Natural units make a lot of things easier to calculate and the Casimir force formla simpliefies to
F = (pi^{2}/240) A/L^{4}
the constant in front is about 1/24, since pisqare is about ten.
With all these formulas if you want to work them in metric you have to put the hbars and cees and maybe some Gees back in first. In other words make them more messy before you can use them. Oh, and sometimes kays too.
However the cost of having radically clean formulas with c=Ghbar=k=1 is that the scale is unfamiliar.
For example E38 is a mile and E35 is a pacea thousandth of a mile, around 1.6 metersand E30 is 16 microns.
E50 is about a micronewton of forcemore exactly 1.2 micronewton.
So you could say "It is hopeless, I will never be able to learn this new scaleE44 is a million miles and E50 is a micronewtonit's impossible" But it's actually OK and one does get used to the scale and the clean simplicity of the formulas is a big plus.
Anyway: think of two plates 1.6 meters on a side so E35 on a side, area is E70. And imagine the separation is 16 microns which is E30.
the force is easy to calculate, just the area (E70) divided by the fourth power of the sep(E120) which gives E50 that micronewton thing, and you have to include the numerical constant which is about 1/24.
Maybe I should include that tiny force of 1.2 micronewtons in the dictionary:
E601.7 billion years
E44a million miles
E38 a mile
E6 22 grams
E501.2 micronewtons
Here's the list of constants etc, for review:
the proton compton wavelength2.103E14 meter13E18
the CMB temperature2.725 kelvin1.93E32
the Hubble time4.35E17 seconds8.06E60
average sunlight photon2E19 joulesE28
the distance to the sun150 million km93E44
earth orbit speed30 kilometer/secondE4
solar constant1380 watts/sq.meterE119
2.701k3.73E23 joule/kelvin2.701
sigma5.67E8 watt/sq.meter kelvin^{4}pi^{2}/60
a (the aT^{4} law)7.565E16 joule/cub. meter kelvin^{4}pi^{2}/15
I see I havent yet used the proton compton. Maybe that should be next.
