When is the flux through a coil from a bar magnet zero?

[aliens123]

Suppose I have a finite sized cylindrical bar magnet of radius $a$ and length $b$. It is coaxial with the $x$ axis and moving from $-\infty$ to $\infty.$ A circular closed wire of radius $r>a$ is in the $y-z$ plane with center at the origin. When the bar center of the bar magnet passes through the origin, what is the total flux through the wire?

The answer is not necessarily zero because note that by increasing the radius $r$ we can increase the total flux through the wire. So what does the flux as a function of the position of the center of mass of the bar magnet look like?

 

[Cutter Ketch]

Lines of magnetic flux form closed loops. That is to say, the flux inside the bar magnet is not zero. There is a very strong B field pointing the opposite direction from the more accessible parts of the loops outside the bar. So unless your loop of wire is very large extending out to where the magnetic field is almost zero, the flux through your wire is not anything like zero even in the static case. Unless it is moving close to the speed of light, the movement of the bar doesn’t change the snapshot of the magnetic field of the bar magnet at that moment when the bar is in the middle.