Current through a bound cross-section

1. Mar 24, 2015

RyanTAsher

1. The problem statement, all variables and given/known data

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)

2. Relevant equations

Cross section of wire (area) =
pi(r)^2

Current Density =
J = I/A

3. The attempt at a solution

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.

2. Mar 24, 2015

Staff: Mentor

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.

3. Mar 26, 2015

RyanTAsher

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

4. Mar 26, 2015

Staff: Mentor

∫ 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.