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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 $E_\mathrm c=\frac{m_\mathrm e c \omega_\mathrm c}{eB}$. 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.

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    $\begingroup$ Why not show your two sets of calculations? $\endgroup$
    – Henry
    Commented Dec 9, 2012 at 20:50
  • $\begingroup$ Alright... I showed my calcs $\endgroup$
    – Atreyu
    Commented Dec 9, 2012 at 20:56

1 Answer 1

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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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  • $\begingroup$ 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) $\endgroup$
    – Atreyu
    Commented Dec 9, 2012 at 21:57
  • $\begingroup$ I am not certain, but I think you need an extra term in the numerator of the CGS calculation $\endgroup$
    – Henry
    Commented Dec 9, 2012 at 22:26
  • $\begingroup$ 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 $\endgroup$
    – Atreyu
    Commented Dec 9, 2012 at 23:01

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