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calculating the solid angle of disc
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(Example)
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We determine the solid angle formed by a disc when one is looking at it on the normal line of its plane set to the center of it.
Let us look the disc from the origin and let the disc with radius $R$ situate such that its plane is parallel to the $xy$ -plane and the center is on the $z$ -axis at $(0,\,0,\,h)$ with $h > 0$ . Into the formula
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of the parent entry, we may substitute the position vector $\vec{r} = x\vec{i}+y\vec{j}+h\vec{k}$ of the directed surface element $d\vec{a} = \vec{k}\,da,$ getting $$\Omega = \int_a \frac{h\,da}{(x^2+y^2+h^2)^{3/2}}.$$ Now we can use a annulus-formed surface element $da = 2\pi\varrho\;d\varrho$ where $\varrho^2 = x^2+y^2$ , whence the surface integral may be calculated as
$$\Omega = \pi h\int_0^R \frac{2\varrho\;d\varrho}{(\varrho^2+h^2)^{3/2}} = \frac{\pi h}{-2}\sijoitus{\varrho=0}{\quad R}\frac{1}{\sqrt{\varrho^2+h^2}}.\\$$ Thus we have the result $$\Omega = 2\pi h\left(\frac{1}{h}-\frac{1}{\sqrt{R^2+h^2}}\right)\!.$$
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"calculating the solid angle of disc" is owned by pahio.
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Cross-references: integral, surface, position vector, parallel, radius, origin, center, plane, normal line, disc, solid angle
This is version 3 of calculating the solid angle of disc, born on 2008-08-22, modified 2008-08-24.
Object id is 10957, canonical name is CalculatingTheSolidAngleOfDisc.
Accessed 883 times total.
Classification:
| AMS MSC: | 51M25 (Geometry :: Real and complex geometry :: Length, area and volume) | | | 15A72 (Linear and multilinear algebra; matrix theory :: Vector and tensor algebra, theory of invariants) |
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Pending Errata and Addenda
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