnetic behavior in the presence of source-free radiation. The discussion makes contact with the ideas of photons in modern physics.

I Introduction

Although textbooks of classical electrodynamics sometimes discuss the scattering and interference of electromagnetic waves in the context of optical behavior, there seems to be little attention paid to the interference arising between a source-free wave and the wave emitted by a transient current source. This omission is parallel to the failure of today’s textbooks of classical electromagnetism to include the source-free solution of Maxwell’s equations in addition to the radiation due to a current source when presenting the general solution of Maxwell’s equations. Inclusion of the source-free solution opens the possibility of understanding further aspects of electromagnetic technology, and also of understanding aspects of modern physics which are not usually treated in terms of conventional classical electromagnetic theory. Here we give illustrations of interference between source-free radiation and radiation from a current source which is appropriate for a junior-level course in electromagnetism. Such interference can give rise to behavior which has some of the aspects usually associated with wave-particle duality and may give insight into photon-like physics.

We begin with a discussion of the general solution of Maxwell’s equation which contains both a source-free term and a source term involving charges and currents at a retarded time. Next we treat an oscillating current sheet in the absence of source-free radiation. We give the plane wave emitted by the oscillating surface current and confirm the balance in power delivered by the source and the outflow of energy in the waves. Next, we consider a pulsed current sheet, and again confirm the energy and momentum conservation ideas for the pulses moving symmetrically outwards from the surface current. At this point, we introduce a source-free solution of Maxwell’s equations corresponding to a standing plane wave throughout space at the same frequency as the frequency of the surface current but with arbitrary amplitude and arbitrary phases in both space and time. We show that the energy and momentum delivered by the pulsed current sheet now depend upon the presence of the source-free standing wave. By varying the phases and amplitude of the source-free standing wave relative to the wave emitted by the pulsed source, we find a variety of behavior, some of which suggests particle-like aspects. In the final section, we remark on the connections of these simple calculations with the ideas of modern physics.

II Source-Free Radiation and the General Solution of Maxwell’s

Equations

The general retarded solution of Maxwell’s equations can be written in terms of the scalar and vector potentials asgaussian

[TABLE]

where $V^{in}(\mathbf{r},t)$ and $\mathbf{A}^{in}(\mathbf{r},t)$ correspond to source-free solutions of Maxwell’s equations. The authors of the electromagnetism texts agree that this in the correct form for the expression for the potentials, but they usually exclude the source-free terms in the final general-solution equations of their texts.Griffiths ,Jackson ,Zangwill ,Garg The source-free potentials $V^{in}(\mathbf{r},t)$ and $\mathbf{A}^{in}(\mathbf{r},t)$ are connected to source-free electric and magnetic fields $\mathbf{E}^{in}=-\nabla V^{in}-(1/c)\partial\mathbf{A}^{in}\mathbf{/\partial t}$ and $\mathbf{B}^{in}=\nabla\times\mathbf{A}^{in}.$

The universe in which we live is not that of the classical electromagnetism texts which take $V^{in}(\mathbf{r},t)=0$ and $\mathbf{A}^{in}(\mathbf{r},t)=0.$ We live in a universe which is filled with radiation which we do not control with our local equipment. Thus in our laboratory rooms, there are radio waves from various transmitters (which our students pick-up on their smart phones), and there is thermal radiation from the walls and objects in the rooms. There is also the random radiation present even at the absolute zero of temperature which gives rise to Casimir forces between uncharged macroscopic objects. This radiation which we do not control with our local equipment can be regarded locally as a source-free solution of Maxwell’s equation which is present when we turn on our local sources. It is the interference between the source-free radiation and the radiation generated by the local sources which we discuss in the following examples.

III Emitted Radiation from an Oscillating Current Sheet Source and No

Source-Free Radiation

We start with the familiar situation where the source-free fields $\mathbf{E}^{in}$ and $\mathbf{B}^{in}$ are taken to vanish. A very simple example of a radiation source is an infinite (charge-neutral) oscillating current sheet which gives rise to plane waves.G499 In empty space, the plane waves spread out symmetrically on either side from the current sheet. The power per unit area delivered to the surface current by an external agent appears in the Poynting vectors of the plane wave pointing away from the current sheet.

