# Introduction to electrodynamics - help with a dipole problem

1. Dec 11, 2011

### Flynndle

1. The problem statement, all variables and given/known data
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?

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

3. 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

Last edited: Dec 11, 2011
2. Dec 13, 2011

### 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