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Published 2011
What is it?
Though dimensional analysis is something you have probably not yet covered, it is actually a very simple and easy way of checking that an expression is consistent. It checks that an expression has the "correct" dimensions. Of course if the dimensions of an expression are not correct, then clearly the expression cannot be correct either. However, an
expression with the right dimensions is not necessarily correct but we can say it could be correct.
Take for example these equations for force:
$$\text{F}= \text{ma}$$
$$\text{F}= \text{pA}$$
$$\text{F}= \frac{P}{v}$$
$$\text{F}= \frac{q^2}{4\pi\epsilon_0r^2}$$
Force is given in units of newtons which is a unit in terms of [mass][length][time]$^{-2}$. 1 newton can be expressed in SI form as 1 kgms$^{-2}$. We can check that the dimensions in the above equations are consistent.
$\text{F}= \text{ma}$
$\displaystyle [mass][\frac{length}{time^2}] \equiv [mass][length][time]^{-2}$
SI units of kgms$^{-2}$
$\ $
$\text{F}= \text{pA}$
$\displaystyle \frac{[mass][length][time]^{-2}}{[length]^2}\times [length]^2 \equiv [mass][length][time]^{-2}$
$\ $
$\text{F}= \frac{P}{v}$
$\displaystyle \frac{\frac{[mass][length][time]^{-2}\times[length]}{[time]}}{\frac{[length]}{[time]}}\equiv {[mass][length][time]^{-2}}$
$\ $
$\text{F}= \frac{q^2}{4\pi\epsilon_0r^2}$
$\displaystyle \frac{({[time]}{[current]})^2}{[mass]^{-1}[length]^{-3}[time]^{4}[current]^{2}\times[length]^{2}} \equiv [mass][length][time]^{-2}$
$\ $
$\ $
Check that the units of the following expressions are consistent: