The instrument used for measuring very small electrical currents is call a Galvanometer.
A galvanometer is the most sensitive apparatus to measure very small current among the all moving-neddle analog meters.Galvanometer is actually a moving coil instrument which measures current by creating a magnetic force proportional to the current and compairing it with restoring force. Other practical moving-coil meters (voltmeter, ammeter etc.) are actually a galvanometer modified to extend measure abilities.
Galvanometer is very sensutive that can measure current of order of magnitude 10 uA while an ammeter can can measure current of order of 1 A current. That's why to measure a very small current we use a Galvanometer.
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