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Homework Help: Introduction to electrodynamics - help with a dipole problem

  1. Dec 11, 2011 #1
    1. The problem statement, all variables and given/known data
    Let the dipole [itex]\vec{m}[/itex] = m[itex]\hat{k}[/itex] 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: [itex]\vec{B}[/itex] =[itex]\frac{μ_{0}m}{4∏r^3}[/itex](2cosθ[itex]\hat{r}[/itex]+sinθ[itex]\hat{θ}[/itex])


    3. The attempt at a solution
    I tried putting the previous equation into the coordinate free form to try if that would help.

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

    I then simplified this to:


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

    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[itex]\frac{z}{(y^2+z^2)^(1/2)}[/itex] = θ
     
    Last edited: Dec 11, 2011
  2. jcsd
  3. Dec 13, 2011 #2

    ehild

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    Homework Helper

    The position vector is written as
    [itex]\vec r = x \hat i+y\hat j +z\hat k[/itex].
    If [itex]\vec r[/itex] 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 [itex]\vec r[/itex] is
    [itex]\hat r = \sin(\theta)\cos(\phi)\hat i+\sin(\theta)\sin(\phi)\hat j +\cos(\theta)\hat k[/itex].

    Use all of these to get [itex]\vec B[/itex].

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

    ehild
     
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