People watch sunsets all the time. You don't see people using special "sunset glasses".
Yet with an eclipse, warnings are posted everywhere not to view it without special glasses.
Why is this?
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During a sunset, the Sun is lower in the sky than during most of the day - much lower. Therefore, light from the Sun travels through about 120 miles of dense atmosphere, compared to the roughly 2 miles it travels through from straight up. Here's a rough sketch (not to scale) to demonstrate this. It is clear that $B>A$:
Light scatters in the atmosphere; in fact, one type of scattering is why the sky is blue. The longer travel distance means that there is much more scattering of ultraviolent light, which in turn means that the light you see is less intense.
Okay, you say. But doesn't the eclipse still block a lot of light? Well, unless there's 100% coverage - totality - there's still plenty of light coming from the uncovered part of the Sun, and that matters. The uncovered part is as bright as it normally is, and looking at that part is just as dangerous with or without the eclipse.
There's one more thing to consider, which is that people watching a sunset don't look at the Sun; they look at the clouds and sky around the Sun. If you look directly at the Sun, your eyes will be damaged, no matter what's happening, an eclipse or a sunset.
There is also a second reason for this. The density of the atmosphere decreases as you go up in the atmosphere. At sunset, the sun's rays hit the atmosphere at an angle and refract through the atmosphere. The refraction is proportional to the incident angle, so it happens more at the horizon than during the day when the sun is high in the sky. The refraction also causes dispersion of light. This is part of the reason why the sun becomes red at sunset. The redness is also caused by the increasing scattering of blue light as the light passes through more atmosphere. The other colors, including ultraviolet radiation are systematically removed from the sun as it drops below the horizon. Because the light is refracted, the sun's image appears above the horizon even though the sun has actually already gone below the horizon. At this point, the damaging ultraviolet light can no longer reach your eyes.