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to understand why the light bulb lights up without being connected.
the high voltage end of the transformer is an antenna. when the light bulb is near the antenna, the energy is received by its filament, causing current to flow through it.
the antenna and the filament and the air surrounding them acts like a capacitor. as electric charge builds up on the antenna, a corresponding charge builds up on the filament. and anything else nearby.
so be careful when playing with these things; they can disrupt or destroy delicate electronic devices nearby, eg. watches, phones, pacemakers. they'll also make a crapload of interference which will annoy neighbours, and possibly your local communications regulatory authority and/or federal law enforcement.
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