Does current density from a wire same in smaller cross section?

Question

To be more simple, If current $i$ is flowing through a long cylindrical wire does the current density of region $A$ and $B$ same?

Answer 1

It depends on the distribution of current within the wire. This post (Does electricity flow on the surface of a wire or in the interior?) suggests that, for DC current, there will be no (or really very little) difference in current density between the two regions. For AC current, this will not be the case and the current density will be biased towards the surface regions of the wire.

Answer 2

The answer goes as follows:

let A be the area of cross section of the cylindrical conductor we have considered then the current density is defined as the current flowing per unit area perpendicular to the direction of the flow.

From the above definition you can tell that the total current passing through area is equal to integral of dot product of current density j and area vector dA. Since violet part has more area of cross section, it sustain greater current density than the green part. Hence current is more in violet part.