Magnetic Vector Potential and conductor

[sikrut]

Homework Statement

Show that, inside a straight current-carrying conductor of radius R, the vector potential is:
$$ \vec{A} = \frac{\mu_{0}I}{4\pi}(1-\frac{s^2}{R^2}) $$

so that $\vec{A}$ is set equal to zero at s = R

Homework Equations

$\vec{A} = \frac{\mu_{0}}{4\pi}\int\frac{\vec{I}}{|r'-r|} dl'$

The Attempt at a Solution

I'm really having a hard time even setting it up.

 

[Simon Bridge]

Start by drawing a picture of a current carrying wire with a significant radius.
How does the current vary with radius?

 

[Simon Bridge]

Hint: cylindrical coordinates.
Some people like to turn the line integral into an area integral too ... there are lots of approaches.
For marking - it is usually best to use the method covered in class.

 

[vanhees71]

Another hint: The Biot-Savart Law is not very efficient in many problems. It's easier to use the local form of the (magnetostatic) Maxwell Equations:
$$\vec{\nabla} \times \vec{B}=\mu_0 \vec{j}, \quad \vec{\nabla} \cdot \vec{B}=0.$$
The second equation ("no monopoles") is already solved by the introduction of the vector potential, cf.
$$\vec{B}=\vec{\nabla} \times \vec{A}.$$
So, just take the appropriate derivatives (in cylindrical coordinates), and check that you get the right current density.