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Flux in spherical coordinates

  1. Apr 5, 2008 #1
    1. The problem statement, all variables and given/known data
    Calculate the rate at which muons pass through a flat plate of area A.

    2. Relevant equations
    [tex]J_{1}=\int_{\theta\leq\frac{\pi}{2}}j(\theta,\phi)cos\theta d\Omega[/tex]
    [tex]d\Omega[/tex] = sin[tex]\theta d\phi d\theta[/tex] in spherical coordinates.
    [tex]j(\theta,\phi)[/tex] is the angular distribution of muons at sea level.
    At sea level [tex]j(\theta,\phi) \approx cos^{2}\theta[/tex]
    More specifically [tex]j(\theta = 0,\phi) \equiv I_{v}[/tex] which equals the flux per unit solid angle per unit horizontal area per second about the vertical direction.
    [tex]I_{v} = 1.1*10^{2}m^{-2}sec^{-1}sterad^{-1}[/tex] for all penetrating particles at sea level of which 75% are muons.

    3. The attempt at a solution
    I'm very confused about spherical coordinates... this equation is just a tiny part of my muon lifetime prac but I'm not sure at all how to go about this integral. Any help would be greatly appreciated.

    My attempt is as follows:
    [tex]J_{1}=0.75* \int_{\theta\leq\frac{\pi}{2}} I_{v} cos\theta sin\theta d\phi d\theta[/tex]
    [tex]J_{1}=0.75* 1/2 \int_{\theta\leq\frac{\pi}{2}} I_{v} sin2\theta d\phi d\theta[/tex]

    this is where I get stuck... I know that the integral of [tex]sin2\theta[\tex] is [tex]-1/2 cos2\theta[\tex]... but I'm confused about what I'm evaluating between... and what to do with the [tex]\phi[\tex] after that step...

    Thanks in advance!
  2. jcsd
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