Why our eyes hurt looking at welding arcs?

People misunderstand the inverse square law: the intensity on your retina does not drop with distance until you cannot resolve the object any longer
When you are close enough to resolve the arc, the intensity on your retina is constant regardless of distance. This is due to the same reason that trees in the distance don’t look really dark compared to those up close. Also the reason that the rock in the photo below does not look darker the farther away it is. You do in fact receive a total amount of light that decreases as 1 over the distance squared. However, at the same time, the area of the object’s image on your retina decreases by the same amount. Therefore it appears to be the same brightness until the object is so far away it is contained in the point spread function of the eye, which is about an arc minute or roughly a basketball a kilometer (half mile) away. (That would be a football 1 km away for folks not in North America.)
Notice that the rock further away from the camera does not look darker according to the inverse square law. In fact, all the rock looks roughly the same lightness, regardless of distance. Rock on right is lighter due to direct sunshine.
So let’s say the arc is effectively 3 inches (7.5 cm) in diameter. The intensity on your retina would not even begin to decrease until you got to 250 meters (yards). So at 500 meters (yards) the intensity would drop to 25%)

The moon and Venus are roughly same same brightness (Venus may be about twice as bright as the Moon at their surfaces). Yet Venus is 160 times further away than the Moon. According to the way people use the inverse square law, Venus should be only 1/25,600 as bright as the Moon. Even allowing for Venus to be about 10 times the projected area of the Moon, Venus should only be 1/2300 as bright as the Moon. Yet they appear to be comparable in brightness. Obviously, people are not using the inverse square law correctly. The total amount of light from the Full Moon (stellar magnitude -13) is in fact 2290 times greater than the total amount of light from Venus (brightest stellar magnitude -4.6.) But the apparent brightness on the retina depends on the size of the image as well as the total light.
Welding arc seen in close proximity.

Same welding arc seen a half block away. These are the same level of irradiance at the retina. The spot on the retina is much smaller in the second case, but the watts per square cm on the retina are the same. The same damage can occur, but only in a smaller spot.
Now it is true that a much smaller spot on the retina is affected, but since your eye is always moving around, shifting the image, you can still damage a lot of retinal tissue. Therefore moving 500 meters away did not reduce the irradiance (brightness in layman terms) on the retina!
You probably need to be several km away to be safe, where the welder’s arc is just a pin point, as small as a star seen from Earth. The Rayleigh scatter of the atmosphere will probably be a significant factor in reducing the ultraviolet intensity in the image in your eye at this distance.
If the welding arc is smaller, say only a couple of cm, (much smaller than in the photos) then the r squared losses begin sooner, say about 100 meters. Up to that point, only Rayleigh scattering is reducing the intensity. That is not enough.
I would never look at a welding arc from less than 10 km away without an appropriate filter. Distance is not a reliable way to reduce the intensity unless you are absolutely certain that the arc is small compared to your eye’s resolution.
Precautions:

 ·        Make sure you have protective clothing  – jacket, no frayed pants, helmet with proper shade, steel toe boots, etc.)

 ·        NO Flammable liquids! (lacquer thinner is great for cleaning dirty metal before welding, but causes really big flames if you don’t wipe off with  a simple green type cleaner)

 ·        Keep a look out for the main power switch to kill the power, if you have a problem. 

 ·        Have some extinguishers handy.

 ·        Welding  should always be done in an area with proper ventilation since welding fumes are not good for your health.

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