It's either math or magic. It seems like the only thing that will explain operation are complex formulas.
If a child were to ask "how does an antenna work" what should one say, it magically changes electricity into invisible photons?
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Say you have a positively charged object, and a negatively charged object. They will be attracted together, like opposite poles of a magnet.
Now if you pull these two objects apart, there will exist an electric field between them. If you were to place an infinitesimal test charge somewhere around your separated charged objects, you'd find if the test charge is positive, then it will be attracted to your negatively charged object, and repelled from the positively charged object.
If your two objects have been in the same position, with the same charge, for an infinite amount of time, then as we get farther away from them the force experienced by the test charge is weaker, but it never goes away. That is, for any point in space, no matter how far, we can calculate some force that will be experienced by the test charge. If we were to calculate the direction and magnitude of the force that would be experienced by the test charge for every possible location in space, this is the electric field.
But what if your two charged objects move? Now the electric field will be something else. If we instantly reversed the positions of your two charged objects, then the electric field around them would be more or less the same, except in the opposite direction.
However, the field can't change instantly -- that would violate causality. Instead, the new field propagates outward at the speed of light. If you are very close to these charged objects, then the field would seem to change instantly. If you are one light-year away, then the field you observe at your location reflects how the charged objects were one year ago.
Now, consider a simple dipole antenna. It has two halves. Our transmitter sucks electrons out of one side (making it positively charged) and shoves electrons in the other side (making it negatively charged). It is very much like the two charged objects above. An electric field begins to propagate out...
And then, the transmitter changes direction, and now the charges of each half of the dipole are reversed. This change in the field is also propagating outwards. This reversing of charges continues at whatever our transmit frequency is. If it is 7 MHz, then the charges reverse 7000000 times per second. If you are not moving relative to the antenna, then at one instant the electric field is pointing one way, and then shortly later the electric field is pointing the other way. The wavelength is the distance traveled by light in the time it takes for the antenna to go through one of these oscillations. It's how far the change in the electric field will have propagated by the time we start another cycle.
Remember, we said the electric field allows us to predict the force that an imaginary test charge would experience at that point. What if the test charge isn't imaginary? Like, the mobile electrons in a receiving antenna?
Not surprisingly, the charge in the receiving antenna experiences a force. Because the antenna is a good conductor, this makes a current in the antenna, which the receiver can then amplify and demodulate.
For further reading, I highly recommend A Visual Tour of Classical Electromagnetism.
You know how sound waves let me talk to you? You can't see them, but you can hear them. Light is similar, you can send it from a lightbulb or flashlight, and receive it with your eyes and cameras, but you can't hear it.
Antennas use electromagnetic waves. You can't hear or see them, but antennas can send and receive them.
Maybe compare an antenna with an electric heater or incandescent bulb? If you feed the heater with electricity, you can 'feel' the radiation. Similar for a light bulb, where you can 'see' the radiation (and feel it, if you're not careful).
For a less correct, but more visual approach: stick a pole in water and move it - you'll see the waves irradiated by the movement. Hey, you can even see the standing waves there, and reflections.
Radio waves travel at the speed of light, and although the electrical current that generates the magnetic and electric fields travel almost as fast, the fields have already escaped so the transmitter cannot change them, only make new fields. It's like once you have said something, you cannot take it back. Dropping a stone in water causes waves to travel away from the spot where the stone was dropped, but dropping another stone cannot change the waves that have already escaped.
An antenna is similar to a lightbulb.
Lightbulbs spit out photons at a wavelength we can see. These are roughly in the range of 400 to 900 nanometers... a SUPER short wavelength compared to our everyday experiences.
A radio wave emits the same photons, at a much larger wavelength, for example about 3 meters when looking at the FM broadcast dial in the US.
In order to generate radio waves, a property that can be referred to as resonance (think about 2 tuning forks that resonate with each other when just the first is struck) greatly controls what wavelength of photons/EMR an antenna of a particular size might be able to spit out and detect.
Over-simplistically, the radios that the antennas are plugged into convert voices (or data) into on/off patterns that the antenna & radio on the other side knows how to convert back into voices (or data). In reality, there's MUCH more to the encoding of information than just on/off patterns, but it's a reasonable analogy.
I like to think of a resonant antenna like a guitar string that vibrates at a given frequency that you can hear. Tighten/shorten the string, the pitch/frequency increases; loosen/lengthen the string, the pitch decreases.
Similarly for a resonant antenna like a dipole: shorten the wire, the resonant frequency increases (wavelengths become shorter); lengthen the wire, the resonant frequency decreases (wavelengths become longer).
This doesn't describe how or why the RF leaves the antenna and propagates through the air, and maybe it's not an entirely correct comparison to make (the guitar string is vibrating air molecules that carries the sound to your ear, but the RF doesn't vibrate the air to carry itself), but since we're talking about different frequencies on the electromagnetic spectrum there's probably some truth in part of this explanation. At least it's a comparison with something that someone would be more familiar with so it helps understand the concept.
One way I have explained to kids (and to myself) is to imagine a jump rope tied to a tree or a post. You start moving it up and down, soon it makes a nice wave. If you continue to move it up and down, it continues to keep waving. That is a radio transmitting into a wire. Now if the rope were dipped in lots and lots of paint, each time the wave peaked, imagine the paint color flinging off straight up into the sky in a wave form. That's kind of what the excited electrons do in the wire, they are so small, they simply fling an image of themselves out into space in the form of photons or light particles. The analogy breaks down after a while, but it is also easy to use this example to teach the principle of a standing wave. Maybe not the most elegant, but for non scientific types it bridges the concepts when too much technical jargon might turn them off. Apologies to the purists. JP
Two things and maybe getting back to the five year old's view of the world.
A tuning fork. the sound waves/vibrations from one tuning fork, when struck, radiate out bumping one molecule into another until reaching the other tuning fork. when the lengths of the elements are the same the energy of the wave is absorbed by the second tuning fork where it then begins to emit the same sound/vibration as the first one.
A soap bubble and the ring used to blow it. They have a relationship. if you use the same size ring you can catch a bubble. If it is not similar enough the bubble pops.
Electro Magnetic Mechanics are not quite the same as either of these structures but this helps communicate a handle that can lead to better understanding.