How are lights glowing at very hot temperatures like 5000K and 6500K kept relatively cool to touch?

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Also, why do their components and things in close proximity not melt or catch fire?

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They are colour temperature not literal temperature.

The colour temperature is obtained using different wavelengths of light.

Incandescent lights can very much burn you. They also don’t get above 2700K at the filament – tungsten has a melting point you know.

Cold colored lights generate light using methods other than just heating themselves to glow. LEDs exploit electrons falling to a lower energy state giving off photons, those tube lights excite gases using high voltage, etc.

To build on what others said, everything that’s not at 0K emits light, more precisely, everything emits EM radiation. The hotter something is, the higher the frequency of the emitted light. Things that are barely hot to touch (~300K) emit only infrared light but as they get progressively hotter they start emitting red, then orange, yellow, and so on up the rainbow.

A 5000K lamp is a lamp that emits the same light color as a 5000K body

The temperature is the equivalent of the light given out by a hypothetical construct, called a *blackbody*, at that temperature.

https://en.m.wikipedia.org/wiki/Black_body

It’s equivalent to asking how hot does a piece of metal – that can never melt – needs to be to glow with those same colours.

Glowing is from electrons changing energy levels. This always happens in a certain way in hot things, meaning the colors of a hot thing’s glow depends on its temperature.

Electrons can change energy levels by other processes in some materials. In LED’s, electrons change energy levels when electricity is applied.

To describe the color of an LED used for lighting a room, the box it comes in usually lists the temperature of a hot thing that would glow with a similar color.

The LED doesn’t actually get 5000K hot. It makes electrons jump energy levels in a way that gives off similar colors to what you would see from a thing that’s 5000K hot.