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Derivation of the potential of a sphere

  1. May 13, 2005 #1
    Hi,

    I've been told in a lecture course that the potential used in the virial theorem (for our application) is 3/5 GM/r. This describes the potential for a sphere of radius r, mass M created by bringing infinitly thin shells from infinity to form the next 'layer' of the sphere. I am having difficulty deriving this for myself, anybody wana give it a try ??
     
  2. jcsd
  3. May 13, 2005 #2

    SpaceTiger

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    For a constant density sphere, that's:

    [tex]E=-\frac{3}{5}\frac{GM^2}{R}[/tex]

    [tex]E=-\int_0^R \frac{GM_r}{r}dm[/tex]

    [tex]M_r=\frac{4}{3}\pi r^3\rho[/tex]

    [tex]dm=4\pi r^2\rho dr[/tex]

    [tex]E=-\frac{16\pi^2G\rho^2}{3}\int_0^R r^4dr=-\frac{16\pi^2G\rho^2}{15}R^5[/tex]

    [tex]M=\frac{4}{3}\pi R^3\rho[/tex]

    [tex]E=-\frac{3}{5}\frac{GM^2}{R}[/tex]
     
    Last edited: May 13, 2005
  4. May 13, 2005 #3
    Spacetiger: Where do the first two lines follow from?
     
  5. May 13, 2005 #4

    SpaceTiger

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    The first one is just correcting the result. I'm pretty sure he had mis-typed it. The second is summing over the potentials of spherical shells at a radius r and with a width of dr (brought in from infinity and assuming the potential is zero at infinity).
     
    Last edited: May 13, 2005
  6. May 13, 2005 #5
    yep i missed the squared out ... thanks for your help :)
     
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