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单词 CalculatingTheSolidAngleOfDisc
释义

calculating the solid angle of disc


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 parallelMathworldPlanetmathPlanetmathPlanetmath to the xy-plane and the center is on the z-axis at  (0, 0,h)  with  h>0.  Into the

Ω=-ada1r=adar|r|3(1)

of the parent entry (http://planetmath.org/SolidAngle), we may substitute the position vector  r=xi+yj+hk  of the directed surface element  da=kda,getting

Ω=ahda(x2+y2+h2)3/2.

Now we can use a annulusMathworldPlanetmath (http://planetmath.org/Annulus2)-formed surface element  da=2πϱdϱ  where  ϱ2=x2+y2, whence the surface integral may be calculated as

Ω=πh0R2ϱdϱ(ϱ2+h2)3/2=πh-2/ϱ=0R1ϱ2+h2.

Thus we have the result

Ω=2πh(1h-1R2+h2).
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更新时间:2025/5/4 10:22:22