# Derivation for the e/m of an electron when moving in a circular path due to a constant magnetic field

I'm attempting to derive a formula for the $e/m$ of an electron moving in a circular path due to a constant magnetic field. Using the relationships $F_B = qvB\sin(θ)$, $F_{net} = ma$, and $a_c = v^2/r$, I was able to get the relationship of $qvB = mv^2/r$. This appears to be in agreement with most of the places I've checked online. However, using this equation, I tried to get the relationship of $q/m$ by rearranging the equation. This gives me $q/m = v/Br$. However, most places online say that this is incorrect, and that the actual relationship is $q/m = 2v/B^2r^2$. Can someone please enlighten me on how this is the case? Where did my derivation go wrong? Thanks for any help!

• Could you provide a reference for the supposedly correct equation? I don't think it is correct. – leongz Mar 30 '18 at 21:39
• Here is a reference I found online. I would give my uni's page which shows the same thing, but anonymity and all that. – That Dalek Mar 30 '18 at 21:46
• That $V$ is the potential difference, not the velocity $v$. – leongz Mar 30 '18 at 21:49
• Thanks for point that out. That would explain it. I think I may well be an idiot. – That Dalek Mar 30 '18 at 21:55

Your formula $$q/m = v/Br$$ is correct! The right hand side also has the correct dimension $$[charge/mass]$$The expression $$q/m = 2v/B^2r^2$$ cannot be correct because the right hand side doesn't have the required dimension $[charge /mass]$ but the wrong dimension $$[charge^2/(mass^2 length)]$$ I wonder which sources give you this wrong formula.