I'm reviewing for an electromag exam and I stumbled upon a problem that's really hard to figure out. Here it is:
A small magnetic dipole with moment $\vec m = m_o \hat z$ is in a region with uniform magnetic field $\vec B = B_o \hat z $. Show that there is a sphere where the net magnetic field is zero. What is the radius of the sphere?
That certain sphere is what I'm referring to as the "blind spot" of a magnetic dipole. It's the first time I've encountered something like this. I believe that it is a sphere with its hemispheres on either side of the x-y plane. Why? I'm not really sure, but my argument is that the fields cancel from both sides of the hemisphere since the field lines in the close vicinity of the dipole point along its direction (you can see it from the usual sketches of the field lines of a dipole, ie a bar magnet)
I also don't think the dipole is centered, or even inside, that sphere.
However the actual calculation is really tricky. I've experimented with using the scalar potential $$\Delta \phi_m = 0 $$ but the geometrical analysis is killing me.
I'm a bit of stuck, so..... anybody out there willing to help me? Even just a rough outline of what to do will be extremely helpful. Thank you very much!!!