a problem about spherical light source

[shuipi]

Given a spherical light source with radius r and uniformly emits radiance L from every point on its surface in all possible directions, what is the irradiance at a point outside the light source that's distance R away from the center of the sphere?

My idea is, every point on the surface of the light source emits light in a hemisphere of directions centered around that point's normal direction (which is the direction from the sphere center to that point). So the solid angle of this is 2Pi, multiplying 2Pi by L gives the irradiance at any point on the sphere surface: 2Pi * L, and multiplying the sphere surface's area with this gives the total power emitted from the light source: 2Pi * L * 4Pi * r * r, then at any point that's R distance away from the light center, the irradiance would be the total power divided by the surface area of the sphere that passes this point, which should be 2Pi * L * r * r / (R * R)

I first saw this problem in the Real Time Rendering book, where it gives an almost same answer except there's not the "2" in the answer.

Any idea where I did it wrong? Thanks

 

[Mize]

nvm... blew the cosine

 

[shuipi]

I understand if there were an arbitrarily oriented plane passing through that point, to calculate the irradiance into that plane, I need to use the cosine factor. But this problem is without that plane, just the irradiance into that point.

 

[Mize]

My guess is that it comes from the time averaging of the Poyting vector. In acoustics you solve in terms of amplitudes (E for EM) and then convert to power with a time average of power flux.

 

[Simon F]

Sorry, I haven't read through this in detail, but are you saying you have an energy-emitting sphere of radius R1 and want to know the energy per unit area falling at distance R2, R2>R1?

It seems to me that you don't need to worry about integration at all, if you just consider the following. Let the total power output of your light source of radius R1 be P. If you now enclose that by another sphere of radius R2 then, by symmetry, the total power radiated by that must also be P. Surely, therefore, you just need to divide the relative surface areas of the spheres. <shrug>

Or have I completely missed the point?

 

[shuipi]

It's true the irradiance ratio between the 2 spheres should be inversely proportional to their areas, but the original problem gave the radiance (watt/sq meter/steradian) L at the light source's surface sphere, and asked for the irradiance(watt/sq meter) at some other point outside it.