Question: I saw this ad on TV for a natural supplement that’s supposed to improve your vision. The ad claims “you will be able to see a candle flame from 10 miles away.” That just doesn’t seem possible to me, and I was wondering if maybe you could “shed some light” on that claim. — CH, Hales Corners, WI
Answer: No problem. But we’ll need to make some assumptions (and do a little math).
- Its a standard beeswax candle, similar to what you put inside pumpkins on Halloween.
- The candle has been burning long enough to get the wax melting, and it’s putting out maximum light.
- It’s totally dark, with no competing lights to affect your vision.
- The air is very clear, with no suspended dust or water vapor.
- Your eyes are dark-adapted, and your pupils are fully dilated.
Let’s first get a handle on just how much light a candle puts out. A burning candle is converting chemical energy into heat and light at a rate of about 80 watts (80 joules/second). But it’s obviously not as bright as an 80 watt bulb. This is because candles are very inefficient light sources. Most of that 80 watts is heat, with only about 5 watts of actual light. That amount of light is comparable to a typical Christmas tree bulb.
Now let’s convert those 5 watts into the actual number (N) of photons being emitted each second. Each photon is a single “particle” (more properly “quantum”) of light carrying an amount of energy proportional to its frequency. For simplicity, we’ll assume all the light is yellow with a wavelength of λ = 570 nm (nanometers). Here’s the formula:
P = ΔE/Δt = N × (hc/λ)/Δt ⇒ N = PΔtλ/hc
P is the power of the candle (5 watts), λ is the wavelength (5.7 × 10-7 m), h is Planck’s constant (6.6 × 10-34 J·s), and c is the speed of light (3 × 108 m/s). When you run all those numbers through your calculator, you’ll discover the candle emits an astonishing 1.4 × 1019 photons of light every second.
Of course, these photons are being emitted in all directions. How many enter your eye depends on how large your pupil is (human dark-adapted average = 8 mm) and how far away the candle is. You can use geometry to calculate the number of photons entering your eye, assuming they spread out evenly from the candle in all directions. At a distance of 16 km (about 10 miles) that expanding “sphere” of light will have a surface area of:
A = 4πR2 = 3,217 km2 = 3.217 × 1015 mm2 *
An 8 mm diameter pupil has an area of A = πR2 = 50 mm2. So that means the number of photons entering the eye will be:
1.4 × 1019 photons × (50 mm2 / 3.2 × 1015 mm2) = 218,000 photons *
Is that enough to “see?” An experiment done at Columbia University in 1941 found that only 5-15 photons were required to experience the sensation of visible light in 50% of the subjects tested. But that was under ideal conditions: the light was aimed at the most sensitive part of the retina, it was blue-green light with λ = 510 nm (the color to which the human eye is most responsive), and the light was transmitted as a brief flash in a totally darkened room.
So for spotting a candle, 10 miles is a believable distance. An online search for “candle + visible + maximum + distance” finds a range of values with a maximum of 30 miles. Even at that distance, the photon count only drops to 24,000.*
But even clear air will scatter some of the light before it reaches your eye. The greater the distance, the greater the loss. And we’ve made other simplifying assumptions. We ignored atmospheric refraction. We also assumed a candle only emits yellow light, but much of its light is in the orange-red wavelength range, to which the human eye is less responsive.
If we ignore these other factors, as we have done, we must temper our interpretation of the results. That’s why working the calculation backward from “5-15 photons” (call it “10” for calculation purposes) to find the candle visibility distance yields an unbelievable result of R = 2.4 billion miles. What we did here is called a “first order” (i.e., “rough”) calculation.
Still, the claim you heard in that ad is believable, but you won’t need any natural supplements to see a candle 10 miles away (under good conditions). The human eye is amazingly sensitive to light, and can do just fine on its own.
By the way, the arrow in the graphic is not pointing to a “hot pixel” on your screen. That’s just my 1-pixel “simulation” of a candle seen from a distance of 10 miles.
* I gratefully acknowledge Terry Toepker, Physics Dept., Xavier University. He noticed that I forgot to square the conversion factor between mm and km. Values marked with an asterisk have been corrected. Good eyes. Thank you sir.
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