This may be a stupid question but I am having trouble getting the same result in CGS units as I would if I used SI units for a unitless calculation.
I have to calculate $E_\mathrm c=\frac{m_\mathrm e c \omega_\mathrm c}{eB}$. I have in CGS units that $\frac{e}{m_\mathrm ec}$ is $1.76\times10^{7}\ \mathrm{s^{-1}\ G^{-1}}$ and $B=10^{-4}\ \mathrm G=10^{-8}\ \mathrm T$ and $\omega_\mathrm c$ is just some frequency. Essentially, when I do this calculation in SI units and in CGS units I get very different results. From this equation it seems clear to me that $\frac{e}{m_\mathrm ec}$ in CGS and SI should only be off by a few factors of $10$ (namely $4$) since $B$ is scaled by $10^{-4}$ from CGS to SI, but this is not the case. What am I doing wrong?
Here are my calculations:
(i) CGS: $$\frac{(1.52\times10^{18}\ \mathrm s^{-1})}{(1.76\times10^7\ \mathrm{s^{-1}\ G^{-1}})(3\times10^{-4}\ \mathrm G)}$$
(ii) SI: $$\frac{(9.11\times10^{-31}\ \mathrm{kg})(3\times10^8\ \mathrm{m\ s^{-1}})(1.52\times10^{18}\ \mathrm {s^{-1}})}{(1.6\times10^{-19}\ \mathrm C)(3\times10^{-8}\ \mathrm T)}$$
Unless the value I'm using for $\frac{e}{m_\mathrm ec}$ in CGS is wrong.
Best Answer
The problem comes with the CGS units of charge. One of them is the statcoulomb with $$1 \mathrm{C} \leftrightarrow 2997924580 \,\mathrm{statC} \approx 3 \times 10^9 \mathrm{statC}$$
It is this factor of $3 \times 10^9$ which seems to be the difference between your two calculations, and needs to be taken into account in the CGS calculation. Your $\frac{e}{m_ec}$ may be about $1.76\times10^{7}$ but its units are not $s^{-1}G^{-1}$.