Everyone knows that formula, but here’s something that not everyone knows, that formula is actually a simplification of this formula:

E^{2} = p^{2}c^{2} + m^{2}c^{4}

Here is what all of the terms are:

- E stands for energy, units: joules [J]
- p stands for momentum, and in this case it’s the momentum of a photon, units: newton metres [Nm] (Nm = Kg m/s)
- c stands for the speed of light, 3*10
^{8}, units: metres per second [m/s] - m stands for mass, units: Kilograms [Kg]

The first collection of terms (p^{2}c^{2}) in the formula only applies to photons, the second collection of terms (m^{2}c^{4}) only applies to matter. Therefore the formula can be simplified to the two following formulas:

- E = pc – for photons
- E = mc
^{2}– for matter

As you might be able to see, those two formulae are the same. If you don’t believe me, check this out:

p (momentum) = mv (mass * velocity)

Therefore we can substitute in mv for p:

E = mvc

However, since this equation applies to photons and photons travel at the speed of light (c), v = c, therefore:

E = mcc

E = mc^{2}

So what are the implications of these two formulae? Well, if you consider E = mc^{2}, it basically means that energy and mass are equivalent. If you were to convert 1 Kg of mass totally into energy, this is how much you’d have:

E = mc^{2}

E = 1 * ( 3*10^{8} )^{2}

E = 9 * 10^{16} J

That’s a stupidly large value, so let me put it into perhaps a more understandable unit, the KiloWatt hour. That’s the unit used to measure the consumption of electical energy in houses. 25,000,000,000 KWh.

That’s still a pretty large value and it’s hard to put it into context, so I google’d and found this web-site. Assuming that his figures are correct and that they haven’t changed much since 1997, here’s some interesting finds. If the average house uses 10,215 kWh per year, that means that 1Kg of mass would be enough to run that average house for almost 2.5 million years.

Or if we look at this extract, “On a peak day in 1999, California used 50,743 megawatt-hours of electricity, or 50,743,000,000 watt-hours.” So California used 50,743,000 KWh in one day, if it used that much every day, then that 1 Kg of mass could run California for 492.7 days, or 1.35 years. That’s a long time and California is a big place.

So what about the other formula, E = pc? Well, there’s something very interesting here, if we re-arrange the formula it becomes a bit clearer:

p = E / c

This means that photons have momentum, therefore if a photon is absorbed by some matter, the matter will receive an impulse, therefore it will gain momentum. This is the principle behind a solar sail, a single photon will not have much of an effect, however the sun gives off rather a lot of photons, so if a space craft has a sail big enough it could use the photons colliding with it to accelerate and save on fuel!

Also, since the momentum of a photon depends upon it’s energy, the higher frequency photons have more momentum. If you know the wavelength or frequency of a particular photon then you can work out the momentum as follows:

E = hf (where h is Planck’s constant and f is the frequency)

E = hc / λ (where λ is the wavelength)

Therefore:

p = hf / c

p = h / λ

So while the energy (and consequently the momentum) of a photon increases with the frequency, it will decrease as the wavelength increases, since the wavelength and frequency are inversely proportional to each other. Oh, and Planck’s constant is 6.63*10^{-34}, units: metre’s squared kilograms per second [m^{2}Kg / s].

Here are a couple of wavelengths for two different colours of light:

- Red – 650 nm
- Violet – 400 nm

And their corresponding momentum:

p_{red} = 6.63*10^{-34} / (650*10^{-9})

p_{red} = 1.02*10^{-27} Nm

p_{violet} = 6.63*10^{-34} / (400*10^{-9})

p_{violet} = 1.66*10^{-27} Nm

So in other words, really f*cking tiny.

On reflection, E = mc^{2} really has the more interesting conclusions and leads to the production of bombs and stuff. It also brings in the idea of a binding energy for nuclei aswell, really fascinating stuff (at least for me it is). I think I’ll write about why in another entry though.

Wow. You’re a nerd, huh? :p

No, but physics is fun.

…

I love physics and all but….

…

I’m a music major.

oh yeah, and the original formula looked like: http://www.pbs.org/opb/einsteinswife/pix/photos/science4.gif