What is the position dependent magnetic field for a Stern-Gerlach experiment?

[M Demov]

Homework Statement

For a Stern-Gerlach experiment, there is a apparatus designed to create a magnetic gradient. There is a dipole magnet. The radius of the convex pole is 5 cm, the radius of the concave pole is 10 cm. The convex pole as a 2 T magnetic field along its surface. (The apparatus is depicted in the attached image)

I need to find the position dependent magnetic field to solve for the separation of the two beams. I know how to proceed once I have the position dependent magnetic field. But I need help with finding the position dependent magnetic field. I found it by placing two imaginary wires at the foci of the pole curves. However, my professor told us we are not allowed to do this and we have to solve for the magnetic field from first principles (maybe Gauss's Law for Magnetism?...) I don't know how to proceed from there and thus don't have any equations to share.

Homework Equations

None thus far, given that I cannot use a two imaginary wire method.

The Attempt at a Solution

My attempt at the solution was using a two imaginary wire method, until I was told I could not use this.

 

[mfb]

Gauß' law is fine.
I don't understand where you want to add imaginary wires.

 

[M Demov]

Gauß' law is fine.
I don't understand where you want to add imaginary wires.

But how do I use Gauss's Law?

 

[mfb]

There is an integral that gives the same value for different surfaces. Or 0 for a suitable closed volume.
The problem in the gap is similar to electrostatics.