I would like to know if the magnetic field is in "the x-y plane". Does this mean the field lines are actually parallel to the visible line x or y? Or does it mean the field lines are parallel to the z-axis and crossing the x-y plane?
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Neither. "In the XY plane" means "perpendicular to the Z axis". It doesn't mean they are either parallel to X or Y - it could be at 45° to either of these axes.
The equation of a plane is sometimes written as
$$(\vec{x} - \vec{x_0})\cdot \vec{n} = 0$$
Where $\vec{x}$ is any point on the plane, $\vec{x_0}$ is a known point on the plane, and $\vec{n}$ is the normal to the plane (in this case, the Z axis). That expression basically shows that any vector in the XY plane is perpendicular to Z (since the dot product is zero).
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1$\begingroup$ @gonenc - I agree. Looking closely, I think that $B_0$ is in the XY plane, with $Z$ sticking out of the page. It shows the $X$ component of the $B$ field... $\endgroup$– FlorisMay 28, 2015 at 19:21