Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Solid angle question

  1. Jul 9, 2010 #1
    Hello,
    it is often written in books that the solid angle [itex]\Omega[/itex] subtended by an oriented surface patch can be computed with a surface integral:

    [tex]\Omega = \int\int_S \frac{\mathbf{r}\cdot \mathbf{\hat{n}} }{|\mathbf{r}|^3}dS[/tex]

    where r is the position vector for the patch dS and n its normal (see also wikipedia).
    However I would like to know how to derive this formula from the definition of solid angle, that is: the area of the the projection of a surface on the unit sphere.


    I can already see that:

    [tex]\frac{\mathbf{r}}{|\mathbf{r}|} \cdot \mathbf{\hat{n}} dS = cos(\theta)dS[/tex]

    where [itex]\theta[/itex] is the angle between the position (unit)-vector for dS and the normal vector for dS

    Unfortunately I don't understand where that [itex]|\mathbf{r}|^{-2}[/itex] comes from.
     
    Last edited: Jul 9, 2010
  2. jcsd
  3. Jul 9, 2010 #2

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    I think the 1/|r|2 factor just scales the area down to its projection on a unit sphere. Your expression

    [tex]
    \frac{\mathbf{r}}{|\mathbf{r}|} \cdot \mathbf{\hat{n}} dS = cos(\theta)dS
    [/tex]

    gives the projection of the surface patch on the sphere of radius |r|. Since area is proportional to the square of the radius, you need the 1/|r|2 to scale it down to the unit sphere.
     
  4. Jul 9, 2010 #3
    Ok, thanks.
    Now I see how it works.

    I was just wondering how to sketch a rigorous proof that the surface area of an infinitesimal "disk" dA is projected onto an infinitesimal spherical cap [itex]d\Omega[/itex] having area [itex]|\mathbf{r}|^{-2}dA[/itex].
     
  5. Jul 10, 2010 #4

    HallsofIvy

    User Avatar
    Science Advisor

    In order to have a rigorous proof of anything involving "infinitesmals" you will need to to "nonstandard analysis" where infinitesmals themselves are rigorously defined! Otherwise you will need to be content with limit proofs. What does the "[itex]\mathbf{r}[/itex]" represent in [itex]|\mathbf{r}|^{-2}dA[/itex]?
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook