I was studying dimensional analysis, which is a technique used in conversion of units, checking the homogeneity of equations and also sometimes deriving unknown equations, if we can guess the factors on which a physical quantity depends and if the dependence is of the product type.
While calculating the dimensional formulae of charge, I faced some difficulties.
Before mentioning my problem related to this, I want to first pose a general problem which was in my mind. The question is that can a physical quantity have more than one dimensional formulae (they are expressed in most simplified way and in terms of the seven fundamental physical quantities)?
This question occurred to me because of considering the following problems when calculating the dimensional formulae of electric charge, Q. I considered the following equation involving electric charge : $$ F=k\frac {Q_1Q_2}{r^2} $$ and then after taking arrived at the following dimensions –
1/2 dimensions in mass, 3/2 dimensions in length and -1 dimensions in time.
( * couldn't write the dimensional formulae using mathjax)
But in my book they arrived at the dimensions of electric charge using the formulae – $$Q=It$$ and thus received dimensions as –
1 in current and 1 in time.
Both these dimensional formulae are contradicting and I'm unable to find a cause for it. Any help is appreciated.
Best Answer
You run into a problem because you are considering $k$ to be dimensionless - and it is not... "the electromagnetic properties of vacuum" influences the relationship (as captured in $k$) and if you don't include that in your analysis the answer doesn't make sense.