Magnetic vector potential of finite wire

[zezima1]

Homework Statement

The problem statement is attached.

The Attempt at a Solution

I know how to solve the problem. However, my teachers solutions notes and my book's do it differently, and I want to ask what the difference is, so I have attached them both. My book does it the way I did it. My teacher's however introduces a dummy variable z', which is integrated over rather than z. Thus we end up with an expression that actually contains z. What is geometrically the difference between these two solutions? Clearly in my book's solutions the origin is placed a distance z1 below the wire - where is the origin in my teachers note's?

Cheers :)

 

[vela]

The book's calculation finds the vector potential in the xy-plane, at a point a distance s from the origin. Your teacher's calculation is more general. It finds the vector potential for any z. If you set z=0 in his or her solution, you'll get the book's solution.

 

[wwang02]

The book's calculation finds the vector potential in the xy-plane, at a point a distance s from the origin. Your teacher's calculation is more general. It finds the vector potential for any z. If you set z=0 in his or her solution, you'll get the book's solution.

I do not understand, how can we do this only in the xy-plane, doesn't the vector field also depends on z? I think the textbook solution is wrong because we have to consider z dependence but not simply set it to 0.