Oh Larmor! Energy in Electromagnetic Waves

Posted in Cute Problems, The Universe and Stuff with tags , , , on April 16, 2021 by telescoper This week I started the bit of my Advanced Electromagnetism module that deals with electromagnetic radiation, including deriving the famous Larmor Formula. It reminded me of this little physics riddle, which I thought I’d share again here.

As you all know, electromagnetic radiation consists of oscillating electric and magnetic fields rather like this: (Graphic stolen from here.) The polarization state of the wave is defined by the direction of the Electric field, in this case vertically upwards.

Now the energy carried by an electromagnetic wave of a given wavelength is proportional to the square of its amplitude, denoted in the Figure by A, so the energy is of the form kA2 in this case with k constant. Two separate electromagnetic waves with the same amplitude and wavelength would thus carry an energy = 2kA2.

But now consider what happens if you superpose two waves in phase, each having the same wavelength, polarization and amplitude to generate a single wave with amplitude 2A. The energy carried now is k(2A)2 = 4kA2, which is twice the value obtained for two separate waves.

Where does the extra energy come from?

Amplitude & Energy in Electromagnetic Waves

Posted in Cute Problems, The Universe and Stuff with tags , , , on September 22, 2015 by telescoper

Here’s a little physics riddle. As you all know, electromagnetic radiation consists of oscillating electric and magnetic fields rather like this: (Graphic stolen from here.) The polarization state of the wave is defined by the direction of the Electric field, in this case vertically upwards.

Now the energy carried by an electromagnetic wave of a given wavelength is proportional to the square of its amplitude, denoted in the Figure by A, so the energy is of the form kA2 in this case with k constant. Two separate electromagnetic waves with the same amplitude and wavelength would thus carry an energy = 2kA2.

But now consider what happens if you superpose two waves in phase, each having the same wavelength, polarization and amplitude to generate a single wave with amplitude 2A. The energy carried now is k(2A)2 = 4kA2, which is twice the value obtained for two separate waves.

Where does the extra energy come from?