Exploding grease, exploding microwave ovens, exploding cheese — it’s a Mythbusters fan’s dream episode :-) Of course, where there are explosions, there’s physics, and the latest episode of Mythbusters is no exception.

Here’s one: you can’t blow up C-4 by microwaving it. Kari explained in the show that this is because C-4 is a plastic explosive, and microwaves are designed to pass through plastics (as well as metal and glass). So how exactly does that work?

Microwaves heat food by a process called dielectric heating, which generally refers to the ability of many materials to absorb energy from electromagnetic radiation passing through them. Physically, an electromagnetic wave consists of rapidly oscillating electric and magnetic fields. These fields (well, primarily the electric field) exert forces on the charged particles that make up all matter — since the fields are oscillating, so do the forces. Essentially, an electromagnetic wave makes atoms and molecules rapidly jiggle back and forth, and as they do so, they bump into other nearby atoms and molecules, transferring kinetic energy to them and raising their temperature. Of course, if the atoms and molecules are gaining energy, that energy must be coming from somewhere, and the electromagnetic wave is the only source — this is how energy gets transferred from the microwaves to, say, food.

But why doesn’t this work just as well for C-4? To answer that question, we need the mathematical description of just how much power gets transferred from the waves to the matter. (This calculation is not for the faint of heart, so feel free to skip to the last paragraph) The behavior of electromagnetic fields in matter is characterized by two numbers, the permittivity (for electric fields) and the permeability (for magnetic fields). Permittivity is defined as the proportionality constant that relates the electric field \(\vec{E}\) to the “displacement field” \(\vec{D}\) (which, roughly speaking, represents the “net” electric field in a material after you account for the material’s polarization). In other words (letters):

This is simple enough when the electric field is static. But when it’s not, as in an electromagnetic wave, the changes in the displacement field lag behind the changes in the electric field due to the amount of time it takes for the molecules of the material to change polarization. We represent this using a phase difference,

\(\delta_E\) is the phase difference, which is generally a function of the frequency \(\omega\). There’s a similar equation for magnetic field and permeability,

These two quantities can be plugged into Poynting’s theorem which describes the propagation of electromagnetic energy. The equation is

where the first two terms tell you how much electric and magnetic energy (density), respectively, get transferred to the material. (The last term tells you how much passes on through.) So for our purposes,

and similarly,

Now, it happens that, for many common materials, \(\mu \approx 1\) and \(\delta_B \approx 0\), which means that the magnetic energy isn’t absorbed very much in the material. So the power per unit volume dissipated by the radiation is pretty much just

Sometimes you may see \(\epsilon\sin\delta\) written as \(\epsilon_I\), the imaginary part of the complex permittivity.

Now, getting back to the big picture: that last formula shows that the power dissipation in a material depends on the intensity \(\abs{E_0}\) and frequency \(\omega\) of the electromagnetic radiation, as well as the permittivity \(\epsilon\) and phase delay \(\delta_E\) of the material. It’s the latter two that explain why C-4 barely heats up in the microwave. Materials like glass and plastic (including C-4) consist of large molecules which (1) are more or less fixed in place, and (2) have very little dipole moment, which means that they don’t polarize very much in an electric field. Remember that the permittivity is the ratio between polarization and electric field, so if a material doesn’t get polarized much, it will have a low permittivity (small \(\epsilon\)), which in turn means that not much power is absorbed from the microwave radiation. On the other hand, liquid water, and to a lesser extent fats and sugars, are more easily polarized, which means they have higher permittivities and will absorb more energy from the microwaves. Metals, as conductors, have the largest responses to an applied electric field — you can get large numbers of electrons zipping back and forth along a piece of wire in an electric field, and this is what caused the sparking, and eventual explosion, seen on Mythbusters.