This question has been asked probably many times, and is inspired by area that appears in denominator of energy flux unit [W/m^2]. From what i have read so far, i come to conclusion, that this post should, first of all, try to explain that this question makes some sense.
First thing that should be explained is FIELD AREA (e.g. electric field area). We can substitute into mathematical expression of Coulomb's law (F = k * q1q2/r^2) any distance (r), and we always get specific force value. That's the reason for which is often assumed that the field is INFINITE. But it doesn't mean that we can not talk about it's shape and size. By assuming some numerical value (E) of electric field, we can draw an outline/contour (bounding surface) by connecting all the points in space around charge having this value [Pic. 1]. So, we've already got shape of the field (spherical, oval, etc.) and it's location relative to electric charge. Additionally, if we repeat this procedure for stronger charge, we can start talking about size, at least in comparative context (in this case: cross-section area of 2 times stronger charge will be 2 times bigger). We can also verify that all of this applies to any boundary value (E). So even if we substitute infinity for distance (r) or 0 for force (F), these rules will be fulfilled - it just mathematiacal apparatus we use, can not cope with such types of values. In other words: infinity of interaction doesn't mean that we can not talk about size of electric/magnetic field.
Let's imagine that we've a simple antenna [Pic. 2a] - piece of straight wire in vertical position. In this antena there is an electron (for this moment just one) which can move up and down. Doing so, it generates an electromagnetic (E-M) wave radiating in all directions (for simplicity let's constraint this model to 2 dimentions, like top-view picture [Pic. 2b]). I'm using this model intentionally because it's the best one i've found so far (unfortunately i don't remember address of that thread), in an attempt to explain absurdity of question about E-M wave's surface. The best doesn't mean correct (more precisely: not complete). Anyway, i use it because it provides a good base for further analysis.
This model shows that in practice we never generate a single E-M wave, but rather we are dealing with radiation which, at greater distances from the source, turns into almost PLANE WAVE (curvature of wave front decreases, area covered by radiation increases). That's why answer to the question "How to convert energy flux [W/m^2] to energy [J]?" often is: Multiply flux by time (this is understandable) and by area. What area? The area we are interested in, the area our illuminated object has, the area of the slit through our plane wave has passed - in other words: ANY area we want. It's easy to understand in context presented before, but usually it's not the intention of those who are asking. The question concerns the spatial size(extension) of field (e.g. electric) of SINGLE E-M WAVE with some given amplitude.
A single E-M wave? What does it mean? Does something like that even exist? Look at the picture, there are two patches/areas [Pic. 2] A1,A2 (at different distances from source) by which a couple of rays (let's consider that they are our single E-M waves) coming through. According to energy conservation law, the same amount of energy has to pass through both of them. So, the only possibility is to lower E-M wave amplitude, and this, in turn, is only possible by moving apart our rays, relative to each other. This process can be seen as opposite to interference. In the latter, when two separate waves approach each other, they E-M fields become indistinguishable and result is treated as a single wave (with doubled amplitude, of course). When moving apart, there should be some minimal distance from which we would be able to see two separate waves again.
But we also know, that all E-M radiation is quantized, it can not divide indefinitely - it can do this just up to the photon level. We should perform a simple experiment: Let's move our electron a little bit (for example: one time up and down - or something like that), just to generate a smallest unit of E-M radiation we can (the shorter the better, in ideal case it will be the shortest unit of E-M wave which can travel freely in space, but it's not a must). We know how much work we have done and how much energy E-M radiation has. We have to divide this by energy of single photon and we've got number of photons in the system. Let's define some minimal distance on projection screen (our measurement equipment) we can distinguish one photon from another and multiply it by photon count. We get some length, this is circumference of a circle. We just have to calculate the radius - distance from radiation source where we should perform our measurements (place projection screen there).
Now, everything depends on what pattern we will see: (1) single photons distributed evenly along screen [Pic. 3a] - it means that photons are generated with uniform spatial distribution of direction - direction of momentum, or if you like: direction of wave vector. It may be partially the result of some hyphotetical process like, let's call it for example: quantum dispersion (photons can diverge sponaneously, with some probability), but in practice it should have different characteristic [Pic. 4a] (be recognizable) then geometrical (dominant) one [Pic. 4b]. Conclusion: indeed, there is no such a thing like single E-M wave/ray in this dimension. In case (2) photons concentrated around some specific points [Pic. 3b] - it means that there are some predefined radiation directions (for example, due to the fact that angles are quantized in some way).
