Dimensional Analysis

  • Thread starter Nusc
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  • #26
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This actually is a fairly subtle point that usually doesn't matter, but seems to be a source of confusion here. If my answer adds more confusion feel free to ignore it.

Division by an arbitrary voltage is the same thing as selecting your units for measuring the potential difference. For example, suppose we are measuring the height of a man. We might use a meter stick and find that the man is 2 m tall. What this actually means is that the dimensionless ratio between the height of the man and the length of the meter stick is 2. I.e. L(man)/L(meter) = 2 -> L(man) = 2 m. In this sense all dimensionful measurements are actually dimensionless ratios to some standard with the same dimensions.

So if you divide by one Volt you are simply stating that you are using base SI units. On the other hand, if you divide by one statvolt then you are using base CGS units. If you divide by some other voltage then you are using some other units.

In this case the equation works out such that the choice of units does not matter, but that is not always the case. In other situations, where the choice of units matters there will be an explicit division by a non-arbitrary voltage and the argument to the transcendental function will be explicitly non-dimensional. The voltage on the bottom of such an equation defines a kind of "natural" unit of potential difference for the system in question.
That is useful.

Thank you!
 
  • #27
DrGreg
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For what it's worth, when I was first taught mathematics at grammar school/secondary school/high school level, many many decades ago at the age of 11, my maths teacher insisted that it was wrong to talk about "a voltage V", but you ought to say "a voltage V Volts". With that understanding, V is a dimensionless number (the ratio of your voltage to 1V), not a voltage, so there's no reason not to take its logarithm.

That's not the way that physicists usually view it, who do usually think of V as a voltage, in which case the units are irrelevant as long as you use consistent units throughout (usually, but not compulsorarily, SI units).
 

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