Finding expression for non-uniform current density of a wire

[phys-student]

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?

 

[TSny]

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?

 

[phys-student]

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

 

[SammyS]

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.

 

[phys-student]

Yep you're right. Totally forgot that I also need to integrate over the angle from 0 to 2 pi

 

[phys-student]

Okay I'm good now, thanks