In case #(2) we've got already our single E-M wave! In case #(1) we have to continue searching in other dimensions. Let's generate again some smallest unit of E-M radiation, if photons hit exactly the same places as before [Pic. 3c], then we have found even something better: a single E-M wave with the smallest possible amplitude (stream of photons following each other with properties of continuous E-M wave). This can be called not only single but rather elementary E-M wave. If this also fails, it's time for last resort: so far our example has operated in 2-D (two dimensions) - it was like one slice. We have to take into account third dimension. There are other electrons in vertical wire (below, above), and they move synchronously with the first one. In this dimension, for this type of antenna photons/rays shouldn't diverge from each other (at least they should do it with different charachteristic - slower then in horizontal plane). If such photons are horizontally aligned (with those generated at other altitudes), we should be able to see vertical stripes on the projection screen [Pic. 3d]. Even if we can't get continuous wave this way, we still get a piece of such single wave. This is something more then a photon. A single photon is to small to perform measurement on it, but this piece of wave can have measurable amplitude (this is still a bunch of photons but with limited horizontal distribution - a "thin" wave) and it's E-M field strength is enough to extend to measurable sizes (E-M field is continuous, it can not just drop instantenously in one place. The stronger the source is, the larger is the area that can be detected using standard methods).
How to test all of this in practice? Maybe we don't have to perform all those measurement hundreds,thousand or millions miles away from the source of radiation (Note: Please, treat the following examples only as a theoretical, they are just to show very general idea, not real techniques). First, we can filter tight slice from radiation emitted in all directions (we can use narrow slit) - this will be our initial ray. Then, we can use for example two opposite mirrors to filter it further [Pic. 5a] (simulate longer distance). In theory, light can travel between them almost indefinitely (attenuation if present should be uniform, and maybe even can be helpful in faster reaching the final conditions). Perfecty flat surface of mirror doesn't change angle of incoming ray, just direction (the angle of incidence equals to the angle of reflection), so all we have to do is to wait for a while. If we worry about the interference of rays (incidented and reflected), we can use different mirror's configuration [Pic. 5b] . We can take advantage of the fact that we are only testing small fragment of wave not continuous one, and this fragment has to be shorter then 4L (ray can travel inside box xN times, and later be released on demand). You should also notice that all those mirrors are small, it allows rays diffracted too much to escape from box and without producing too much mess (E-M noise) inside. As i said, it was just theoretical example. Maybe there are better ideas? Anyone knowns about any similar experiments or can recall any knowledge/facts i've missed?
First thing that should be explained is FIELD AREA (e.g. electric field area). We can substitute into mathematical expression of Coulomb's law (F = k * q1q2/r^2) any distance (r), and we always get specific force value. That's the reason for which is often assumed that the field is INFINITE. But it doesn't mean that we can not talk about it's shape and size. By assuming some numerical value (E) of electric field, we can draw an outline/contour (bounding surface) by connecting all the points in space around charge having this value [Pic. 1]. So, we've already got shape of the field (spherical, oval, etc.) and it's location relative to electric charge. Additionally, if we repeat this procedure for stronger charge, we can start talking about size, at least in comparative context (in this case: cross-section area of 2 times stronger charge will be 2 times bigger). We can also verify that all of this applies to any boundary value (E). So even if we substitute infinity for distance (r) or 0 for force (F), these rules will be fulfilled - it just mathematiacal apparatus we use, can not cope with such types of values. In other words: infinity of interaction doesn't mean that we can not talk about size of electric/magnetic field.
Let's imagine that we've a simple antenna [Pic. 2a] - piece of straight wire in vertical position. In this antena there is an electron (for this moment just one) which can move up and down. Doing so, it generates an electromagnetic (E-M) wave radiating in all directions (for simplicity let's constraint this model to 2 dimentions, like top-view picture [Pic. 2b]). I'm using this model intentionally because it's the best one i've found so far (unfortunately i don't remember address of that thread), in an attempt to explain absurdity of question about E-M wave's surface. The best doesn't mean correct (more precisely: not complete). Anyway, i use it because it provides a good base for further analysis.
This model shows that in practice we never generate a single E-M wave, but rather we are dealing with radiation which, at greater distances from the source, turns into almost PLANE WAVE (curvature of wave front decreases, area covered by radiation increases). That's why answer to the question "How to convert energy flux [W/m^2] to energy [J]?" often is: Multiply flux by time (this is understandable) and by area. What area? The area we are interested in, the area our illuminated object has, the area of the slit through our plane wave has passed - in other words: ANY area we want. It's easy to understand in context presented before, but usually it's not the intention of those who are asking. The question concerns the spatial size(extension) of field (e.g. electric) of SINGLE E-M WAVE with some given amplitude.
