What energy is transformed into when I stop two magnets from repeling each other?

Question

If I take two magnets and I try to make them touch in the way they repel each other, I transform the energy from my muscles into potential energy. But where does the energy go when I keep the magnets at the same distance?

Answer 1

The energy is not going anywhere, because you are not doing work. Recall that

$\text{Work done by force} =\int_{\textbf{r}_1}^{\textbf{r}_2}\textbf{F} \bullet d\textbf{r}$

When you just try to keep the magnets at the same distance, no energy is being transferred. Only if you try to move them closer together or further apart is energy transferred. In the first case the energy from your muscles is converted kinetic energy and ultimately into potential energy in the magnetic field (and some heat energy due to friction in your muscles and joints). In the second case the magnetic field does work on your hands- the potential energy in the magnetic field is converted into kinetic energy and ultimately heat energy due to friction.