Dimensional analysis in the formula for gyration frequency

[jonggg]

I have a question about dimensional analysis in the formula of the gyration frequency of synchrotron radiation of a relativistic particle (electron of charge e and mass m) in a magnetic field B. Leaving aside the adimensional Lorentz factor γ and numerical factors, the formula reads v ~ (e * B) / (m * c) (see for example eq. 9 in https://www.astro.utu.fi/~cflynn/astroII/l4.html). The dimensions of frequency are already given in the expression (e * B) / m, so how does the speed of light c in the denominator fit in here? I am afraid this must be something obvious about units system or convention, but I am confused. Help would be appreciated.

 

[TSny]

Yes, it has to do with the system of units being used. See the comments starting at the bottom of the second page here.

 

[vanhees71]

Indeed, the formula refers to Gaussian (or Heaviside-Lorentz) units for the electromagnetic quantities, which is much nicer when formulating electrodynamics in a relativistically covariant way (as it should be done, because it's after all the paradigmatic example of a relativistic field theory).

In these units the Lorentz force reads (in the non-covariant formulation with $m$ the invariant mass of the particle)
$$m \mathrm{d}_t (\gamma v)=\vec{F}=q \vec{v} \times \vec{B}/c.$$
For the particle moving on a circle of radius $r$ this must be the centripetal force, i.e.,
$$m \gamma r \omega^2 = q v B/c=q \omega r B/c \; \Rightarrow \; \omega=q B/(m \gamma c).$$

 

[WernerQH]

Indeed, the formula refers to Gaussian (or Heaviside-Lorentz) units for the electromagnetic quantities, which is much nicer when formulating electrodynamics in a relativistically covariant way (as it should be done, because it's after all the paradigmatic example of a relativistic field theory).

It is a pain that the cgs-system is still in use. (Incidentally, you are deviating from your cherished Lectures on Theoretical Physics by Arnold Sommerfeld. ;-)
We could have gotten rid of it at least half a century ago, but some people (for example Edward Purcell in the Berkeley Physics Course) thought otherwise, that the cgs-system is "inherently" better. Generations of students have stumbled over this, and apparently this will continue forever, because there will always be some who think one system is superior to the other. Thus you cannot copy an equation from a book without checking whether the magnetic field is $B_\text{cgs} = c B_\text{SI}$ or $B_\text{SI} = B_\text{cgs} / c$.

Of course, in relativistic work you can always set $c = 1$. And I have no problem with that. But in the end you must be able to put back the factors you set to $1$. Dimensional analysis helps, but only if you remember what the dimensions should be. I have met people using cgs units who found checking dimensions too tiresome. They had a formula for a plasma effect that was off by ten orders of magnitude. It's now obvious that a factor of $c$ was missing, but they didn't bother to check the dimensions.

 

[vanhees71]

It is a pain that the cgs-system is still in use. (Incidentally, you are deviating from your cherished Lectures on Theoretical Physics by Arnold Sommerfeld. ;-)

He committed two sins. The lesser is the use of the SI in electrodynamics and the really severe one is the use of the $\mathrm{i} c t$ convention in special relativity.

We could have gotten rid of it at least half a century ago, but some people (for example Edward Purcell in the Berkeley Physics Course) thought otherwise, that the cgs-system is "inherently" better. Generations of students have stumbled over this, and apparently this will continue forever, because there will always be some who think one system is superior to the other. Thus you cannot copy an equation from a book without checking whether the magnetic field is $B_\text{cgs} = c B_\text{SI}$ or $B_\text{SI} = B_\text{cgs} / c$.

A system of units, where components of a field have different dimensions is flawed from a physical point of view. From a practical point of view the SI is of course better to use.

Of course, in relativistic work you can always set $c = 1$. And I have no problem with that. But in the end you must be able to put back the factors you set to $1$. Dimensional analysis helps, but only if you remember what the dimensions should be. I have met people using cgs units who found checking dimensions too tiresome. They had a formula for a plasma effect that was off by ten orders of magnitude. It's now obvious that a factor of $c$ was missing, but they didn't bother to check the dimensions.

You can do this in the SI too, i.e., you set $\mu_0=\epsilon_0=1$, and you use the Heaviside-Lorentz system in its natural form. In the SI you have $c$'s at very intuitive places. Dimensional analysis is simpler in the SI, because of the additional base unit A, i.e., you avoid the fractional powers of c, g, and s in the units of the electromagnetic quantities.

 

[WernerQH]

A system of units, where components of a field have different dimensions is flawed from a physical point of view.

That's a silly argument. From this point of view all of physics is flawed that uses different units for time and space. Too bad that you are helping to perpetuate a "schism" that confounds students (and sometimes professionals too).

 

[vanhees71]

Well, that's why we set $c=1$ in theoretical studies involving relativity. It's "more natural". That's all I'm saying. Formulating electromagnetism in terms of SI units spoils the natural beauty of the theory when written in its most natural form, i.e., as a relativistic (Q)FT.