The famous Rule of 72 says that to find out how many years it takes an investment to double in value, divide 72 by the annual percentage rate. I’ll come back to that in a little bit.
This morning I read a really good article, Fifty Things you can do with a Software Defined Radio. The article includes a rule of thumb for how long an antenna needs to be.
My rule of thumb was to divide 72 by the frequency in MHz, and take that as the length of each side of the dipole in meters [1]. That’d make the whole antenna a bit shorter than half of the wavelength.
Ideally an antenna should be as long as half a wavelength of the signal you want to receive. Light travels 3 × 108 meters per second, so one wavelength of a 1 MHz signal is 300 m. A quarter wavelength, the length of one side of a dipole antenna, would be 75 m. Call it 72 m because 72 has lots of small factors, i.e. it’s usually mentally easier to divide things into 72 than 75. Rounding 75 down to 72 results in the antenna being a little shorter than ideal. But antennas are forgiving, especially for receiving.
Just as the Rule of 72 for antennas rounds 75 down to 72, the Rule of 72 for interest rounds 69.3 up to 72, both for ease of mental calculation.
The approximation step comes from the approximation log(1 + x) ≈ x for small x, a first order Taylor approximation.
The last line would be 2 rather than 2.05 if we replaced 72 with 100 log(2) = 69.3. That’s where the factor of 69.3 mentioned above comes from.
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[1] The post actually says centimeters, but the author meant to say meters.