Wednesday 18 March 2015

Speed of light, squared

It's been a while since I managed to post anything, mostly because I have been going round and round in circles!

This post is actually two posts! This post is on one of the more commonly known constants in physics, the speed of light, or rather the speed of light squared. In the first part I am going to derive a couple of equations and in the second I will discuss them in a little more detail.

The most common version of $C^2$ as got to be Einstein's very famous equation,

$ E = mc^2 ... (1)$

E - energy
m - mass

probably the most famous equation in the world. There a a fair number of others, from Maxwell's equations we have

$ \frac {\delta E} {\delta t}$ $= c^2 \bigtriangledown$ x $ B  ... (2) $

E - Electric field
B - Magnetic field

from General relativity we have the Schwarzschild radius

$ 2GM = r_s c^2 ...(3) $

G - gravitational constant
M - mass
$r_s$ - Schwarszchild radius

If you have a play with the Planck Units you can derived a whole bunch, here are a couple of them

$\sqrt {F_p G} = c^2 ... (4) $

$F_p$ - Planck Force
G - gravitational constant

$\frac {v_p q_p} {m_p}$ $= c^2 ...(5) $

$v_p$ - Planck voltage
$q_p$ - Planck charge
$m_p$ - Planck mass

From equation (2) though there is another version that prompted this post, it is this

$ c^2 =$  $\frac {1} {\epsilon_0 \mu_0}$ $ ...(6) $

$\epsilon_0$ - permittivity of free space
$\mu_0$ - permeability of free space = $4\pi . 10^{-7} \ kg \ m \ q^{-2}$

So, a while back I did an introductory post on the idea of Planck units. These are values for mass, length, force, energy etc based on five constants of nature. See my earlier post for more details. The 5 constants are

c - speed of light
G - Gravitational constant
$\hbar$ - reduced Planck's constant
$4 \pi \epsilon_0$ - Coulomb's constant
$k_b$ - Boltzmann's constant

The $\epsilon_0$ in Coulomb's constant and the $\epsilon_0$ in equation 6 are the same, which got me thinking, are there similar equations for the other 3 natural constants? ie,

$c^2=$ $\frac {1} {\hbar \mu_\hbar}$ $ ...(7) $
$c^2=$ $\frac {1} {G \mu_G} $ $ ...(8) $
$c^2=$ $\frac {1} {k_b \mu_{k_b}} $ $ ... (9) $

$\mu_\hbar, \mu_G, \mu_{k_b}$ - something to be determined.

Let's start with equation (7) $\hbar$ is the reduced Planck's constant, but what is $\mu_\hbar$? is there anything that corresponds to this? Rearranging (7) and multiplying by $4\pi^2$ we get

$\frac {4\pi^2} {\mu_\hbar} $ $= 2 \pi h c^2   ...(10)$

note that the $\hbar$ is now h - Planck's constant.  In physics there is something called the first radiation constant ($c_1$) which has the value

$c_1 = 2\pi h c^2  ...(11) $

with a little substitution from (10) & (11) and some minor rejigging we have

$\mu_\hbar = $ $\frac {4 \pi^2} {c_1}$  $ = 1.055073 . 10^{17} \ s^3 \ kg^{-1} \ m^{-4}...(12)$

That's a good start, and is probably to be expected. There is a very close relationship between h and c that we will explore else where, but what about (8) and (9)? Repeating the rearranging we did for (7) we get

$\mu_G =$ $\frac {1} {G c^2}$ $ = 1.66718.10^{-7} \ kg  \ s^4  \ m^{-5} ...(13) $

$\mu_{k_b} =$ $\frac {1} {k_b c^2} $ $ = 805,889 \ K \ s^4 \ kg^{-1} \ m^{-4} ...(14)$

It looks like we have drawn a blank here. (13) and (14) do not appear to have any recognised physical significance. I shall pick this up in a later post. For now dividing  $\mu_0$ by (13) gives

$\frac {\mu_0} {\mu_G} = \frac {G} {\epsilon_0} $ $=$ $\frac {4 \pi .10^{-7}} {1.66718 .10^{-7}} $ $\approx 2.4 \pi \ m^6 \ q^{-2} \ s^{-4} ... (15) $

and,

$\frac {G} {4 \pi \epsilon_0} $ $\approx 0.6    ...(16)$

and also

$\frac {\mu_G} {\epsilon_0} $ $=$ $\frac {\mu_0} {G}$ $\approx 6000 \pi \ kg^2 \ s^2 \ q^{-2} \ m^{-2}  ...(17)$

Finally, taking

$z_0 = \sqrt {\frac {\mu_0} {\epsilon_0}} $ 

and

$z_G = \sqrt {\frac {\mu_G}{G}} $ $\approx 50 $

$\frac {z_0} {z_G} $ $\approx 2.4 \pi$, as expected

So, just to summarise this post. We have shown that equation 7 is actually something already known, the first radiation constant. For (8) and (9) we do not yet have a clear understanding, but it has been shown that in the case of G, the gravitational constant and $4 \pi \epsilon_0$, the values are very similar.

More on this in the next post.


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