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I've been looking at crystal radios recently, and I noticed that many of them had quite big coils (maybe around 5cm on average from what I can see), but wouldn't they want quite a low value inductor to get the right frequency without using an absolutely miniscule capacitor? I know the size would have been quite useful for tapping, but wouldn't they have still wanted them as small as possible?

These are some of the images I found, they are all modern recreations and kits, but ones made to look like the old ones:

Crystal radio with big coil 1

https://steampunk.wonderhowto.com/how-to/complete-guide-build-crystal-radio-plus-they-work-0141117/

Crystal radio with big coil 2

https://en.wikipedia.org/wiki/Crystal_radio

Crystal radio with big coil 3

hackaday.com/2020/01/28/eric-talks-crystal-radios/

Crystal radio with big coil 4

https://www.mtcradio.com/the-pickard-crystal-set-radio-kit/

This is the equation I was using to calculate the frequency:

LC oscillator frequency equation

https://www.elprocus.com/lc-oscillator-circuit-working-and-its-applications/

These were the formulas I was using as a reference for inductors (I know they're air-core, but they showed me that increasing the radius of the coil increases the inductance, which should be the same for a ferrite-core inductor as well)

Air core inductor formulas

https://www.allaboutcircuits.com/uploads/articles/round-wire-formula.jpg

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  • \$\begingroup\$ The capacitors are tiny (value wise.) Those humongous variable capacitors had values in the picofarad range. That goes together with the relatively low inductance of the large air cored coils. \$\endgroup\$
    – JRE
    Feb 24, 2021 at 10:08

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Note how these inductors are air filled so even though they're physically large in size, that doesn't mean they also have a large inductance.

Indeed tapping off a signal is much easier with large inductors and that is I think the actual reason why they're this large.

With a smaller inductor tapping will be more difficult, the wires are not isolated, if they were, tapping off would be impossible (unless you remove the isolation first). Also the un-isolated wires cannot touch as that would short the inductor.

Also, with a smaller inductor you would get a smaller voltage (at a higher current but that doesn't help) so the output signal would be smaller resulting in a lower volume in the (high impedance, crystal) headphones.

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