Core of sun: 27 million ºC
Surface of sun: 6,000 ºC
"The leading idea among experts is the sun's magnetic field is actually bringing energy from inside the sun up through its surface and into its atmosphere."
The "napkin math" will involve gathering some data about RF power harvesting, and some assumptions (e.g. cosmic rays)
It's even more interesting that you mention the magnetic field, could we use solar panels to draw more energy from the sun to power a tesla coil and microwave the core?
It's surprising how often that movie comes up, especially given how it's often accompanied by a note that it's not good. I suspect it's hit a pseudo-cult classic film status. The last few times I've seen it mentioned I've had the desire to seek it out and watch it again since I only remember bits and pieces.
This sounds like the beginning of a great sci-fi novel.
A future where Earth's core has cooled because of excess draws to power humanity. Now, the planet is losing it's magnetic field.
To save humanity, humans need to re-heat the core, build biospheres, or leave the planet.