# Compare light volume of two light bulbs with different degree spreads

I got two light bulbs, they are the same, but one spreads the light with an angle of 45 degrees, while the other 60 degrees.

I have measured their lux from a height of 1 meter, starting right below the light bulb an, then measured again, walking 20 cm to the right, and again at 40 cm and so on. By doing this I was able to measure how well the light bulb spreads the light. Here are the actual measurements.

         0 cm       20 cm      40 cm      60 cm     80 cm    100 cm
45°    624 lux     371 lux     82 lux     18 lux     6 lux     3 lux
60°    327 lux     307 lux    152 lux     54 lux    17 lux     7 lux


Now my question, how do I verify that the light volume is in fact the same only spread differently?

I'm thinking about giving each measuring point a value, to help calculate their importance. A lux value at distance of 100 cm, would count more at than at 0 cm.

Hope it's clear what I want. But I'm looking for a way to validate that the two light bulb are in fact sending out the same amount of light, but with different spreads.

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For those unfamiliar with photometric units, "lux" is the same "as lumens per square meter", and "lumen" is a unit of photometric flux. So we can also think of the lamp as a device that shoots "bullets" randomly in different direction, where the density of bullets landing on different infinitesimal areas of the floor is as in the OP's table. The OP is then asking whether the two sources shoot the same number of bullets in total. – Henning Makholm Nov 11 '11 at 14:49
I don't understand the downvote. It seems like a good question to me. – Ross Millikan Nov 11 '11 at 15:14

## 1 Answer

It would be better to take the data over a sphere centered on the light bulb. You want to take your data far enough away that the size to the bulb doesn't matter, but close enough that "all" the light falling on your meter comes from the bulb. Then it shouldn't matter what distance you use. The solid angle only depends upon the central angle. You would hope the pattern is symmetric around the axis of the bulb and all you need to do is map the intensity as a function of polar angle. Then if you integrate lux$(\theta)\cdot \theta$ out to where the intensity goes to zero you should get the same value for each bulb. The factor of $\theta$ reflects the fact that the solid angle between $\theta$ and $\theta + \Delta \theta$ is proportional to $\theta$, it is like the circumference of a circle.

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Almost surely the angles are not the half cone. If the angles were the half cone, at 100cm, even if the light source is non-uniformly emitting the light in the solid angle, its intensity should be closer to 1/2 the value at 0cm, and not 1/40. At 1m below the bulb, a sharp half-cone of 22.5 degrees will be completely dark beyond ~42cm, and half-cone of 30 degree will be so beyond ~58cm, which I think corresponds better with the data. – Willie Wong Nov 11 '11 at 15:54
@WillieWong: I didn't read carefully enough to see the way the data was taken. I'll update the answer. – Ross Millikan Nov 11 '11 at 15:59