I'm studying an old E&M textbook which uses cgs unit system. I'm re-writing the formulas in SI unit.
The book says $\vec{H}=\vec{B}-4\pi\vec{M}$.
So I guessed that $H$ should have same unit as $B$ in cgs system.
But I found that $H$ has unit of Oersted, which is same as $1cm^{0.5} g^{-0.5} s^{-1}$.
How is it possible?
In essence in the cgs system of units $\mu_0 = 1$
B is the magnetic flux density (or magnetic induction)and has the derived unit gauss which in base cgs units $\rm cm^{-\frac 12}g^{\frac 12}s^{-1}$
H is the magnetic field strength and has the derived unit oersted which in base cgs units $\rm cm^{-\frac 12}g^{\frac 12}s^{-1}$
You may also be interested in the fact that E is the electric field with the derived unit $\rm statV\, cm^{-1}$ which in terms of cgs base units is $\rm cm^{-\frac 12}g^{\frac 12}s^{-1}$, the same as that for B and H.
There are very many examples of cgs units and Gaussian units being explained and discussed on the Internet and in textbooks. For example look at this PDF.
Dimensional analysis with the standard cgs or SI dimensions will not reveal the nature of where the $4\pi$s ought to go. Instead, you have to use an extra dimension, which turns the $4\pi$ into a variable, which becomes either $4\pi$ or $1$, according to the system.
I use the 'rule of substance' here. It is in my physics pdf, but i shall describe it here.
Space, time, fields and fluxes of all kinds, potentials of all kinds, are not substances, and are left unaltered.
Mass, and mechanical quantities with Mass in the dimension (forces, energy, pressure, density, power), charge, dipoles of all kinds, and their respective densities, capacitances and conductances, susceptances and susceptabilities, are quantities of substance and represent a dimension $S^1$.
Resistances and inductances represent a dimension of $S^{-1}$.
In order to change a formula, you tick the substances in the equation, and cross the inverse substances (ie $S^{-1}$). If there is an inbalance of ticks and crosses, this is corrected by a ticked $4\pi$ (cgs->si), or cross $4\pi$ (si->cgs).
So in the equation of the question: $H=B-4\pi M$
$H$ is a field, and thus not a substance. Likewise $B$ is a flux density, and is also not a substance. The $M$ is a magnetic polarisation and is thus a substance. To counter the imbalance of the substabce dimension, you need to divide the $M$ by a 'ticked $4\pi$', which cancels out the $4\pi$, giving $H=B-M$.
This can be done at reading speed.
$ sign, not [math] and [\math]. I've edited it here. A quick Mathjax tutorial can be found here.
$\endgroup$
As you will have found, in SI units $B$ is measured in teslas ($T$) and magnetic flux $\Phi_B$ (magnetic flux) is measured in webers ($Wb$), thus a flux density of $1\ Wb/m^2$ is $1\ T$. The SI unit of tesla is equivalent to newton·second/coulomb·metre.
In Gaussian-cgs units, $B$ is measured in gauss ($G$). The conversion is $1\ T = 10000\ G$. One nanotesla is equivalent to 1 gamma ($\gamma$). The $H$-field is measured in amperes per metre ($A/m$) in SI units, and in oersteds ($Oe$) in cgs units. It is equivalent to 1 dyne per maxwell. And, as you have found, its Gaussian base is $1 cm^{−0.5}\cdot g^{0.5}\cdot s^{−1}$. [1]
The oersted is closely related to the gauss as you might have inferred from the units. In a vacuum, if the magnetizing field strength is $1\ Oe$, then the magnetic field density is $1\ G$, whereas, in a medium having permeability $\mu_r$ (relative to permeability of vacuum), their relation is:
$$B({\text{G}})=\mu _{r}H({\text{Oe}})$$
Because oersteds are used to measure magnetizing field strength, they are also related to the magnetomotive force (mmf) of current in a single-winding wire-loop:
$$\displaystyle 1{\text{ Oe}}={\frac {1000}{4\pi }}{\text{A}}/{\text{m}}$$
Since Amperes are a measure of unit-charge per unit-time, you can probably work out from units of $A/m$ how to get units of $cm^{-0.5}\cdot g^{0.5}\cdot s^{-1}$ :-)
