Spherical to cartesian?

  1. 1. The problem statement, all variables and given/known data
    A field is given in spherical coordinates as F=[cos(θ)/r2]∙ar+[sin(θ)/r]∙aθ. Express F in terms of x, y, z, ax, ay, az


    2. Relevant equations

    ar∙ax=sin(θ)cos(∅)
    ar∙ay=sin(θ)sin(∅)
    ar∙az=cos(θ)
    aθ∙ax=cos(θ)cos(∅)
    aθ∙ay=cos(θ)sin(∅)
    aθ∙az=-sin(θ)
    x=r*sin(θ)*cos(∅)
    y=r*sin(θ)*sin(∅)
    z=r*cos(θ)
    r=√(x2+y2+z2 )
    cos(θ)=z/r
    ∅=tan-1(y/x)

    3. The attempt at a solution
    cos(θ)/r2*[sin(θ)cos(∅)ax+sin(θ)sin(∅)ay+cos(θ)az]+sin(θ)/r*[cos(θ)cos(∅)ax+cos(θ)cos(∅)ay-sin(θ)az]

    z/r3*[sin(θ)cos(∅)ax+sin(θ)sin(∅)ay+cos(θ)az]+sin(θ)/r*[cos(θ)cos(∅)ax+cos(θ)cos(∅)ay-sin(θ)az]

    (z*r)/r4*[sin(θ)cos(∅)ax+sin(θ)sin(∅)ay+cos(θ)az]+sin(θ)/r*[cos(θ)cos(∅)ax+cos(θ)cos(∅)ay-sin(θ)az]

    z/r4*[xax+yay+zaz]+sin(θ)/r*[cos(θ)cos(∅)ax+cos(θ)cos(∅)ay-sin(θ)az]

    z/(x2+y2+z2)3*[xax+yay+zaz]+sin(θ)/r*[cos(θ)cos(∅)ax+cos(θ)cos(∅)ay-sin(θ)az]

    That's about as far as I've gotten. I'm not even sure if what I've done so far is on the right track or not :/ I'm not sure what to do with the 2nd half of this equation?
     
    Last edited: Sep 4, 2011
  2. jcsd
  3. vela

    vela 12,628
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    Use the fact that [itex]\vec{F} = (\vec{F}\cdot\hat{a}_x)\hat{a}_x + (\vec{F}\cdot\hat{a}_y)\hat{a}_y + (\vec{F}\cdot\hat{a}_z)\hat{a}_z[/itex].

    Calculate [itex]\vec{F}\cdot \hat{a}_x[/itex] using the various dot products you listed above. Then convert from the spherical variables to the Cartesian variables.
     
  4. Isn't that what I did above? I didn't originally show the ax, ay, az in my work but I just added them in there for clarity.
     
  5. vela

    vela 12,628
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    Oh, OK. I didn't see the unit vectors in your original attempt, so I figured you were doing it all wrong and didn't bother to look too closely.
     
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