Vector potential inside an infinite wire

[Jean-C]

Homework Statement

The problem statement is simply to find the vector potential inside and outside an infinite wire of radius R, current I and constant current density j using the Poisson equation.

Homework Equations

The Poisson law can be written A = μ₀ /4π *∫(I/r*dl) or A = μ₀ /4π *∫(i/r*dV)

The Attempt at a Solution

I already found out that A=-μ₀ /2π * ln(r/R) outside of the wire. The problem is that I can't figure out how to calculate it inside the wire using the Poisson equation. Using the Biot-Savard law and the definition B=∇ x A, I found out that the vector potential should be μ₀ j/4 * (R²-r²), but I can't find how to get this answer using the Poisson equation. By placing the wire on z-axis and using the second definition for Poisson, I have :
A = μ₀ j/2 * ∫∫r/(r²+z²)^1/2 *dz dr

Here, I have some trouble finding the borders of my integrals. Also, this equation seems false, since it doesn't seem to get me anywhere the expected result.

 

[mark726]

Thank you for your post. It seems like you have made some progress in solving the problem, but are still struggling with finding the vector potential inside the wire using the Poisson equation. I will try to guide you through the process and hopefully help you find the solution you are looking for.

First, let's review the Poisson equation for the vector potential A: A = μ0 /4π *∫(j/r*dl) or A = μ0 /4π *∫(j/r*dV). Here, j is the current density and dl or dV is the differential element of length or volume, respectively. In this case, we are dealing with a wire of radius R, so we will use the second form of the Poisson equation.

To find the vector potential inside the wire, we need to integrate over the volume of the wire. Let's assume that the wire is centered at the origin and oriented along the z-axis. This means that the limits of integration for z will be from -R to R and the limits for r will be from 0 to √(R^2-z^2), since we are dealing with a cylindrical volume.

So, our integral becomes: A = μ0 /4π *∫∫∫j/r*dV. We can rewrite this as: A = μ0 /4π *∫∫∫j/(√(r^2+z^2))*r*dz dr dθ, where θ is the angle around the z-axis. Since j is constant, we can take it out of the integral and we are left with: A = μ0 j/4π *∫∫∫r/(√(r^2+z^2))*dz dr dθ.

Now, we can solve this integral using cylindrical coordinates. The integral over θ will give us a factor of 2π, and the integral over z will give us a factor of 2R. This leaves us with: A = μ0 j/4 *∫r/(√(r^2+z^2))*r*dr. This integral can be solved using a substitution, where u = r^2+z^2 and du = 2r dr. This gives us: A = μ0 j/4 *∫√(u)*du = μ0 j/4 * (2/3