What is the relationship between current density and wire radius in a wire?

[freshcoast]

1. Problem statement
The current density from the axis in the wire of radius R is given by J = cr^3/2

A) draw the current density as a function of r graph
B) determine the constant c in terms of I_total

C) determine the current as a function of r.

D) draw the graph of the current as a function of r.

2. Known equations

3. Attempt

For part a) I think the graph would just look like a gradually increasing graph, since r grows exponentially.

For part b) I am just confused on this one, I really don't know just how to start this, I don't know which equation to include to give me c in terms of I

Any input is greatly appreciated, thanks!

 

[UltrafastPED]

You need to calculate I_total; this is the integral of J over the cross sectional area - this will relate c and I_total.

 

[freshcoast]

Ok so I am using the equation.. J = I / A,

The area, I think would be r * d(theta) since we are looking at the area of a wire, and using the J as given above, I will get

Cr^3/2 = I / r * d(theta)

I move r * d(theta) to other side, then integrate from 0 to 2pi, then I solve for c leaving me

C = I / 2pi * r^5/2

 

[UltrafastPED]

But r is the radial distance from the wire axis, and R is the wire radius; to get the total current:
I_total = ∫J r dr dθ

The angular factor is, as you found, 2 pi.
So I_total = 2 pi ∫(cr^3/2 x r) dr = 2 pi c∫r^5/2 dr.

After you integrate this you can solve for c in terms of I_total.

 

[freshcoast]

Oo I see, the bounds for the integral will then be from 0 to R correct?

Now since I know the equation of I_total isn't that the answer for part c aswell?

I(R) = 4pi * c * R^(7/2) / 7

 

[UltrafastPED]

You should be okay with the rest.