Scale; Some Numbers



Have you ever wondered about the numbers we casually toss about? Do you understand their significance, or realise how big or how small they really are? A typical example is the ratio of power levels between the smallest detectable signal which a radio receiver can resolve and the largest, corresponding to the overload or “de-sense” level. This can easily be 130dB or more. This represents a range of received Radio Frequency power of 10,000,000,000,000 to 1. Similarly, the dynamic range of human hearing is roughly 150 dB, between the “threshold of hearing” and the onset of discomfort and damage. Fortunately, we usually only use “the middle bit”, about 40 to 80dB of this. The numbers used in other branches of science are even more extreme.

We humans live in the middle of extremes. Consider first some of the smallest numbers. These are the province of scientists like those working at the Large Hadron Collider, (LHC), under the French-Swiss border. They deal with the tiny sizes of the building blocks of matter such as protons, quarks and the like. Here “small particles” means about 10 15 m. (For those not familiar with “scientific notation”, 103 is shorthand for 1000, 106 for 1,000,000 etc. It is very convenient for dealing with huge numbers. Similarly, 10 3 means one thousandth, 10 6 means one millionth etc. It enables us to write extreme numbers without filling the line or the page with noughts). Radio amateurs tend to think of 1 micro-second as rather short and 1 nano-second as very short. The LHC aims to reproduce conditions within 10 31 seconds after the “Big Bang”. At this time, the energy of particles “banging into each other”, was so high that electrons, protons and neutrons could not exist as complete entities but, it is believed, were broken down into even simpler particles. It is these that these scientists are getting a glimpse of in high energy collision experiments.

Going from small numbers to very large numbers, our telescopes, (radio and optical), can see out to the farthermost reaches of our universe, to nearly 13,000,000,000 light years or 8 × 1022 Miles. That is what I mean by “we are roughly in the middle” of the smallest measurable sizes and the largest.

To return to more familiar territory, (i.e. amateur radio and electricity, and more modest numbers), do you realise that at room temperature, the electrons in a conductor are wandering about in a random manner at a speed of about 106 metres per second, but with no general drift, unless a voltage is applied across the conductor? (This random motion is what causes electrical noise in resistors). When a voltage is applied, the drift superimposed on the random motion of the electrons is almost infinitesimal compared to their random motion velocity. For an electric field of a few tenths of a volt per metre along a copper wire of 1mm2 cross section, causing a current of 5 Amps to flow, the drift velocity is about 10 4 m/s, almost, (but not quite), insignificant compared to their thermal velocities. In the case of mains frequency alternating current of 5 Amps, where the direction of travel is reversed every 100th of a second, the distance electrons travel about their average position, (which doesn’t move), is only 2.3 × 10 6 metres. It is of course even less for higher frequencies.

How is it then, that signals can travel so fast, (at up to the speed of light at 3 × 108m/s), when the electrons carrying the signal move so little and slowly? To answer this it is useful to make an analogy. Now electrons repel each other very strongly and don’t need to actually touch each other to have an influence on their neighbour although they often behave as if they do. Imagine electrons in a wire to be like marbles or ball bearings tightly packed into a cardboard tube of just over the diameter of the marble or ball bearing. If you push the protruding ball at one end of the tube, the ball at the other end moves almost instantly although the individual balls hardly move. Hence the movement is conveyed fast although the movement is small. The understanding of many branches of science is aided by analogies.

Inside an atom analogies are usually of less use, as hinted at in the article on the electron of several months ago. However, it is often useful to scale up some dimensions to really appreciate their significance. For example, we are told that the interior of atoms is mostly empty space, with a nucleus in the centre and electrons in appropriate orbits surrounding it. Typical dimensions are: the diameter of the nuclei ranges between 1 and 7 × 10 15 metres, and the diameter of each atom ranges from 1 to 3 × 10 10 metres. These figures are for the lightest and the heaviest atoms respectively, i.e. for hydrogen and uranium. (If you are wondering why the uranium atom, (which is about 238 times heavier than the hydrogen atom), is only 3 times bigger it is because the massive positive charge in the nucleus of the heavier atom pulls the electrons into closer “tighter” orbits). However, if the figures 10 10m and 10 15m are translated into more “imaginable” sizes, say football fields, we can visualise more completely the emptiness of the space inside an atom. Imagine two 100m times 50m football pitches placed side by side to represent a hydrogen atom. (100m × 100m is a hectare). On this scale the nucleus would be about the size of a 1mm diameter pin-head in the centre of this Hectare. This scaling applies to the height dimension too, so you have to imagine a 100m by 100m by 100m cube with a pin-head at its centre. Then you start to appreciate emptiness!

Scaling to everyday sizes and using analogies are not a complete answer in understanding science but they often help.


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