Regarding difference in dimensions of a physical quantity in different unit systems So, Coulomb's Law states that $F_e = k\dfrac{q_1q_2}{R^2}$, where $F_e$ is the force of attraction between two charged particles of magnitude $q_1$ and $q_2$, where the distance between them is $R$. $k$ is Coulomb's constant here.
In SI units, $k$ has some dimensions whereas in CGS/Gaussian units, it doesn't. This would imply that the dimensions of charge in both of these unit systems is different, right?
But, shouldn't a certain physical quantity have the same dimensions in every single unit system? Please clarify this.
 A: I will simply quote at length from John David Jackson's Classical Electrodynamics, Appendix A part 1

The arbitrariness in the number of fundamental units and in the dimensions of any physical quantity in terms of those units has been emphasized by Abraham, Planck, Bridgman$,^{1}$ Birge,$^{2}$ and others. The reader interested in units as such will do well to become familiar with the excellent series of articles by Birge.
The desirable features of a system of units in any field are convenience and clarity. For example, theoretical physicists active in relativistic quantum field theory and the theory of elementary particles find it convenient to choose the universal constants such as Planck's quantum of action and the velocity of light in vacuum to be dimensionless and of unit magnitude. The resulting system of units (called "natural" units) has only one basic unit, customarily chosen to be mass. All quantities, whether length or time or force or energy, etc., are expressed in terms of this one unit and have dimensions that are powers of its dimension. There is nothing contrived or less fundamental about such a system than one involving the meter, the kilogram, and the second as basic units. It is merely a matter of convenience.$^{3}$
A word needs to be said about basic units or standards, considered as independent quantities, and derived units or standards, which are defined in both magnitude and dimension through theory and experiment in terms of the basic units. Tradition requires that mass ($m$), length ($l$), and time ($t$) be treated as basic. But for electrical quantities there has been no compelling tradition. Consider, for example, the unit of current. The "international" ampere (for a long period the accepted practical unit of current) is defined in terms of the mass of silver deposited per unit time by electrolysis in a standard silver voltameter. Such a unit of current is properly considered a basic unit, independent of the mass, length, and time units, since the amount of current serving as the unit is found from a supposedly reproducible experiment in electrolysis.
On the other hand, the presently accepted standard of current, the "absolute" ampere "is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross section, and placed one metre apart in vacuum, would produce between these conductors a force equal to $2\cdot 10^{-7}$ newton per metre of length." This means that the "absolute" ampere is a derived unit, since its definition is in terms of the mechanical force between two wires through equation (A.4) below.$^{4}$ The "absolute" ampere is, by this definition, exactly one-tenth of the em unit of current, the abampere.
Since 1948 the internationally accepted system of electromagnetic standards has been based on the meter, the kilogram, the second, and the above definition of the absolute ampere plus other derived units for resistance, voltage, etc. This seems to be a desirable state of affairs. It avoids such difficulties as arose when, in 1894, by act of Congress (based on recommendations of an international commission of engineers and scientists), independent basic units of current, voltage, and resistance were defined in terms of three independent experiments (silver voltameter, Clark standard cell, specified column of mercury).${}^{5}$ Soon afterward, because of systematic errors in the experiments outside the claimed accuracy, Ohm's law was no longer valid, by act of Congress!
The Systeme International d'Unites (SI) has the unit of mass defined since 1889 by a platinum-iridium kilogram prototype kept in Sevres, France. In 1967 the SI second was defined to be "the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium-133 atom." The General Conference on Weights and Measures in 1983 adopted a definition of the meter based on the speed of light, namely, the meter is "the length of the distance traveled in vacuum by light during a time 1/299 792 458 of a second." The speed of light is therefore no longer an experimental number; it is, by definition of the meter, exactly с = 299 792 458 m/s. For electricity and magnetism, the Systeme International adds the absolute ampere as an additional unit, as already noted. In practice, metrology laboratories around the world define the ampere through the units of electromotive force, the volt, and resistance, the ohm, as determined experimentally from the Josephson effect ($2e/h$) and the quantum Hall effect ($h/e^2$), respectively.$^{6}$
$^{1}$ P. W. Bridgman, Dimensional Analysis, Yale University Press, New Haven, CT (1931).
$^{2}$ R. T. Birge, Am. Phys. Teacher (now Am. J. Phys.), 2, 41 (1934); 3, 102, 171 (1935).
$^{3}$ In quantum field theory, powers of the coupling constant play the role of other basic units in doing dimensional analysis.
$^{4}$ According to Ampère, the force per unit length between two infinitely long, parallel wires separated by distance $d$ and carrying currents $I$ and $I'$ is
$$\frac{dF}{dl} = 2k_2 \frac{II'}{d} \tag{A.4}$$
The proportionality constant $k_2$ in (A.4) is thereby given the magnitude $k_2 = 10^{-7}$ in the SI system. The dimensions of the "absolute" ampere, as distinct from its magnitude, depend on the dimensions assigned $k_2$. In the conventional SI system of electromagnetic units, electric current ($I$) is arbitrarily
chosen as a fourth basic dimension. Consequently charge has dimensions $It$, and $k_2$ has dimensions
of $mlI^{-2}t^{-2}$. If $k_2$ is taken to be dimensionless, then current has the dimensions $m^{1/2}l^{1/2}t^{-1}$. The question of whether a fourth basic dimension like current is introduced or whether electromagnetic quantities have dimensions given by powers (sometimes fractional) of the three basic mechanical
dimensions is a purely subjective matter and has no fundamental significance.
${}^{5}$ See, for example, F. A. Laws, Electrical Measurements, McGraw-Hill, New York (1917), pp. 705-706.
${}^{6}$ For a general discussion of SI units in electricity and magnetism and the use of quantum phenomena to define standards, see B. W. Petley, in Metrology at the Frontiers of Physics and Technology, eds. L. Corvini and T. J. Quinn, Proceedings of the International School of Physics "Enrico Fermi," Course
CX, 27 June-7 July 1989, North-Holland, Amsterdam (1992), pp. 33-61.

I made a slight modification to footnote 4 to include the equation (A.4), but I do suggest checking out the rest of the appendix for a further discussion and elaboration of these points.
