First, allow me to offer some general guidance regarding decibels.
Decibels
always imply power ratios. Always. No matter what.
The decibel is defined, relating the ratio of two powers
P2 and
P1 as:
10 \log_{10} \left( \frac{P_2}{P_1} \right)
So one might ask, "well then how can we use decibels with voltage? Voltage isn't power," or "where does the '20' come from when we use ratios of voltages?"
The answer is that we make the pseudo-assumption that the input and output impedances of a given circuit are equal; i.e.,
Rin =
Rout =
R. (By the way, this is a realistic pseudo-assumption, since it is often the case for circuits that are optimized for maximum power transfer, signal to noise ratio, etc.)
Also we know that the power through a resistor is
V2/R.
Plugging that into our decibel equation, and noting that log(
x2) = 2log
x,
10 \log_{10} \left( \frac{ \frac{V_2^2}{ R}}{ \frac{V_1^2}{ R}} \right) = 10 \log_{10} \left( \frac{V_2^2}{V_1^2} \right) = 10 \log_{10} \left( \left[ \frac{V_2}{V_1} \right]^2 \right) = 20 \log_{10} \left( \frac{V_2}{V_1} \right)
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Now allow me to move on to something more specific: Sound/acoustics.
Sound decibels are defined in terms of power flux* ratios.
*(Power flux is power through a unit area -- you may think of the unit area as the cross sectional area of your ear canal if it helps visualize it -- or if you want to stick with SI units, the sound power propagating through a square meter.)
The reference power flux is typically defined as the power flux that corresponds to the human threshold of hearing.
If we define the reference power flux, that of the threshold of human hearing, as \Phi_{ref}, and the power of the source as
P, then we know from the problems statement that
90 \ \mathrm{dB} = 10 \log_{10} \left( \frac{\frac{P}{A_1}}{\Phi_{ref}} \right)
where A_1 is the surface area of a circle with a 1 meter radius. So now the question is, what is
10 \log_{10} \left( \frac{\frac{P}{A_{10}}}{\Phi_{ref}} \right) \ ,
where A_{10} is the surface area of a circle with radius 10 meters?
Hint: There is an easy way to do this such that you won't even need a calculator to solve this problem (you might even be able to solve it in your head). Do this by noting that log(
x/y) = log
x - log
y, and asking yourself, Does the area of a circle increase proportionally with the radius, or does it increase proportionally with the
square of the radius?"
