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How does a speaker and a microphone work in opposite ways?

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A microphone converts sound into electrical impulses. A speaker converts electrical impulses into sound.

Answer

Because they both work on the same principle, a changing electrical field (the audio signal) in a coil causes a magnet within the coil to vibrate in tandem. The magnet is attached usually to a paper cone which allow our ears to hear the sound. The converse is also true. For example if you talk into a headphone it causes the magnet to vibrate which produces an electrical signal in the coil which is connected to the headphone cable.

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