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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 $E_c=\frac{m_e c\, \omega_c}{eB}$ I have in CGS units that $\frac{e}{m_ec}$ is $(1.76\times10^{7}s^{-1}G^{-1})$ and $B=10^{-4}G=10^{-8}T$ and $\omega_c$ 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 $\frac{e}{m_ec}$ 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?

Edit: Here are my calculations.

(i) CGS: $\frac{(1.52\times10^{18}s^{-1})}{(1.76\times10^7s^{-1}\,G^{-1})(3\times10^{-4}G)}$

(ii) SI: $\frac{(9.11\times10^{-31}\mathrm{kg})(3\times10^8\mathrm{m \, s^{-1}})(1.52\times10^{18}s^{-1})}{(1.6\times10^{-19}C)(3\times10^{-8}T)}$

Unless the value I'm using for $\frac{e}{m_ec}$ in CGS is wrong...

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Why not show your two sets of calculations? – Henry Dec 9 '12 at 20:50
Alright... I showed my calcs – Atreyu Dec 9 '12 at 20:56
up vote 1 down vote accepted

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 $s^{-1}G^{-1}$.

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Thanks very much for your help – Atreyu Dec 9 '12 at 21:48
Does this mean the calculation in CGS units is wrong? I mean how can it convert differently if the above calculation is supposed to give a unitless value? (It is supposed to be proportional to the $\gamma$ factor in SR) – Atreyu Dec 9 '12 at 21:57
I am not certain, but I think you need an extra term in the numerator of the CGS calculation – Henry Dec 9 '12 at 22:26
The problem is I think that the answer from the CGS calculation gives me the right answer (as is)... But I don't see why the SI calculation would be wrong – Atreyu Dec 9 '12 at 23:01

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