Here we take the surface current as

[TABLE]

in the plane $z=0,$ giving rise to two plane waves moving outwards from the sheet,gaussian

[TABLE]

where $\omega=ck,$ and the subscripts $r$ and $l$ correspond to “right” and “left.” Plane waves are solutions of the source-free Maxwell equations,G397 so that we need only check the boundary conditions at $z=0$ to confirm that we have a valid solution of the full Maxwell equations. At the surface $z=0,$ the electric field $\mathbf{E}$ is continuous at all times $t,$ and the magnetic field $\mathbf{B}$ in Eqs. (4) and (5) is discontinuous, giving

[TABLE]

so that from Eq. (3), we have the relation between the amplitude of the emitted wave and the amplitude of the surface current

[TABLE]

The power per unit area $\mathcal{P}/\mathcal{A}$ delivered by the external source which is driving the surface current is given by the negative of the electromagnetic power per unit area delivered by the electromagnetic fields. Then from Eqs. (3)-(5) and (7), we have

[TABLE]

where we have taken the electric field “at” the current sheet as the average $[\mathbf{E}_{r}+\mathbf{E}_{l}]/2$ of the electric fields on the two sides of the sheet. The Poynting vector on the right-hand side of the current sheet is

[TABLE]

and on the left-hand side is

[TABLE]

so that the outgoing power per unit area in the waves balances the power per unit area delivered by the driving source

[TABLE]

IV Oscillating Current Sheet and In-Coming Plane Waves

In the situation above, we considered the current sheet as generating the out-going plane waves on either side. However, if we simply add an appropriate source-free plane standing-wave solution, then we can obtain the situation corresponding to in-coming waves which are absorbed by the oscillating current sheet. Thus we introduce the source-free standing wave

[TABLE]

throughout the spacetime in addition to the retarded fields generated by the oscillating current sheet. Then the full electromagnetic fields are $\mathbf{E=E}_{S}+\mathbf{E}_{c},$ $\mathbf{B=B}_{S}+\mathbf{B}_{c},$ where $\mathbf{E}_{c}$ and $\mathbf{B}_{c}$ are the retarded fields generated by the current sheet, given in Eqs. (4) and (5). But the standing wave fields in Eq. (12) can be rewritten as running waves using

[TABLE]

We see that the out-going plane waves generated by the current sheet as given in Eqs. (4) and (5) serve to cancel the outgoing part of the standing plane wave leaving only the in-coming part of the standing wave. Thus by adding an appropriate source-free solution to the retarded field generated by the current sheet, we have obtained an entirely different situation where the signs of the energy in Eqs. (8), (9), and (10) are all reversed. The surface current is now absorbing the energy of symmetrical in-coming plane waves.

V Pulsed Oscillating Current Sheet and No Source-Free Radiation

We now go back to the situation where the source-free solution of Maxwell’s equations is assumed to vanish. If the driving source acts only during a time interval corresponding to half a cycle so that equation (3) holds not for all times $t$ but rather only for time $-\pi/(2\omega)<t<\pi/(2\omega),$ then there are two plane-wave pulses of radiation emitted,G499 so that equations (4) and (5) become

[TABLE]

We still have the energy balance involving the energy per unit area delivered by the external agent and the energy per unit area appearing in the external pulses. Thus the energy per unit area $U/\mathcal{A}$ delivered by the external agent corresponds to the time-integral of the power per unit area in Eq. (8)

[TABLE]

The energy per unit area in the right-hand pulse is

[TABLE]

and there is an equal energy per unit area in the left-hand pulse.

The momentum per unit area $\mathbf{P}_{r}/\mathcal{A}$ carried in the right-hand pulse is given by

[TABLE]

The momentum per unit area $\mathbf{P}_{l}/\mathcal{A}$ in the left-hand pulse has equal magnitude but is opposite in direction so that the total electromagnetic momentum vanishes.

The calculation which we have carried out corresponds to a traditional calculation in classical electromagnetism. What we want to emphasize is that the energy emitted by the surface current is symmetric on either side of the emitting source and that there is no recoil of the radiation source. This is the traditional view relevant when there is no interference with source-free radiation.

