My hands burn like crazy when I use lotion! Have talked to 3 dermatologists and all they can say is its some form of "contact dermatitis"
Which I figured out when I was ~10 years old. The best theory I've come up with, and my doctors have supported, is that it isn't really burning. Its more like when you use allergy eye drops and your cells rapidly change shape. Can feel like a cigarette being put out in your eye. Same deal with your hands, but from a different cause.
When you use allergy eye drops your cells go from huge and swollen to almost immediately "normal" sized, as the swelling and fluid leave the cells it agitates the surrounding nerves and causes a harsh burning feeling.
I theorized that when the lotion hits your hands it causes the cells to immediately pick up moisture and effectively swell to a more "normal" size. Most of us have very very dry hands, simply because we work with them constantly and leave a small amount of their vital moisture on everything we touch. As the cells take moisture from the lotion they too would mildly agitate the extremely sensitive nerves in the hands, which could cause the burning sensation. This would be more likely to happen on the palms of the hands, as that skin has a higher moisture transfer rate than the skin of the backs of the hands etc. [Palms, arm pits, scalp, groin, backs of the knees and bottoms of the feet all lose moisture more rapidly than the rest of the skin; these are the areas that your body uses to produce larger amounts of sweat to keep itself at an optimal temperature! /end factoid LOL
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I occasionally get the same sensation when I wash my hands in cool or cold water. It doesn't happen with warm water because the heat soothes the nerves.