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Good absorbers are good emitters: An ideal absorber is often called a black body. It absorbs all the radiation that hits it. It is easy to see that an ideal absorber of a particular wavelength of radiation is also the best possible emitter at that wavelength. That is, no object at temperature T can emit more radiation than a black body.
The cup will never be warmer than the kettle at best they'll equalise and the kettle is being warmed by electricity so it's entirely losing heat and no perpetual motion to worry about.
Making tea, thermogram
The shiny surface also has a lower surface area