Light from a Hairbrush

Question from a reader:

Pick up a comb, rub it with your hair and you have got some electric charge. Now shake it and you are generating an electromagnetic wave. Am I right?

Yes indeed. So why don't we see light emitted when we brush our hair? Let's run some numbers. If you wiggle around an electric point charge, electromagnetic radiation is emitted. The power carried by this radiation is given by the Larmor formula:

$latex \displaystyle P = \frac{e^2 a^2}{6 \pi \epsilon_0 c^3}&s=1$

Well, a comb isn't a point charge. But if we're just interested in an order-of-magnitude estimate, we can pretend it is. How much charge is on a comb? It's probably a substantial overestimate, but the human body has a capacitance of around a hundred picofarads, which is part of why you can get slightly shocked when you rub your feet on carpet and touch a doorknob. For a purpose-built capacitor in an electronic circuit that's pretty small, but a comb isn't a purpose-built capacitor either so it's not unreasonable to say that as an order of magnitude it has a capacitance of 100 picofarads. That doesn't tell us how much charge it holds though, we also need to know the voltage. The voltage required to zap you with static electricity when you touch a doorknob is a surprisingly high number on the order of 10,000 volts, so we'll say the comb is charged to that potential since a comb can hold enough charge to produce a spark. 100 picofarads multiplied by 10kV gives 1 microcoulomb.

That gives us the charge e for the Larmor formula. How about the acceleration a? Earth's surface gravity is about 9.8 m/s^2, and I think a person can probably fling a comb around faster than that. Let's be generous and call it 100 m/s^2. Plug all that into the formula:

$latex P = 2.2 \times 10^{-24}\, \mathrm{W}&s=1$

So around a trillionth of a trillionth of a watt. That's why combs don't glow when you shake them. Well, the first of two reasons.

Say you then went to Wal-Mart, bought a bucket of electrons, and dumped a million coulombs worth of charge on your comb. (This would in fact blast the comb to bits, but let's pretend.) Now you've got a pretty bright 2.2 watts, but it would in fact still be invisible. You're waving the comb around a few times per second, and the resulting electromagnetic wave will tend to have a similar frequency. These are extremely long-wavelength radio electromagnetic waves, which are invisible to our eyes.

Nonetheless, this is more or less how actual radio devices work. Waves are generated by moving electrons back and forth within the metal wire antenna. However, the quantity of charge that's being moved around is large (the bazillion conduction electrons in the metal are all moving back and forth), and the acceleration that can be produced on an individual conduction electron by an applied electric field is pretty large as well.

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Very nice order-of-magnitude analysis, but I think your second half is slightly off-point. The reader asked about EM waves in general. You interpreted that as, "why don't we see light emitted", and referred to "light" specifically a second time.

As you correctly point out, the reason we don't see _light_ emitted is that a human being shaking the comb does so at a frequency of maybe a few hertz. If you could waggle the comb at a petahertz or so, then it _would_ emit a few optical photons!

The estimate using Larmor's power formula was right on, and shows that, at 10^-24 watts, the ultralongwave radio emission is completely undetectable.

By Michael Kelsey (not verified) on 15 Mar 2013 #permalink