From here

To generate a long wavelength requires an aerial of roughly one wavelength in size.

and here

One of the difficulties posed when broadcasting in the ELF frequency range is antenna size, because the length of the antenna must be at least a substantial fraction of the length of the waves.

I gather that a charge has to be moved along a straight distance similar to the wavelength of an EM wave to generate an EM wave of that frequency.

If a straight conductor has wavelength in size, does an electron have to move from one end to the other and back to generate the wave ?

This seems totally different than the distances involved in electron jumps between electron orbitals for generating a given wavelength.

Regardless of the above assumptions being true or false please detail: What are the movement, speed and accelleration of a charge required to generate an EM wave of a given wavelength?

related: How do near-field EM fields change to far-field EM waves at an antenna

EDIT: Although answers here shed light on the mistakes of my assumptions, none provides a clear answer to the question in bold. I had to to ask another question to get the proper answer : a charge oscillating in space at a given frequency will generate an EM wave of the same frequency

  • $\begingroup$ I'm not sure why you're asking about the movement of a single charge in the context of an antenna. The conductor that forms the antenna is full of mobile charge. When the antenna is driven, the (time varying) density and drift velocity of the mobile charge varies along the length of the antenna. So a single point charge moving back and forth doesn't have much in common with a long wire antenna. And the EM waves from an antenna have little to do with photons emitted by "electron jumps". It just isn't clear to me what your exact question is. $\endgroup$ – Alfred Centauri Jan 8 at 14:10
  • $\begingroup$ im trying to get to the underlying mechanism of EM generation. I would expect a bunch of charges to behave like the sum of its constituents. The electron jump was just there to show that the antenna length requirement is weird. I now believe it's just an optimization (for resonance) and not a hard requirement. $\endgroup$ – Manu de Hanoi Jan 9 at 3:08

You are confusing two frameworks. The classical electromagnetic wave generation , and the quantum mechanical, where electrons live and are modeled.

An antenna works mathematically well with the classical Maxwell equations which predict radiation when the fields across it change.

What are the movement, speed and accelleration of a charge required to generate an EM wave of a given wavelength?

It is the maxwell equations that have to be solved for a speciific antenna design (boundary conditions) and will have the quantities you ask directly or one can derive them from the solutions.

Electrons in a conductor are modeled with the quantum mechanical band theory of solids. Electrons in the conduction band over the whole metal lattice, building up the charge distributions that are necessary for the classical theory to work, when the fields are changing. The classical emerges from the quantum mechanical , but one needs mathematical treatments in order to demonstrate this. See the complication of demonstrating how classical electromagnetic fields emerge from quantum ones here.

  • $\begingroup$ thanks for your answer, would it be correct to say that any charge oscillating in space (regardless of the spacial amplitude) at a given frequency will emit a wave of the same frequency? $\endgroup$ – Manu de Hanoi Jan 9 at 2:56

Accelerating a charge in an oscillatory fashion at a given frequency will generate electromagnetic waves regardless of the length of the path over which the charge is accelerated.

The main reason for selecting a particular antenna length is to make it resonant to the frequency it's designed to transmit or receive. See, for example, Wikipedia - whip antenna:

To reduce the length of a whip antenna to make it less cumbersome, an inductor (loading coil) is often added in series with it. This allows the antenna to be made much shorter than the normal length of a quarter-wavelength, and still be resonant, by cancelling out the capacitive reactance of the short antenna. The coil is added at the base of the whip (called a base-loaded whip) or occasionally in the middle (center-loaded whip). In the most widely used form, the rubber ducky antenna, the loading coil is integrated with the antenna itself by making the whip out of a narrow helix of springy wire.

  • $\begingroup$ If I understand this correctly, the resonnance length of the antenna is just there as an optimzation factor to help amplification, but in theory any length could be used. Just like an electric guitar doesnt need a resonance cavity of a classic guitar because it has powered amplification. I still have trouble understanding why resonance matters for an electrically powered emitter, especially a low frequency one as shown in the 2nd link to ELF ( why need resonance when you have amplificators). Also why would a fraction of a wavelength help ? I though a wavelength was needed for standing waves $\endgroup$ – Manu de Hanoi Jan 9 at 3:15

The answers provided so far have provided some insight into the challenge of the question as posed. But it's not a challenging question. It just has to be posed properly; that is to say, it should be stated not in terms of macroscopic charge and current, but in terms of electromagnetic force.

