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Q: Can a person get sunburn on a cloudy day and how much damage is caused on skin?
Krishna: When I was very young I used to think that clouds protect you from sunburns.  I have very fair skin, with only a minimal quantity of melanin (skin pigment) and my skin burns whenever I go out in the hot Sun without any protection. But I realized clouds don't protect you from Sun-burn when my skin became red and started burning even on cloudy days. Now I know why. 
You can  get a sunburn on cloudy or cooler days because  Ultra-violent (UV) rays, not the temperature, damage your skin and clouds do not block UV rays. Believe it or not, up to 80 per cent of the sun's UV rays can penetrate through clouds and fog, so even though you might feel more protected because you can't feel the heat of the sun, you still have to wear a sun-screen.

The effect of clouds on UV radiation is as varied as the clouds themselves. Fully overcast skies lead to reductions in surface UV irradiance. On average, scattered or broken clouds also cause reductions, but short-term or localised UV levels can be larger than for cloud-free skies if direct sunlight is also present. Clouds tend to randomise the directions of the incoming radiation (because of scattering) so that a hat may provide less protection on a cloudy day relative to a clear day.

Clouds sometimes increase, rather than reduce, levels of ultraviolet radiation! Clouds can have paradoxical effects on incident UV radiation. Although the mechanisms aren't yet entirely clear, the degree of enhancement can be significant. Forrest Mims III with the Sun Photometer Atmospheric Network and John Frederick with the University of Chicago reported in a 1994 Nature article measurements of UVB at the Mauna Loa Observatory as much as 29.8 percent above modeled clear-sky levels. In various other studies, the range has been reported as a few percent up to 50 percent.  According to some studies the highest UVI [an index of skin reddening] enhancement  found was 25 percent." (1)

Several studies suggest that reflection off the sides of cumulus clouds is one mechanism by which UV radiation can become focused. Some researchers have also postulated that refraction and scattering of direct and diffuse radiation could result in markedly increased enhancement. Thus cloud conditions that include  cirrus clouds thin enough not to completely obscure the solar disk, along with lower-altitude cumulus clouds, may lead to the perfect UV storm (1). 

But those values are with respect to  expected clear-sky UV. Compared with the level of attenuation usually seen when clouds are present, such measurements can actually be 50 to 75 percent higher than predicted. And therein lies a conundrum for those who work or recreate outdoors and depend on UV forecasts. 

Just how common is cloud enhancement? The various studies have found that between 1.4 and 8 percent of all measurements show cloud enhancement compared with clear-sky values, depending on geographic location, but as many as 25 percent of those made on partly cloudy days may show it. Most often the enhancement lasts for 10 minutes or less—but it has been known to persist for an hour.

Furthermore, people often change their behaviour on cloudy days. If they spend more time out in the open, or don't use  sunscreen, they may end up with a very bad sunburn. In general, less UV radiation is received per hour under an overcast sky than under a clear sky, but extending one’s stay outside because of cloudiness may easily compensate for this effect. A completely cloud-covered sky may still transmit substantial amounts of UV-B radiation. In principle, any amount of UV-B radiation exposure contributes to the skin cancer risk.

The answer to the question 'how much damage' is 'it depends'. It depends on the amount of melanin pigment of your skin,  the time of exposure to UV radiation, amount of cloud cover,  and your geographical area!

Footnotes:

1. https://www.americanscientist.org/article/sunshine-on-a-cloudy-day

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