# A Calculating a Solid Angle

#### MrTy

Hi,

I am trying to simulate necessary flux values on a car due to solar radiance. I'm trying to attain the necessary flux values using a lamp setup. I have the lamp specifications in Watts but I need to convert them into radiance values (W/m2/sr) for my application. I would like to know how to do that!

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#### Dale

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The solid angle for a circular aperture is given by $\Omega=2\pi(1-\cos(\theta))$ where $\theta$ is the angle from the center of the aperture to the edge of the aperture as seen by an observer at the center of the solid angle. Although it is hard to tell without a drawing, I assume this would be the center of the light bulb in your lamp.

#### haruspex

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Please explain in more detail what you are trying to achieve.
You mentioned parabolic reflector (other thread), which may mean that the light intensity will not be uniform across the angle. Will that matter, or do you just need the average radiance over the angle?

#### MrTy

I'm using UV lamp and the setup is shown in the figure below. I'm trying to focus this on to a surface, where I want a specified flux value. I want to calculate the radiance of the lamp that gives me my required flux value.

Edit: If I consider the transparent surface as my source which is face of a cuboid, how is the solid angle calculated in this case?

#### hutchphd

My guess is you really want irradiance (watts/square meter) at the surface in question.
Measure the area of the spot from the lamp and divide the 2 kW by that number.
This says nothing about the spectral distribution of the energy relative to sunlight.

#### MrTy

My guess is you really want irradiance (watts/square meter) at the surface in question.
Measure the area of the spot from the lamp and divide the 2 kW by that number.
This says nothing about the spectral distribution of the energy relative to sunlight.
I have given only lamp figure in the previous reply. The below is the complete picture. The lamp is kept inside a box which has a concentrator and I want to calculate the solid angle of the lamp in this scenario.

#### hutchphd

I believe you want to compare the irradiance of the sun to the irradiance of the lamp. If that is not true you need to describe your purpose in greater detail.

#### haruspex

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You are showing the light source at the apex of the parabola. It should be at the focus. E.g. if you take a line from the lamp at right angles to the parabola's axis, it should strike the parabola at 45 degrees. Maybe it's just the way you have drawn it.

According to my algebra (messy, so unreliable), of the light that is reflected, the intensity at radius y from the centre of the target area varies as $\frac{4f^2}{(y^2+4f^2)^2}$, where f is the focal length. To that you need to add the light that arrives directly.
So as suspected, the intensity diminishes as you move away from the centre of the target area.

Consider a small area radius y at the centre of the target, the target being distance x from the lamp. Of all the light emitted by the lamp the fraction that arrives directly is approximately $\frac{\pi y^2}{4\pi x^2}=(\frac y{2x})^2$.
The light arriving on the same area after reflection is the light that falls inside a circle radius y on the mirror. If this light was emitted in a cone of half angle $\theta$ then $\sin(\theta)=\frac{4fy}{y^2+4f^2}$, where f is the focal length. As @Dale noted, that is a solid angle $2\pi(1-\cos(\theta))=4\pi\frac{y^2}{y^2+4f^2}$, so of all the light the lamp emits the fraction arriving on the target by reflection is $\frac{y^2}{y^2+4f^2}$.

To get the total light on the target, add these fractions together.

Edit: more thoughts...
The analysis above only gives illumination in an area up to the aperture of the mirror. Beyond that, no reflected light hits the target. It looks like you want to brighten an area significantly larger than the aperture of the mirror, so it might be better to position the lamp somewhere between the focal point and the apex. Or use a different shape of mirror.

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"Calculating a Solid Angle"

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