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?
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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.
Ignoring transient effects, a simple conductor is not able to store any charge - unlike for example a capacitor. Hence if an electron flows into the wire, another electron has to leave. So, regardless of how the cross section of the conductor changes, the same number of electrons will flow through it. In other words, the current will stay the same, but the density will vary with the cross-section.
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 $$I=J A$$