Finding expression for non-uniform current density of a wire

AI Thread Summary
The discussion focuses on deriving an expression for the current density J in a wire with a non-uniform current density proportional to the distance s from the center. The proposed solution is J = 2*s*I/R², which the poster believes integrates correctly to yield the total current I. However, there are concerns about the correctness of the expression and the need for a proper derivation. It is highlighted that integrating over the cross-sectional area, including the angle, is necessary for accuracy. The conversation emphasizes the importance of showing work to support the derived expression.
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Homework Statement


A long, straight wire, radius R, carries total current I. The current is distributed in the wire so that the current
density is proportional to s, the distance from the center of the wire.
(a) Write an expression for the current density J in the wire, as a function of s.

Homework Equations


J=dI/da

The Attempt at a Solution


J=2*s*I/R2
I'm pretty sure my attempted solution is correct because when you integrate J from 0 to R with respect to s you get the total current I. However I kinda just pulled this out of thin air and I'm pretty sure I won't get full marks without showing my work. Is there a different way to do this besides just making up an expression for J like I did?
 
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phys-student said:
The current is distributed in the wire so that the current
density is proportional to s, the distance from the center of the wire.

How would you express this statement in mathematical form?
 
I don't know, that's what I'm trying to understand. I was trying to express that statement in mathematical form but couldn't figure it out, so I settled on the expression that I have simply because integrating it over the cross section of the wire results in the total current I
 
phys-student said:

Homework Statement


A long, straight wire, radius R, carries total current I. The current is distributed in the wire so that the current
density is proportional to s, the distance from the center of the wire.
(a) Write an expression for the current density J in the wire, as a function of s.

Homework Equations


J=dI/da

The Attempt at a Solution


J=2*s*I/R2
I'm pretty sure my attempted solution is correct because when you integrate J from 0 to R with respect to s you get the total current I. However I kinda just pulled this out of thin air and I'm pretty sure I won't get full marks without showing my work. Is there a different way to do this besides just making up an expression for J like I did?
I'm pretty sure that's not right. Even the units are wrong.

Likely you're not integrating over an area.
 
Yep you're right. Totally forgot that I also need to integrate over the angle from 0 to 2 pi
 
Okay I'm good now, thanks
 

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