VI Interference of a Source-Free Wave with a Wave Generated by a Current

Pulse

VI.1 Source-Free Standing Plane Wave

Now we come to the essential calculation of this article. In addition to the radiation pulse generated by the pulsed current sheet, we will assume that there is a source-free plane wave present in the space. Although the use of a running plane wave gives extremely interesting interference behavior, we are particularly interested in particle-like behavior for waves. Therefore we will assume the presence of an electromagnetic standing plane wave which has no preferred direction of propagation. The source-free standing wave

[TABLE]

exists throughout spacetime, where $\omega=ck,$ and the phases $\phi$ and $\theta$ are fixed constants.

We can easily check that the standing wave fields in Eq. (19) indeed satisfy the source-free Maxwell equations. The source-free standing electromagnetic wave involves no time-average transfer of energy or momentum in space. The energy density $u(\mathbf{r},t)$ does indeed oscillate in time, since

[TABLE]

However, no energy crosses the nodes in the standing wave electric field, since

[TABLE]

which vanishes at the electric field nodes at $kz+\phi=n\pi,$ for integer $n.$ The energy per unit area oscillates back and forth locally within half a wavelength.

VI.2 Superposition of the Fields of the Source-Free Standing Wave and

the Pulsed Current Sheet

VI.2.1 Energy Aspects

If the external agent causes a pulse in the surface current $\mathbf{K}(t)$ at $z=0$ corresponding to Eq. (3) during the time interval $-\pi/(2\omega)<t<\pi/(2\omega)$ in the presence of the standing wave, then the total electromagnetic fields in spacetime will correspond to the superposition of the source-free fields in Eq. (19) plus the fields generated by the surface-current pulse as given in Eqs. (14) and (15).

The connection between the surface current amplitude $K_{0}$ and the amplitude $E_{0}$ of the radiated fields remains that given in Eq. (7). However, we notice that the power per unit area $\mathcal{P}/\mathcal{A}$ delivered by the external agent now involves not only the pulse fields but also the fields of the source-free standing wave. Thus we find

[TABLE]

The work done by the external agent which produced the surface current involves the time integral of the power per unit area in Eq. (22), giving

[TABLE]

where we have replaced $K_{0}$ by $(-c/2\pi)E_{0}$ from Eq. (7).

The electromagnetic fields in space involve the superposition $\mathbf{E}_{S}+\mathbf{E}_{r,l},~{}\mathbf{B}_{S}+\mathbf{B}_{r,l}$ of the standing wave fields $\mathbf{E}_{S},~{}\mathbf{B}_{S}$ and the pulse fields $\mathbf{E}_{r},$ $\mathbf{B}_{r},~{}\mathbf{E}_{l},~{}\mathbf{B}_{l}$ emitted by the surface current. The energy density in space corresponds to that of the standing plane wave, except in the spatial region where the pulse fields are present. The net energy per unit area above the energy per unit area of the standing wave is given by

[TABLE]

on the right-hand side of the current sheet. The integral involving the interference term for the electric fields can be evaluated as

[TABLE]

and the integral involving the magnetic fields can be evaluated in a similar fashion. We find that on the right-hand side of the current sheet, the electromagnetic field energy per unit area above the energy per unit area of the source-free standing wave is constant and is given by

[TABLE]

Similarly, the electromagnetic field energy per unit area above the energy per unit area of the source-free plane wave on the left-hand side of the current sheet is

[TABLE]

We notice that, as expected, the energy per unit area delivered by the external agent given in Eq. (23) fits exactly with the energy per unit area moving out on either side of the current source,

[TABLE]

VI.2.2 Momentum Aspects

In addition to providing energy, the external agent must now provide an external force per unit area acting on the surface current so as to balance the electromagnetic force per unit area on the surface current. The net force per unit area on the surface current arises from the standing wave alone, since the magnetic field due to the current pulses averages to zero at $z=0.~{}$ The force per unit area due to the external agent must be given by

[TABLE]

where we have used the magnetic field $\mathbf{B}_{S}(0,t)$ from Eq. (19). The net impulse $\mathbf{I}/\mathcal{A}$ per unit area delivered by the external agent is given by the time integral of the external force per unit area in Eq. (29), giving from Eq. (7)

[TABLE]

We can compare this impulse with the linear momentum carried in the pulses generated by the current sheet in the presence of the source-free standing wave. The momentum per unit area carried in the region of the right-hand electromagnetic pulse is given by

[TABLE]

Once again, it is easy to evaluate the integrals. We find

[TABLE]

so that the standing wave itself carries no net linear momentum when summed over half a wavelength. Next we have

