Current through a bound cross-section

[Destroxia]

Homework Statement

[/B]
The magnitude J of the current density in a certain wire with a circular cross section of radius R = 2.20 mm is given by J = (3.07 × 108)r2, with J in amperes per square meter and radial distance r in meters. What is the current through the outer section bounded by r = 0.917R and r = R?

Givens
R = 2.20 mm = 2.20E-3
J = (3.07E8)r^2
r = .917R (inner bound)
r = R (outer bound)

Homework Equations

Cross section of wire (area) = [/B]pi(r)^2

Current Density = J = I/A

The Attempt at a Solution

[/B]
Since we are attempting to find the current in a bounded section we need to subtract the outer bound area from the lower bound area:

pi(R)^2 - pi(.917R)^2 = bounded section area

Since we have the current density we can use I = JA:

(3.07E8)r^2* ( pi(R)^2 - pi(.917R)^2) = 736.8

It wants the answer in mA so = 736000 mA

It says my answer is incorrect, and I'm not sure where I went wrong.

 

[gneill]

Apparently the current density depends upon the radial position within the conductor. So you can't simply deal with the areas involved, you need to take into account the current density over the cross section. You'll need to set up an integral to compute the current in the desired region.

 

[Destroxia]

Apparently the current density depends upon the radial position within the conductor. So you can't simply deal with the areas involved, you need to take into account the current density over the cross section. You'll need to set up an integral to compute the current in the desired region.

So am I only going to be integrating the area, or will the current density be involved? Is it the equation ∫ J ⋅ dA ?

 

[gneill]

So am I only going to be integrating the area, or will the current density be involved? Is it the equation ∫ J ⋅ dA ?

∫ J ⋅ dA is the appropriate notion. You'll have to work out the details since J is a function of r, and you'll need to express the differential area element dA in terms of r, too.