Could one use wireless recharging technology (see http://home.howstuffworks.com/question292.htm) to power a light-emitting diode (which does not have a filament that gets hot (see http://oaicorp.com/digital-led-faq.aspx)) to display this week's parsha in shul on Shabbos? This is a theoretical question.
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Assuming the words on the sign are a sort of stencil, and this only gives it back lighting...
According to the poskim who say that all eletricty usage is an issur of Binyan, it seems that this would not be allowed.
However, according the poskim, (such as Rav Aurbach) who see 'the normal usage' as not falling under the category of binyan (such as a door or a window in a house) it seems like there would not be any real issur with the charging itself.
The charging is really no different than static eletricity which appears to be allowed. Shemiras Shabbos Kehilchaso 15:72. "Clothes made of synthetic materials may be worn, even though they give off sparks when being put on or taken off".(Vol.1, Chapter 15, No.72)