First, to quickly define the notion of the electric and magnetic fields. Early observations in electricity showed charged objects (e.g. the famous rubbing of amber, causing the amber to pick up dust and things) attracting and repelling one-another, without being in contact, thus the electric force was defined. Magnetism was of course observed as a force between something like lodestone and iron ore. Currents produce similar force. Thus we have magnetic force. It is useful to abstract the forces, for mathematical reasons, to vector fields. The electric field is the force per charge at a point in space, and is thus represented by a force vector. The magnetic field is dependent on current, the macroscopic flow of charge we now know, and in fact the force on an object is not absolute, but depends on the movement of the object. All force is restricted to one plane of force, and the magnetic field vector defines this plane by pointing normal to it with a magnitude indicating the magnitude of the force per charge per velocity. Thus the magnetic field vector is not a force vector, but an area vector, representing the plane of forces possible.

Okay, we now have those cleared up. From Maxwell's equations in differential (point) form in free space, that is, in the absence of charge or current densities, we arrive at the Helmholtz wave equation, $$ \nabla^2 \mathbf{E} = \frac{1}{c^2} \frac{\partial^2 \mathbf{E}}{\partial t^2} $$ Which, as has appeared at various points in mathematical physics, is a 3D wave equation. It holds any time $\rho = 0, \mathbf{J}=0$. That is there is no charge or current at a given point. An analogous equation holds for $\mathbf{H}$, the magnetic field.

That means every electromagnetic field obeys a 3D wave equation when in free space. This includes electrostatic solutions, electrodynamic solutions, everything in free space.

While understanding what the wave equation says is nice and enlightening to some extent, the point I'm hoping to make is that the idea of the electromagnetic field is almost like a pool of water. Every change causes a "wave" in that it is a solution to the wave equation. It essentially implies that changes in the electromagnetic field are not instantaneous, but propagate at speed $c$ (the speed of light). So what causes electromagnetic waves? Anything that results in electromagnetic fields in space. That is, any current or charge distribution (macroscopically speaking).

By making simplifying assumptions, most often by assuming that at any given point the direction of wave propagation is well defined, and perhaps by imposing boundary conditions, we can make a very insightful study of electromagnetic wave propagation in terms of sinusoidal waves in space and time. Combining this with Fourier analysis gives a convenient formulation of electromagnetic waves in terms of frequency content, which can be helpful. We also discover the requirements for the propagation of different propagation "modes", where a propagation mode can be considered any arrangement of the fields which produces changes in time according to the real part of $e^{j\omega t}$ at each point and changes in space according to the real part of $e^{-j\beta z}$ for propagation in the $\hat{z}$ direction with phase/propagation constant $\beta$ ($\gamma$ is the general propagation constant, with real and imaginary/phase components). This means that not only does the field at any given point change magnitude and direction cyclically, but the same thing occurs as we move in space.

With such a simplification, the wave equation above gives a number of special cases, namely transverse electromagnetic (TEM) waves, with the electric and magnetic field vectors both transverse to propagation direction, as well as transverse electric (TE) and transverse magnetic (TM) waves, which must propagate in between particular boundaries to exist. We can also look at spherical waves, which are quite common, as you can imagine. Even among these special cases there are further special cases, which allow us to calculate exact field solutions analytically (without computer calculation, numerical solutions) for some geometries. This is really the domain of radio (RF), microwave, and optical engineers and scientists.

However, it is the antenna engineers and scientists who study the radiation of electromagnetic waves. I do a lot of microwave frequency work, and I thought I had it bad with vector calculus, but the classical treatment of antennas is downright tedious.

The topic of radiation, then, focuses on what happens at an interface between e.g. a conductor and free space. Moreover, what exactly do the distant fields look like, and how efficient is the transfer of power/energy? That's a harder question. It's best tackled using the scalar and vector potentials and a lot of vector calculus. That's how it was done by Maxwell, and thats how it's taught today, there's not much getting around it. It's a difficult topic.

All this comes about in terms of classical electrodynamics, which was developed mostly in the 19th century by mathematical physicists and mathematicians including Laplace, Poisson, Green, Gauss, Hamilton, d'Alembert, and Maxwell. Experimentally, Hertz was the first to demonstrate electromagnetic waves. It's worth noting, however, that even Hertz, performing the first experiments in EM wave behavior, noticed the inability of classical electrodynamics to explain some phenomena (in this case, the photoelectric effect). So in essence, when you look closer, it's worth modifying classical electrodynamics into QED and relativistic classical electrodynamics. This is where the idea of charge falls apart, as mentioned by previous answers. Anyway, I hope this is helpful. Just remember that the true insight is in the mathematics, not the stories we tell about the mathematics.

