# Relationship between dynes and abamperes?

I've been reading about Ampère's law and the historic CGS systems of units, as part of some general reading on the history of electromagnetism.

The Wikipedia article states that:

The EMU unit of current, biot (Bi), also known as abampere or emu current, is defined as that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed one centimetre apart in vacuum, would produce between these conductors a force equal to two dynes per centimetre of length. Therefore, in electromagnetic CGS units, a biot is equal to a square root of dyne.

But I don't quite understand how you get from this:

$$\frac {F_m} {L} = 2 k_A \frac {I_1 I_2 } {r}$$

… to this: $\mathrm{1\,Bi = 1\,\sqrt{dyne}=1\,g^{1/2} \cdot cm^{1/2} \cdot s^{-1}}$.

I know that $I_1 = I_2$, and from the definition of the abampere, $F_m = 2 \,\mathrm{dyne}$ and $k_A = 1$. The 2 on both sides of the equation cancel out, but where do the other terms go?

Note that the definition is 2 dynes per centimeter of length; so if $F_m = 2$ dynes, then you must have $L = 1$ cm. (Or, more accurately, we have $F_m/L = 2$ dynes/cm.) Similarly, the definition requires the wires to be 1 cm apart, so $r = 1$ cm.
Once you have gotten rid of these quantities, it should be fairly easy to prove that $1 \text{ Bi} = 1 \, \sqrt{\text{dyne}}$.