Calculating Current Through a Cylindrical Conductor

[DODGEVIPER13]

Homework Statement

The current density in a cylindrical conductor of radius a placed along the z-axis is J=10e^-(1-ρ/a)Uz
Find the current through the cross-section of the conductor.

Homework Equations

I=JA
A=(pi)a^2

The Attempt at a Solution

So far I have this I=10e^-(1-ρ/a)Uz((pi)a^2) which I assume I am to integrate but what should I integrate with respect too and what should the limits be I am guessing from 0 to a?

 

[UltrafastPED]

You integrate the flow normal to the cross section through which it is flowing.

 

[DODGEVIPER13]

So that would be along the x or y?

 

[DODGEVIPER13]

The normal is the perpendicular vector right?

 

[UltrafastPED]

Yes, the normal is perpendicular to the surface.

 

[DODGEVIPER13]

ok so a normal vector equals to Uz/Uz (where the bottom Uz is the length of Uz it is in between those two absolute value mark things). Do you have any more hints on how to set this up

 

[UltrafastPED]

Your problem states that the current is flowing along the axis Uz, hence it is perpendicular to the cross section formed by the x-y plane.

So you just have to integrate the density given as it crosses the circle of radius a defined by the cylinder of the wire. Since the axis passes along the center of the wire it would be easiest to switch to cylindrical coordinates for this integration.

 

[DODGEVIPER13]

∫JdS=Ienc where dS=ρd(phi)dρ in cylindrical right?