[TABLE]

The integral involving $\mathbf{E}_{r}$ and $\mathbf{B}_{r}$ was given in Eq. (18). Thus we have finally

[TABLE]

so that the momentum on the right-hand side of the current sheet is constant in time and involves an interference between the source-free plane wave and the pulse created by the current sheet. An analogous calculation involving the pulse on the left gives

[TABLE]

As expected, there is momentum balance; the net impulse per unit area delivered by the external agent in the presence of the standing plane wave was given in Eq. (30), so that

[TABLE]

VI.3 Examples of Non-uniform Spreading of the Radiation Pulse Energy in

the Presence of Source-Free Radiation

VI.3.1 Symmetrical Outgoing Pulses of Positive Energy for Large

Radiation Pulses, $E_{0}>>E_{S}$

In the absence of the standing plane wave, the current sheet generates pulses (14) and (15) which carry energy and momentum away from the sheet in a symmetrical fashion, as given in equations (17) and (18). However, in the presence of the standing plane wave, the energy and momentum carried in the pulses depends upon the phases in both space and time of the standing wave relative to the oscillation of the current sheet, and also on the relative magnitude of the standing wave field and of the field generated by the surface current. If the magnitude $E_{0}$ of the field generated by the surface current is very large compared to the field $E_{S}$ of the standing wave, then the standing wave can be ignored; the pulses generated by the surface current carry equal amounts of energy and momentum, spreading out symmetrically from the current source. There is no recoil of the sources of radiation. This corresponds to the same situation where the source-free plane wave is absent, and is the situation generally assumed in the physics literature.

VI.3.2 Small Radiation Pulses, $E_{S}>>E_{0}$

If the magnitude $E_{S}$ of the plane wave is much larger than the field $E_{0}$ generated by the current source, then the interference terms involving $E_{S}E_{0}$ dominate the situation, while the terms in $E_{0}^{2}$ are negligible. The interference terms in $E_{S}E_{0}$ depend upon the relative phases $\phi$ and $\theta.$

Case $\phi=0,~{}\theta=0,~{}E_{S}>>E_{0}$

If $\phi=0$ while $\theta=0,$ then the current source introduces pulses of large positive energy which move symmetrically outwards from the current sheet. The external agent introduces an energy per unit area in Eq. (23) $U/\mathcal{A\approx[}E_{S}E_{0}/(4k)],$ but no net momentum in Eq. (30). If the spatial phase is $\phi=0$ while the temporal phase is taken as $\theta=\pi,$ then the current source generates large pulses of negative relative energy per unit area (holes in the standing wave energy) which travel outwards symmetrically. The energy per unit area delivered by the external agent from Eq. (23) is $U/\mathcal{A\approx-[}E_{S}E_{0}/(4k)],$ but again no net momentum is delivered. The pulses of radiation carry negative relative energy (holes in the standing wave) from Eqs. (26) and (27) $U_{R}=U_{L}\approx-E_{S}E_{0}/(8k)$ and momentum in Eq. (34) $\mathbf{P}_{R}/\mathcal{A\approx-}\widehat{z}E_{S}E_{0}/(8\omega)$ and in Eq. (35) $\mathbf{P}_{L}/\mathcal{A\approx}\widehat{z}E_{S}E_{0}/(8\omega);$ the negative energy moves outwards from the current sheet, but the momentum travels in the opposite direction from the direction of negative energy motion. The energy behaves as a “hole” in the standing wave energy.

Case $\phi=\pi/2,$ $\theta=\pi/2,$ $E_{S}>>E_{0}$

If the spatial and temporal phases of the plane wave are given by $\phi=\pi/2$ and $\theta=\pi/2,$ then the current pulse has introduced no net electromagnetic energy per unit area into the system since $U/\mathcal{A}\approx 0$ in Eq. (23), but the current pulse has introduced net linear momentum since $\mathbf{I}/\mathcal{A}=\widehat{z}E_{S}E_{0}/4\omega$ in Eq. (30). In this case, a positive energy per unit area travels to the right as in Eq. (26) with $U_{R}/\mathcal{A\approx}E_{S}E_{0}/(8k)$ carrying linear momentum as in Eq. (34) $\mathbf{P}_{R}/\mathcal{A\approx}\widehat{z}E_{S}E_{0}/(8\omega)$ , while a negative relative energy per unit area as in Eq. (27) with $U_{L}/\mathcal{A\approx-}E_{S}E_{0}/(8k)$ travels to the left acting like a “hole” in the plane wave carrying negative relative energy to the left and momentum from Eq. (35) as $\mathbf{P}_{L}/\mathcal{A\approx}\widehat{z}E_{S}E_{0}/(8\omega)$ to the right. If the sign of the temporal phase angle is reversed so $\theta=-\pi/2,$ then the directions taken by the positive and negative energy pulses are reversed.

