Introduction to electrodynamics - help with a dipole problem

[Flynndle]

Homework Statement

Let the dipole \vec{m} = m\hat{k} be at the origin, and call a certain horizontal axis the y axis.
a) On the z axis, what is the angle between the z axis and the magnetic field?
b) On the y axis, what is the angle between the z axis and the magnetic field?
c) On the cone θ=45 degrees, what is the angle between the z axis and the magnetic field?
d) What is the angle of the cone on which the magnetic field is horizontal?

Homework Equations

I believe there is some relevance to the equation: \vec{B} =\frac{μ_{0}m}{4∏r^3}(2cosθ\hat{r}+sinθ\hat{θ})

The Attempt at a Solution

I tried putting the previous equation into the coordinate free form to try if that would help.

\vec{B} =\frac{μ_{0}}{4∏r^3}[3(\vec{m}\bullet\hat{r})\hat{r}-\vec{m}]

I then simplified this to:


\vec{B} =\frac{μ_{0}}{4∏r^3}m[3cosθ-1]\hat{k}

I was not sure what to do after this so I tried:

r^2=x^2+y^2+z^2

set x=0 so it's in the yz plane,

r=(y^2+z^2)^(1/2)

arccos(z/r)=θ

arccos\frac{z}{(y^2+z^2)^(1/2)} = θ

 

[ehild]

The position vector is written as
\vec r = x \hat i+y\hat j +z\hat k.
If \vec r encloses the angle θ with the z axis and the angle φ with the positive x axis, x=r sinθ cosφ, y=r sinθ sinφ, and z=rcosθ. The unit vector along \vec r is
\hat r = \sin(\theta)\cos(\phi)\hat i+\sin(\theta)\sin(\phi)\hat j +\cos(\theta)\hat k.

Use all of these to get \vec B.

(\vec m\cdot \hat r)=m\cos(\theta), and it is multiplied by \hat r, so \vec B has x, y, and z components. Try to find it.

ehild