Proof on Magnetic Dipole in an homogeneous magnetic field

[nexus89]

Homework Statement

A magnetic dipole of strength m is placed in an homogeneous magnetic field of strength Bo, with the dipole moment directed opposite to the field. Show that, in the combined field, there is a certain spherical surface , centered on the dipole , through which no field lines pass. What do the field lines inside the sphere look like? what is the strength of the field immediately outside the sphere, at the equator?

Homework Equations

Equations for the field of a magnetic dipole in spherical coordinates:

-Br=2*m*cos(θ)/r^3
-Bteta = m*sin(θ)/r^3

The Attempt at a Solution

It looks to me as if this is a very easy problem. I just find the combined field by adding B=-Bo (r hat direction) to the field of the magnetic dipole. Then I see when the field is equal to zero. However, the math doesn't make much sense. In fact, apart from the condition B0=2*m*cos(θ)/r^3, the other component of the combined field is zero only for obvious values of θ. It seems clear to me that there should be no restrictions on θ for a spherical surface where no field lines pass. I am very confused by this situation and your help would be really appreciated! Forgive me for not writing with the appropriate notation. However, it is the first time that I write on this website and I was really in hurry.

 

[Asim Riaz]

The field lines inside the sphere will look like concentric circles, since the field is radial. The field strength immediately outside the sphere at the equator is equal to Bo.