Not long ago I wrote about one of the conceptual problems between intro mechanics and intro E&M from the freshman physics standpoint: developing a sense of the size of units between the two subjects. For instance, accelerating a spacecraft to escape velocity is no easy feat, but accelerating an electron to escape velocity only requires a tiny fraction of a volt. Let’s do a similar calculation with power radiated from an accelerating point source.

As we know, a particle with an electric charge produces an electric field. A *moving* charged particle also generates a magnetic field. And though we won’t worry about the details yet, it turns out that an *accelerating* charge actually causes its own electric and magnetic fields to interact in such a way as to cause electromagnetic waves to ripple out in the form of light. Electromagnetic waves carry energy, and so accelerating a charged particle will require a bit of additional force in order to supply that power. The actual amount of power is given by the Larmor formula:

As usual, a is acceleration, q is charge, c is the speed of light, and μ_{0} is the magnetic constant. We can use this to answer a conceptual question of the type that beginning students can use to develop a sense of scale. This is a classic question, which I first saw in Griffiths’ book.

An electron is released from rest and falls under the influence of gravity. How much power does it radiate?

We can look up the charge of the electron and plug it into the Larmor formula with the acceleration equal to the usual 9.8 m/s^{2}. I get:

This is not a lot of power. Even if you get a macroscopic glob of electrons (say, Avogadro’s number of them) it’s still an absolutely tiny total radiated power.

In retrospect it had to be tiny. Hairbrushes don’t burst into glow when dropped. But as always, having a good order-of-magnitude grasp of the general concept can be the difference between understanding and confusion.