These teaching questions are great, so I wish we had better answers.
Connecting force and energy intuitively for gravity may not be too hard. You can say that separating two objects gives them gravitational potential energy. The really familiar case is the one of pulling objects up from the earth's surface or dropping them. That gives potential energy mgh, where m is the mass of the object, h is the height from the surface, and g is the earth's gravitational acceleration field near the surface. So as something falls, it loses mgh and that energy turns into kinetic mv2/2. You can express the same process by saying there's a force mg on the object so that force causes the changes in mv2/2. If your students know derivatives, you can take the time derivative of mv2/2 and show that it goes as the force times the velocity, the same as the rate at which work is being done. I think with a little practice most students will become comfortable with this example, and maybe able to generalize to other cases with other force laws, such as springs.
Magnetism is somewhat trickier. It's especially tricky to describe the effect of a magnetic field on a moving charged particle (ignoring intrinsic spin magnetism, if any). Here there is a force, but it's at right angles to the motion so it doesn't change the energy. It's an example where really the energy picture doesn't replace the force picture. For the forces (and torques) between two intrinsically magnetic objects (spins, bar magnets,...) you can work from an energy picture, but since the energy depends not just on the distance but also on how the objects are oriented, it's too complicated for initial development of intuition.
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(published on 03/10/2017)