"on earth" or "on the earth"

Be careful with the word 'earth'. Sometimes it means the planet, and sometimes it means soil. "On the earth" might be taken to mean something is sitting on the soil or on the dirt. Yes, they can mean the same thing, but not always.

Why not come back with the complete sentence

Also, there's a British electrical reference that British use 'on earth' could mean on electrical grounding (terminal).

I think we need some context.

Yes, they are virtually synonymous. The first is more common.

They are used in slightly different ways.

examples

"The greatest show on earth"

These animals no longer exist anywhere on the earth.