Magnetostatics - Magnetic Flux + Force

[jegues]

Homework Statement

See figure attached.

Homework Equations

The Attempt at a Solution

I can follow how he applies amperes law to obtain the magnetic field produce by the wire but I'm extremely confused how he writes the expression for the flux flowing through S.

What I understand to be the magnetic flux flowing through a surface S.

$$\int_{S}\vec{B}\cdot\vec{dS}$$

Since the wire is a cylinder I assume cylindrical coordinates.

$$B = \mu_{0}H$$

$$dS = rdrd\phi$$

My integral seems to be quite a bit different than his.

Can someone clarify the steps to getting his integral, and why he chooses to work in the yz plane?

Thanks again!

 

[Simon Bridge]

He exploits the phi-symmetry from the beginning while your surface element implies you do not. Perhaps if you showed your integral as how you constructed it I could be specific.

He does not do any of his work in the z-y plane. He shows you a cross-section in the z-y plane because that shows the symmetry of the situation most clearly. Try drawing the situation in the z-x ir x-y planes and you'll see this. He does his work entirely in cylindrical coords - you can see this in his integral over r. The z coord is the factor of d and phi is accounted for in the expression for H_phi.