VI.3.3 Radiation Pulses Matching the Source-Free Radiation:

Particle-Like Behavior

Case $\phi=\pi/2,$ $\theta=\pi/2,$ $E_{S}=E_{0}$

Perhaps the most interesting situation involves the case when the field $E_{0}$ of the pulse is of the same order as the amplitude $E_{S}$ of the standing wave, so that $E_{0}\approx E_{S}.$ In this case, if we have $\phi=\pi/2$ and $\theta=\pi/2,$ while $E_{S}=E_{0},$ then we can have one pulse of positive energy from Eq. (23) with $U/\mathcal{A=}$ $E_{0}^{2}/(4k)$ which is moving to the right, according to Eq. (26) with momentum given by Eq. (34) $\mathbf{P}/\mathcal{A=}$ $\widehat{z}E_{0}^{2}/(4\omega).$ There is no pulse to the left since Eqs. (27) and (35) both give vanishing contributions. Under this situation, the current sheet current recoils to the left and can be held in place by the impulse per unit area to the right exerted by the external agent, as in Eq. (30) with $\mathbf{I}/\mathcal{A=}\widehat{z}E_{0}^{2}/(4\omega)$ . If the phases are $\phi=\pi/2$ and $\theta=-\pi/2,$ then, the equations indicate that there is one electromagnetic pulse moving to the left while the current sheet recoils to the right. Qualitatively, this is the sort of behavior which is interpreted in terms of photons for the electromagnetic pulses. Energy and momentum are conserved, but the asymmetry of the radiation emission in association with the corresponding source recoil depends upon the existence of source-free radiation which is present at the time of the radiation emission.

VI.3.4 Average Behavior for Random Source-Free Radiation

In the discussions above, we have considered fixed phases in space and time for the standing wave. However, for thermal radiation or zero-point radiation, the appropriate description involves random phases. If we regard the phases $\phi$ and $\theta$ as independent random phases and average over the random phases, then the interference terms in Eqs. (23), (26), (27), (30), (34), and (35) all vanish. Thus when averaged over random phases, the average effect corresponds to electromagnetic waves spreading symmetrically out from the current source and gives no evidence of the particle-like behavior which holds for specific choices of relative phase.

VII Discussion

Although the current textbooks of classical electromagnetism do not mention the source-free radiation terms appearing in Eqs. (1) and (2), the accurate classical description of nature must include the source-free terms. The source-free radiation is crucial for understanding the behavior of a broadcasting antenna located in the radio waves from another source. Also, the source-free random radiation corresponding to classical zero-point radiation provides classical insights into aspects of modern physics. It has been pointed out that the presence of source-free classical electromagnetic zero-point radiation (as random classical radiation with a Lorentz invariant spectrum) allows us to understand a number of phenomena which are usually claimed to have explanations only in terms of energy quanta.any Thus if classical zero-point radiation is included as source-free random radiation, then there are natural classical explanations for Casimir forces, van der Waals forces, diamagnetism, blackbody radiation, and the absence of “atomic collapse.”

