I know that antennas can be capacitive or inductive depending on the relation of antenna length to the \$\lambda\$. I know that crystal radios use both antenna and capacitor for tuning and I don't understand why because for me it would seem the antenna should only be capacitive and you should only need inductor for tuning. Why is there a capacitor in crystal radios and why is it used for tuning?
4\$\begingroup\$ Just like a pendulum has two modes of energy storage ( movement, and height) to store energy, so does any other way to store energy, that is, to achieve a resonance. In electronics, the capacitor's electric field and the inductor's magnetic field are the two energy storage mechanisms usually used in radios. Your cellphone also uses quartz, with its own special two mechanisms, to precisely get the channels used. We can acid-etch the quartz more precisely to set a frequency, than we can build capacitors or inductors. \$\endgroup\$– analogsystemsrfNov 22, 2018 at 23:44
You are very astute. You can indeed construct a crystal set without a physical capacitor, relying instead on the capacitance of the antenna alone and tuning by varying the inductance. Of course, the value of inductance required will vary with the length of the antenna. Reckon on 10 pF per metre as a starting point. Perhaps a variable capacitor is popular because it can be arranged to dominate the antenna capacitance and make tuning less antenna length dependent. A wide range, reliable variable inductor can also present challenges. Go ahead and experiment.
It's a transmission-line thing. The electric current propagates down the antenna and, approximately, reflects from the antenna's end back toward the transmitter. Upon reflection, the current (but not the voltage) reverses.
By the time the reflected wave reaches the transmitter again—importantly, this does not happen instantaneously—the reflected wave (with current reversed) superimposes itself upon the signal the transmitter is generating at that moment.
Are you handy with your phasors? The impedance is the ratio of the sum of the voltages to the sum of the currents, phasor style. If the imaginary part of the ratio happens to be positive—and this depends on the wavelength compared to the antenna's length—then the antenna is inductive. Otherwise capacitive.
If you aren't sure that this makes sense, then here is a smaller idea to get you started: at the end of the antenna, the sum of the primary and reflected currents (but not voltages) must be zero because no current flows out the end of the antenna. This smaller idea does not explain the whole problem but it does serve to orient your mind so that, after you have slept on it, you might have a better frame to start to grasp the rest of the problem.
For the AM broadcast band, few long-wire antenna builders erect a full quarter-wavelength wire (about 75 m) and settle for a shorter wire. A shorter antenna benefits from added inductance to compensate for its capacitive reactance.
Once you include all the resistive components (don't forget ground losses), this inductor-compensated antenna has very low Q . To effectively select one station from another, an LC resonator should have loaded Q of at least 100. So you often see an antenna loosely coupled to a high-Q LC resonator.
Why is "C" varied rather than "L" for tuning?
Compare prices for each. A variable capacitor maintains high Q over its very large capacitance range. Variable inductors seem mechanically complex - some air-core types are bulky. Both tuning methods have been employed - there was a time when car radios used ferrite-slug-tuned inductors for tuning. These were not crystal radios.
The standard crystal set has an antenna that is electrically short so it is capacitive depending on antenna length and wire thickness, wire insulation, ground proximity etc. So precise Capacitance is install dependent. The ground resistance will generally be much higher than the radiation resistance because of non ideal ground conductivity.
The hobbyist is not likely to do 360 buried ground radials. Hence a loosely coupled antenna with a known high Q tuned circuit will give more predictable results. Remember that it was common in the 1930s for radio shops to make and sell crystal sets whilst just selling the more complex superhets. This LC tuned circuit can have a variable cap and a fixed inductor which is what you normally see. You can do a fixed cap and a variable coil. I have seen this as a child in 1969 where a tuning knob was attached to a slug tuned coil and it worked fine. For mobile stuff like car radios there is a vibration advantage to using permeability tuning but crystal sets are almost always stationary.