A single E-M wave? What does it mean? Does something like that even exist? Look at the picture, there are two patches/areas [Pic. 2] A1,A2 (at different distances from source) by which a couple of rays (let's consider that they are our single E-M waves) coming through. According to energy conservation law, the same amount of energy has to pass through both of them. So, the only possibility is to lower E-M wave amplitude, and this, in turn, is only possible by moving apart our rays, relative to each other. This process can be seen as opposite to interference. In the latter, when two separate waves approach each other, they E-M fields become indistinguishable and result is treated as a single wave (with doubled amplitude, of course). When moving apart, there should be some minimal distance from which we would be able to see two separate waves again.
But we also know, that all E-M radiation is quantized, it can not divide indefinitely - it can do this just up to the photon level. We should perform a simple experiment: Let's move our electron a little bit (for example: one time up and down - or something like that), just to generate a smallest unit of E-M radiation we can (the shorter the better, in ideal case it will be the shortest unit of E-M wave which can travel freely in space, but it's not a must). We know how much work we have done and how much energy E-M radiation has. We have to divide this by energy of single photon and we've got number of photons in the system. Let's define some minimal distance on projection screen (our measurement equipment) we can distinguish one photon from another and multiply it by photon count. We get some length, this is circumference of a circle. We just have to calculate the radius - distance from radiation source where we should perform our measurements (place projection screen there).
Now, everything depends on what pattern we will see: (1) single photons distributed evenly along screen [Pic. 3a] - it means that photons are generated with uniform spatial distribution of direction - direction of momentum, or if you like: direction of wave vector. It may be partially the result of some hyphotetical process like, let's call it for example: quantum dispersion (photons can diverge sponaneously, with some probability), but in practice it should have different characteristic [Pic. 4a] (be recognizable) then geometrical (dominant) one [Pic. 4b]. Conclusion: indeed, there is no such a thing like single E-M wave/ray in this dimension. In case (2) photons concentrated around some specific points [Pic. 3b] - it means that there are some predefined radiation directions (for example, due to the fact that angles are quantized in some way).
In case #(2) we've got already our single E-M wave! In case #(1) we have to continue searching in other dimensions. Let's generate again some smallest unit of E-M radiation, if photons hit exactly the same places as before [Pic. 3c], then we have found even something better: a single E-M wave with the smallest possible amplitude (stream of photons following each other with properties of continuous E-M wave). This can be called not only single but rather elementary E-M wave. If this also fails, it's time for last resort: so far our example has operated in 2-D (two dimensions) - it was like one slice. We have to take into account third dimension. There are other electrons in vertical wire (below, above), and they move synchronously with the first one. In this dimension, for this type of antenna photons/rays shouldn't diverge from each other (at least they should do it with different charachteristic - slower then in horizontal plane). If such photons are horizontally aligned (with those generated at other altitudes), we should be able to see vertical stripes on the projection screen [Pic. 3d]. Even if we can't get continuous wave this way, we still get a piece of such single wave. This is something more then a photon. A single photon is to small to perform measurement on it, but this piece of wave can have measurable amplitude (this is still a bunch of photons but with limited horizontal distribution - a "thin" wave) and it's E-M field strength is enough to extend to measurable sizes (E-M field is continuous, it can not just drop instantenously in one place. The stronger the source is, the larger is the area that can be detected using standard methods).
How to test all of this in practice? Maybe we don't have to perform all those measurement hundreds,thousand or millions miles away from the source of radiation (Note: Please, treat the following examples only as a theoretical, they are just to show very general idea, not real techniques). First, we can filter tight slice from radiation emitted in all directions (we can use narrow slit) - this will be our initial ray. Then, we can use for example two opposite mirrors to filter it further [Pic. 5a] (simulate longer distance). In theory, light can travel between them almost indefinitely (attenuation if present should be uniform, and maybe even can be helpful in faster reaching the final conditions). Perfecty flat surface of mirror doesn't change angle of incoming ray, just direction (the angle of incidence equals to the angle of reflection), so all we have to do is to wait for a while. If we worry about the interference of rays (incidented and reflected), we can use different mirror's configuration [Pic. 5b] . We can take advantage of the fact that we are only testing small fragment of wave not continuous one, and this fragment has to be shorter then 4L (ray can travel inside box xN times, and later be released on demand). You should also notice that all those mirrors are small, it allows rays diffracted too much to escape from box and without producing too much mess (E-M noise) inside. As i said, it was just theoretical example. Maybe there are better ideas? Anyone knowns about any similar experiments or can recall any knowledge/facts i've missed?
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