Proving Total Current through Insulating Wire Using Spherical Coordinates

[iontail]

i am trying to solve this problem which states that

J(p) = (I/pi) p^2 e^-p^2 in z direction
is the current density flowing in the vicinity of insulating wire.
pi = pie

in standard spherical polar coordinates.

J is the current density.

I need to prove that the total current flowing through the wire is I.

I have tried to used the idea J.dS = I where

and integrate(i have taken the scale factor into consideration) but it does not yield the right result. Any suggestion on a way to move forward will be appreciated.

 

[Born2bwire]

What are the limits of integration here? Still, I can't see how you can get a current of I because of your exponential, except if your relationship is incorrect.

$$J(\rho)=\frac{I}{\pi}e^{-\rho^2}$$

If we have a wire of infinite size then this one would work, but this is just playing around.

 

[jtbell]

You missed a factor of $\rho^2$:

$$J(\rho)=\frac{I}{\pi} \rho^2 e^{-\rho^2}$$

 

[Born2bwire]

You missed a factor of $\rho^2$:

$$J(\rho)=\frac{I}{\pi} \rho^2 e^{-\rho^2}$$

That's his original equation yes, whose integral over the cross-sectional area of the wire I think will be

$$I(\rho)=-I_0e^{-\rho^2}(\rho^2+1)+I_0$$

Seeing as the OP has not given us the dimensions of the wire we can't go any further than that but I do not see how any normal choice of radius would allow the current to come out to be exactly I_0. I was just mentioning that if the \rho^2 dependence was dropped then, the integral would come out provided an infinite radius but as I stated I was not seriously suggesting that was an answer.

Another question is what does Stokes' Theorem have to do with the problem. I feel that there were some steps leading up to this point that the OP may have left out.