For voltage variation with length, in using a slide meter, the wire resistance varies linearly with length as Rx = resistivity/Area. Since the voltage and resistance are related with ohm's law, Vx= iRx, the current is the same alog all pts. along the slide wire. The current is set by the battery's voltage Vin and the total resistance of the wire. Putting all that together write a proof to show Vx = Cx where C is a position along the slide meter.
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