# Current or Current Density?

If a conductor of varying cross sectional area is connected across a battery, does the current $I$ or the current density $J$ remain constant throughout the conductor? Why?

Thanks.

It's the current density $J$ that changes.
Current is defined as an amount of charge per seconds, $I=Q/T$, i.e. a number of electrons flowing through the wire every second. Once your conductor is connected to the battery, the battery emf forces a current $I=V/R$ through the conductor. This current only depends on $V$ and $R$, the total resistance of the conductor.
The current $I$ remains constant irrespective of the wire cross section. This follows from the conservation of charge. Thus, when the cross section area $A$ changes along the wire, the product of current density $J$ and area $A$ always stays the same $$I=J A$$