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In a slide wire generator, when the slide wire attains a final constant velocity, is the current in the circuit 0?

It does not make sense if it is 0 as the area keeps increasing, allowing more flux to pass, meaning more current would need to be generated to counteract the increasing flux.

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The whole (pedagogical) point of the slide wire generator is to illustrate that not only do changes in the magnetic field generate current in the loop, changes in the area of the loop - in a constant magnetic field - also generate a current. It's the change in magnetic flux that matters.

As long as the wire is moving with some velocity, the magnetic flux is changing, and there is an induced EMF or voltage in the circuit, and therefor a current.

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