I How to ensure an equation is dimensionless when it includes "Debye"

[bumblebee77]

TL;DR Summary

Can anyone please help me figure out how to break down "Debye" into base units so that I can check if an expression is dimensionless in the CGS system?

I am trying to check if an expression is dimensionless. If it is, then I have done things correctly. However, I am stuck on how to deal with a (Debye^2) term. How can I break it down to find out if it cancels out with the other units I have left? I know this is probably a trivial question, but just cannot figure it out.

I have this ("^2" means "squared"):

(Debye^2) (s^2) / (cm^5) g

 

[vanhees71]

An expression is of course dimensionless in any system of units. It's the cgs-unit for the electric dipole moment having the dimension Franklin times cm. Now $1 \text{Fr} =1 \text{statC}=1 \sqrt{\text{g} \; \text{cm}^3/\text{s}^2}$. So $\text{Debye}^2 \text{s}^2/(\text{cm}^5 \text{g})=1 \text{g} \; \text{cm}^5/(\text{cm}^5 \; \text{g})=1$

 

[bumblebee77]

An expression is of course dimensionless in any system of units. It's the cgs-unit for the electric dipole moment having the dimension Franklin times cm. Now $1 \text{Fr} =1 \text{statC}=1 \sqrt{\text{g} \; \text{cm}^3/\text{s}^2}$. So $\text{Debye}^2 \text{s}^2/(\text{cm}^5 \text{g})=1 \text{g} \; \text{cm}^5/(\text{cm}^5 \; \text{g})=1$

@vanhees71, thank you so much. I just could not get my head around this. Really appreciate it. I voted you up and if there's any other way I can give you credit, just let me know.