Let's say we have a coaxial cable with a 2d rectangular surface lying between the inner and outer conductors and running the length of the cable. I'm trying to understand why the magnetic flux through this surface only includes the magnetic field generated by the current flowing through the center conductor and not the outer conductor. I'm assuming the fields from the current running through the outer conductor cancel out somehow. Is it just that simple?

Ampere's law in integral form is perhaps the easiest way to show the magnetic field from the outer conductor is zero. A circular cross section works best, so that the radial symmetry can be applied. Ampere's law in integral form says: ## \oint B \cdot dl =\mu_o I ## where ## I ## is the current through the loop and the integral is around the loop. If there is no current passing through the circular loop (from the outer conductor), by symmetry ## B=0 ##. (At least the tangential component.) Meanwhile, ## B_z=0 ## because the motion of the moving electrical charges is in the z-direction. (by Biot-Savart's law, the ## v \times r ## in the numerator of the Biot-Savart equation for ## B ## tells us that any ## B ## will be perpendicular to ## v ##. Finally ## B_r=0 ## can be shown by using ## \nabla \cdot B=0 ## along with Gauss's law.