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The above pictures show how this problem is typically setup and how the field lines are typically shown.

The math is messy but every textbook yields the following:

β = ∇xA = (m / (4⋅π⋅R

^{3})) ⋅ (2⋅cos(θ) r + sin(θ) θ)

The issue I am having is seeing how the above equation yields the field lines from the above picture.

If θ is referenced from the Z axis, and the loop of wire is on the X-Y axis, in my mind the field lines are 90 degrees shifted. In other words, when θ=0, the radial component is at it's max straight up the Z-axis, and as θ approaches π/2 the radial component approaches 0. The above picture shows that β is max at π/2 and not 0. In Matlab I plotted a few different Radii for all θ = 0 to 2π and Φ=0:

This picture is 90 degrees shifted from how I think it should be. Can someone help me understand this? Why does the equation not align up with the way the typical picture is shown? Am I incorrect in assuming the Z-axis is perpendicular to loop? Am i missing something? Am i not even close? D: