Because UV radiation, particularly UVB, doesn’t cause damage by ionization, but by excitation of atoms within molecules.
Your DNA is made up of molecules that we call nucleobases (those are the A G C T and U you hear about when you’re learning about DNA or RNA). What UV light in enough intensity can do is that it will cause two neighboring T’s (thymines) to fuse together, thus rendering them unreadable by the molecules that read stuff from your genome. There’s no ionization needed there, it’s just that under certain conditions, neighboring thymines are close enough together to have some of their atoms excited and conjoin with the other thymine (specifically by exciting carbon atoms within the aromatic core of thymine)
It is a common misconception that the frequency of an electromagnetic wave is the only thing that can make said radiation dangerous. Cosmic, Gamma, and X-ray radiation are obviously incredibly dangerous because they are ionizing. However, there are other characteristics of electromagnetic waves and the nature of the exposure that can absolutely still cause harm
Take, for example, visible light and infrared light. Visible lasers and even just bright lights can absolutely cause sensitive tissues like those found in your eyes to become damaged. Other factors matter, like amplitude, dosage, and duration, absolutely can make a difference.
Think about a bright spotlight. If shined in one’s eyes for long enough, it could cause blindness. Simply glancing at it momentarily may be uncomfortable but not dangerous.
Even far lower frequency EM waves such as radio waves or microwaves could cause harm in the right conditions. For example, the 5G radio in your phone is relatively low power and would likely never hurt you. However, if you strapped a dozen high-powered 5G antenna to your body, there is a possibility that it could burn you if you were exposed for long enough at a high level of power.
Now, back to the UV example, UV light can absolutely cause burns and tissue damage without adequate protection or if simply exposed for a long enough time. UV radiation, however, differs somewhat in that it is powerful enough to damage the DNA in your cells. Prolonged exposure to UV light, like with repeated sunburn, could absolutely cause cellular damage.
TL:DR all electromagnetic waves can cause damage if they have the right power level and exposure conditions.
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