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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.

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    $\begingroup$ The units of $k$ are $[\rm N m^2 C^{-2}]$. I think you forgot to include them in your dimensional analysis. $\endgroup$ – ja72 May 17 '17 at 12:10
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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.

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  • $\begingroup$ thanks a lot.. I got the point. Now a little concern of mine is that how do we know that a given constant in an equation is having dimensions. I'm aware that G also has dimensions and similarly k. But other constants like 1/2 in the formulae of kinetic energy is dimensionless. So what is the deciding factor. Thanks again.. $\endgroup$ – Abhinav Dhawan May 17 '17 at 13:52
  • $\begingroup$ @Abhinav Dhawan It'll be given in the question, or you'll have to familiarise yourself with the known equations of your syllabus. I believe that by the time you read many other chapters, this will be clear to you. You needn't do tension. Which class or year in college are you in? If school, CISCE or CBSE? $\endgroup$ – Wrichik Basu May 17 '17 at 14:02
  • $\begingroup$ As I mentioned in my answer - the "medium" (in this case, vacuum) is one of the factors that might affect the relationship, and that is likely to have certain dimensions. Similarly with gravity. With practice you will learn to figure out whether you have "all variables present and accounted for". If you leave something out, you will get a garbage result. $\endgroup$ – Floris May 17 '17 at 14:06
  • $\begingroup$ I'm in 11th in a CBSE affiliated school. I was not asking from examination viewpoint. Anyways, thanks for concern. @Floris , so you want to say that the dimensions of such constants cannot be predetermined and varies from case to case $\endgroup$ – Abhinav Dhawan May 17 '17 at 15:06
  • $\begingroup$ @AbhinavDhawan yes that is what I am saying. Not every problem is amenable to dimensional analysis- learning the limitations of a method is part of your journey in physics. Good luck! $\endgroup$ – Floris May 17 '17 at 15:08

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