It was shown over 45 years ago that the fluctuations of thermal radiation could be understood purely classically, without photons, in terms of classical waves when blackbody radiation included classical zero-point radiation.fluct The present simple model calculation raises the possibility that what is currently regarded as individual photon behavior may perhaps also be understood in terms of conventional classical electrodynamics. Thus if zero-point radiation is assumed to be present, then any current source will interact with this source-free radiation, just as illustrated in our simple example. Since zero-point radiation is random radiation, the total angular radiation pattern corresponds to that found by averaging over all the random phases, and will behave symmetrically, as though the zero-point radiation interference were not present. However, the individual pulses of radiation will indeed be altered by the presence of zero-point radiation. If the current source generates an electromagnetic pulse where the electric field is much larger than the radiation fields of the random zero-point radiation in the same frequency region, then the presence of zero-point radiation can be ignored. This is the situation for macroscopic radiating antennas. However, if the current source generates an electromagnetic pulse where the electric field is comparable to the radiation fields of the zero-point radiation in the same frequency region, then the interference between the zero-point radiation and the radiation generated by the current source will be important. The idea of photons was taken up convincingly in connection with x-ray behavior which showed both classical interference effects in connection with Bragg scattering but also photon-like behavior in connection with Compton scattering. Of course, x-rays involve high frequency radiation where the zero-point radiation per normal mode $U_{0}(\omega)=(1/2)\hbar\omega$ is large and the wave-trains of emitted radiation are small. These are exactly the conditions where our simple example suggests one would find convincing particle-like behavior for the classical radiation pulses.

In this article, we have given a very simple example for the interference of source-free electromagnetic radiation with the radiation generated by a current source. The physics and mathematics of the example are at the level of an undergraduate text. However, the calculation may suggest a way of understanding the particle-like behavior of radiation pulses purely within the conventional wave theory of classical electrodynamics.

VIII Acknowledgement

I wish to thank Professor David J. Griffiths for helpful suggestions regarding this work.

Bibliography9

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1(1) We are using gaussian units which are natural to electrodynamics.
2(2) D. J. Griffiths, Introduction to Electrodynamics 4th edn (Pearson, New York 2013), p. 445.
3(3) J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, New York, 1999), 3rd ed., p. 246.
4(4) A. Zangwill, Modern Electrodynamics (Cambridge U. Press, 2013), p. 725.
5(5) A. Garg, Classical Electromagnetism in a Nutshell (Princeton U. Press, Princeton, NJ 08450, 2012), p. 204.
6(6) See, for example, Griffiths in ref. 2, p. 499, Problem 11.28.
7(7) See, for example, Griffiths in ref. 2, p. 397.
8(8) T. H. Boyer, “Any classical description of nature requires classical electromagnetic zero-point radiation,” Am. J. Phys. 79 , 1163-1167 (2011); ”Understanding zero-point energy in the context of classical electromagnetism,” Eur. J. Phys. 37 , 055206(14) (2016).

TL;DR

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Interference Between Source-Free Radiation and Radiation from Sources:

Abstract

I Introduction

II Source-Free Radiation and the General Solution of Maxwell’s

III Emitted Radiation from an Oscillating Current Sheet Source and No

IV Oscillating Current Sheet and In-Coming Plane Waves

V Pulsed Oscillating Current Sheet and No Source-Free Radiation

VI Interference of a Source-Free Wave with a Wave Generated by a Current

VI.1 Source-Free Standing Plane Wave

VI.2 Superposition of the Fields of the Source-Free Standing Wave and

VI.2.1 Energy Aspects

VI.2.2 Momentum Aspects

VI.3 Examples of Non-uniform Spreading of the Radiation Pulse Energy in

VI.3.1 Symmetrical Outgoing Pulses of Positive Energy for Large

VI.3.2 Small Radiation Pulses, ES>>E0E_{S}>>E_{0}ES​>>E0​

Case ϕ=0, θ=0, ES>>E0\phi=0,~{}\theta=0,~{}E_{S}>>E_{0}ϕ=0, θ=0, ES​>>E0​

Case ϕ=π/2,\phi=\pi/2,ϕ=π/2, θ=π/2,\theta=\pi/2,θ=π/2, ES>>E0E_{S}>>E_{0}ES​>>E0​

VI.3.3 Radiation Pulses Matching the Source-Free Radiation:

Case ϕ=π/2,\phi=\pi/2,ϕ=π/2, θ=π/2,\theta=\pi/2,θ=π/2, ES=E0E_{S}=E_{0}ES​=E0​

VI.3.4 Average Behavior for Random Source-Free Radiation

VII Discussion

VIII Acknowledgement

VI.3.2 Small Radiation Pulses, $E_{S}>>E_{0}$

Case $\phi=0,~{}\theta=0,~{}E_{S}>>E_{0}$

Case $\phi=\pi/2,$ $\theta=\pi/2,$ $E_{S}>>E_{0}$

Case $\phi=\pi/2,$ $\theta=\pi/2,$ $E_{S}=E_{0}$