# Using Current Density to find Total Current

## Homework Statement

Suppose a wire has a radius $R$ and a current density of $J = I_0 \frac{x^2} {R^4}\hat{k}$. Taking the wire to have cylindrical geometry, calculate the total current flowing down the wire, defined as $I = \int J \cdot \vec{dA}$.

## Homework Equations

$J = I_0 \frac{x^2} {R^4}\hat{k}$
$I = \int J \cdot \vec{dA}$

## The Attempt at a Solution

Here, I assumed that dA = R dR dx . I then attempted to do the Integral. I assumed Io was a constant, so I pulled that out of the Integral, giving me...

$I = I_o\int x^2\cdot \vec{dx} \int R^{-3} \cdot \vec{dR}$

Evaluating, I got...

$I = I_o\frac{x^3}{3}\frac{R^{-2}}{-2}\hat{k}$

I have no idea if this is right or wrong. I don't even know if I'm on the right track. Please, any sort of help you could give me would be greatly appreciated!

tiny-tim
Homework Helper
welcome to pf!

hi slavito! welcome to pf!

i don't understand what x is

does x mean distance from the centre (which we would usually call "r")?

or is x one of the usual x,y,z coordinates (with the wire pointing along the z axis)?

(and R is a constant, so you can't have dR)

rude man
Homework Helper
Gold Member
Assuming the wire axis and the current run along the z axis and the wire axis is centered at x=y=0:

what is a sliver of area of width dx and height = distance between 2 points on the circle at x? What is then the current within that area?

Tiny-tim is of course right about R and he's right about the need to define your problem fully. Most posters for some reason don't post the problem as handed to them.