If the wood is totally dry (hasn't absorbed moisture from the air), 2cm is just over one inch, and that's about the limit for Qi. Depending on the charger, you might be a mm or two too far for it to work. If you could make the wood about 17mm or thereabouts, it will probably work. As for the width, if the wood is dry, 4mm or 4 meters wouldn't make any difference - it's the thickness and the electrical conductivity of the wood that matters. (I've gotten close to an inch [by eye - I never measured it] by holding my phone [in a pretty thick case, an Otterbox] above the charging plate, so 0.787 inches (2 cm) would probably work with dry wood. The only way to find out is to put the wood on the charger plate and see. (The coil can be melted into the top plate, giving you less than 1mm minimum distance, or it could be mounted a few mm under the plate, putting it just too far to work reliably.)
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Brayden McGraw, if the wood is dry, there's nothing to catch fire - the wood needs heat, not electricity, to catch. But if the wood gets wet, the water in it will heat up - possibly enough to catch. Soak the wood in a waterproof varnish (polyurethane varnish or marine spar varnish) overnight to make sure that doesn't happen.