  • $\begingroup$ I believe stories come first then the math follow (and that's why math has proven effective in physics, it was created based on physics), that's why im having such trouble making progress in physics, most academics teach the other way around. I found your intro on the fields & wave definitions enlightening, but doubt very much your conclusions regarding charge existence and QM. My postulate is that EM fields arent quantized, but the devices used to measure them are. $\endgroup$ – Manu de Hanoi Jan 19 at 13:46
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    $\begingroup$ The relationship between theory and experiment in physics is definitely not clean cut, there's a lot of room for stories and math. However, it's important to understand when one comes from another. In early theories of electrostatics, people proposed mechanisms and listed observations, no math formalization required. Faraday was arguably the last successful non-mathematical physicist, and his work made a huge impact on Maxwell and his mathematical formulation. However, the most fascinating results came from putting the math down, and saying "if this is true, what else must be true?" [Contd..] $\endgroup$ – Sam Gallagher Jan 19 at 14:05
  • $\begingroup$ [Continuing...] The logical aspects are represented most generally in the math, not the stories. It was from math that maxwell derived the possible existance of EM waves (building on mathematical work of those listed in my answer) among other fascinating results. The fact was, given everything they knew, the math implied that such fields must obey the same type of relation as 3D waves. Without a physical explanation, only general math, the idea was proposed, and interpretation followed. The practical aspect came from Hertz, after much theoretical work. $\endgroup$ – Sam Gallagher Jan 19 at 14:10
  • $\begingroup$ In general, classic electrodynamics assumes continuity, no quantization. However even Hertz, the first person to experiment with EM waves, noticed the inability of classical ED to explain certain phenomena, namely, particle physics. The classical ED still holds, but only for more macroscopic scales, sort of like an "average" behavior, and in this it is extremely accurate. No quantization required, the effects of quantization are often noticed only when looking at particle experiments (cathode rays for example) in the presence of EM waves. $\endgroup$ – Sam Gallagher Jan 19 at 14:16
  • $\begingroup$ I hear you, and dont really disagree as it doesnt really contradict my postulate. The math that I am vaguely aware of regarding QM probably doesnt either. But think about einstein & relativity, he had the story first then worked 15 years on ironing the math. Breakthough dont come from math because any mathematician can unroll the equations. Maxwell, I heard, just formalized and grouped stuff from other people. Quantization can emerge from discrete resonance modes of EM waves in certain conditions. That doesnt mean EM waves must always come quantized. $\endgroup$ – Manu de Hanoi Jan 19 at 15:28

If a straight conductor has wavelength in size, does an electron have to move from one end to the other and back to generate the wave?

No, this is not the case. To explain it, let’s see what happens with the emission of EM radiation in a DC circuit. (It’s possible to run such a wire with a powerful DC source so that the wire glows. This is a strong evidence that the wire emits EM radiation. Of course the wire emits EM radiation even with a low power source, but this time only in the invisible range.)

The emission happens from the moving free electrons in the metal. These electrons are bouncing forth and back on the atomic structure of the wire, but in sum are moving with the drift velocity. The conclusion is that although not using an alternating current, it is an EM radiation induced. The emission happens in between the free path of the electrons. The wavelengths of the emitted EM radiation depends from the accelerations of the electrons (from the voltage, the wire material, the temperature ...).

Interrupting periodically the DC source one get nearly a square wave signal. Running the switch on high frequencies the signal becomes more and more a sine wave.

I gather that a charge has to be moved along a straight distance similar to the wavelength of an EM wave to generate an EM wave of that frequency.

The above said could be applicated to an AC device and an open circuit. Instead of moving the electrons periodically step by step along the closed circuit with a triggered AC device, the electrons move forth and back with less the drift velocity of around 0,1 mm/s and not at all with a velocity of 300,000 km/s. So again, the electrons don’t travel along a straight distance similar to the wavelength.

But of course, the smaller the antenna frequency, the more electrons reach the end of the open circuit and not having the right length, the antenna is not resonant and the efficiency of the EM radiation drops down. Installing a inductive coil, we can damp the current. Or adding a capacity hat or top load we can temporarily collect the arriving on the rods end electrons in the capacitor.

  • $\begingroup$ thanks for highlighting that my assumption would require electrons to move at the speed of light $\endgroup$ – Manu de Hanoi Jan 19 at 13:30

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