Energy and more energy

I recently did a post comparing forces. This is a similar exercise where I am comparing energies. So to start with the usual suspects, Einstein's classic

$ E_E = m c^2$    ...(1)

$E_E$ - Einsteins version of energy
m - mass
c - speed of light in vacuum

Next Planck's version

$ E_P = h \nu $    ...(2)

$E_P$ - Plancks version of energy
h - Planck constant
$\nu$ - a frequency

We are going to re-arrange this slightly using

$ \nu = c / r$     ...(3)

c - speed of light in vacuum
r -  a wavelength

so now Planck's version becomes

$E_P = $ $ \frac {hc} {r}$   ...(4)

Next we have the gravitational potential energy, this is typically represented by U and is negative, but today I have made it E and dropped the -ve sign, so

$E_G = $  $ \frac {G m_1 m_2} {r}$   ...(5)

$E_G$ - gravity version of energy
G - Newton's gravitational constant
$m_1   ,   m_2$ - mass of object 1 and object 2
r - distance between

let us set $m_1 = m_2 = m$ , so (5) becomes

$E = $ $ \frac {G m^2} {r}$   ...(6)

Finally we have the electric potential, given by

$E_C = $ $ \frac {e_1 e_2} {4 \pi \epsilon_0 r}$   ...(7)

$E_C$ - Coulombs version of energy
$e_1   ,  e_2$ - charges on two separate objects
$\epsilon_0$ - permittivity of free space
$\pi$ - 3.141592.....
r - distance between the charges.

Part 1: Gravity, Planck and Einstein

Let's re-arrange (1) in terms of mass and (4) in terms of r, so we have

$m = $ $\frac {E_E} {c^2}$    ...(8)

$\frac {1} {r} = \frac {E_P} {h c}$    ...(9)

lets assume that the distance between the masses in equation 6, is equal to the wavelength, r, given in (4) put these two results into (6)

$ E_G = $ $\frac {G E_E^2 E_P} {c^4 h c }$    ...(10)

Let us set all the energy values to be equal

$E_G = E_P = E_E = E$    ...(11)

so (10) becomes

$ E =$ $ \frac {G E^3} {c^5 h}  $   ... (11)

dividing by E and rearranging gives us

$ E^2 = $   $ \frac {c^5 h} {G}  $    ...(12)

which turns out to be the definition of the Planck Energy! OK, let's think about that for a minute. We set the masses equal, we set the energies equal and we said that the distance between the masses was equal to a wavelength. Everything balanced and worked out.

Part 2: Coulomb, Planck and Einstein

In the Coulomb version of energy there is no mass, so Einstein gets left out of this one. Let us do the same thing as we did for gravity, re-arrange Plancks version to give us (9) again and pop this into (7), in addition, set the charges to be equal, so

$E_C = $ $ \frac {e^2 E_P} {4 \pi \epsilon_0 h c}$    ...(13)

Let's set the energies to be equal as before this time giving

$E = $ $ \frac {e^2 E} {4 \pi \epsilon_0 h c}$    ...(14)

this time we can cancel E to give

$1 = $ $ \frac {e^2} {4 \pi \epsilon_0 h c}$   ...(15)

this however is incorrect, the correct value is

$\frac {\alpha} {2 \pi} = \frac {e^2} {4 \pi \epsilon_0 h c}$   ...(16)

$\alpha$ - is the fine structure constant. So where have we gone wrong?

Part 3: Planck Energy

In part (1), when we set the energies equal, what we had actually done, although it was not apparent, was to set them all equal to the Planck energy, so equations (1), (4) and (6) had become

$E = m_p c^2$    ...(17)

$E = $ $ \frac {hc} {2\pi l_p}$   ...(18)

$E = $ $ \frac {G m_p^2} {l_p}$   ...(19)

$m_p$ - Planck mass
$l_p$ - Planck length

These are just variations of the same equation

$ E = $   $\sqrt {\frac {c^5 \hbar} {G}}  $    ...(20)

Ok, so why didn't this work for part (2), isn't it likely that we are setting them both to the Planck energy again? Let's see, take the Planck version of equation (9)

$\frac {1} {l_p} = \frac {E_P} {h c}$   ...(21)

and let's have the Coulomb equation with r replaced with $l_p$ so

$E_C = $ $ \frac {e^2} {4 \pi \epsilon_0 l_p}$    ...(22)

but this is not $E_P$, the Coulomb equivalent of $E_P$ is

$E_{CP} = $ $ \frac {q_p^2} {4 \pi \epsilon_0 l_p}$    ...(23)

and $q_p \not= e$, so (22) and (23) are not equal. Does this mean we have the wrong choice of $q_p$? Should it be just the charge of an electron? This seems to be moving us away from the idea of Planck units though. If we are going to do that why not use the mass of the electron for Planck mass etc.

For Gravity, Einstein and Plancks relation we were able to find a common point where they all balanced which in itself is remarkable and I shall ponder this further. For Coulomb energy there was no common distance that worked. What does that tell us? I'm not sure at the moment, but if I figure it out I'll update this page. (This isn't strictly true, there is a common distance that works. I'll be covering it in a later post and will put in a link when I publish it.)

I wrote this while listening to this.