Exploring Vector Potentials and the Curl of Magnetic Fields

[DieCommie]

Homework Statement

A magnetic field of a long straight wire carrying a current I along the z-axis is given by the following expression:

$$\mathbf{B} = \frac{\mu_0I}{2\pi} \{\frac{-y}{x^2+y^2} \hat{x} + \frac{y}{x^2 + y^2} \hat{y} \}$$

Find two different potentials that will yield this field. Show explicitly that the curl of the difference between these two potentials vanishes.

Homework Equations

$$\nabla \times \mathbf{A} = \mathbf{B}$$

The Attempt at a Solution

I took a cross product to get this system...

$$\frac{\partial A_y}{\partial x} - \frac{\partial A_x}{\partial y} = 0 \\$$

$$\frac{\partial A_z}{\partial y} - \frac{\partial A_y}{\partial z} = \frac{-y}{x^2+y^2} \\$$

$$\frac{\partial A_x}{\partial z} - \frac{\partial A_z}{\partial x} = \frac{y}{x^2+y^2} \\$$

I don't know what to do! Any ideas?

 

[DieCommie]

Maybe nobody knows how to do this problem...

But does anybody know how to, in general, find a vector 'A' given its cross product 'B'?