MadSci Network: Physics

Re: How effective is energy creation through positron/electron annihilation?

Date: Sun Jun 17 03:48:54 2001
Posted By: Randall Scalise, Faculty, Physics, Southern Methodist University
Area of science: Physics
ID: 992493050.Ph

Dear James,

The Law of Conservation of Energy prohibits the reaction that you
describe.  If the math is done carefully, one will find that just
as much energy comes out as went in.

> If you could create a full gram of positrons and electrons (.5 grams
> each), according to my calculations, this would take approximately
> 9.0 x 10^16 Joules 90 megaGigaJoules (I realize that this is a huge
> amount, but roll with the theory for a second).  Then if you were to
> place them together again, it would re-release that 90megaGigaJoules.

The total is off by a factor of 1000, but the idea is correct.
The energy required to create 0.5 gram of electrons and 0.5 gram of
positrons is, by Einstein's famous formula E=mc^2,
(0.001 kg) * (3 x 10^8 m/s)^2 = 9 x 10^13 joules = 90 kilo-gigajoules.
Putting the electrons and positrons together would release the same
amount of energy.

> However, if one used a particle accelerator to ram them into each
> other at a VERY high velocity (for example, 2.9 x 10^8 m/s), their
> masses would theoretically alter to about 1.4 grams each.

Again, the details are off, but the essence of the argument is fine.
One of the consequences of Einstein's Special Relativity is the
increase in mass with speed.  If an object has rest mass m_o, then
when traveling at speed v it's mass will be

    m = m_o / (1-v^2/c^2)^(1/2)

For the speed in your example, the mass will increase by a factor
of about 3.9, so 0.5 gram of electrons at rest will have a mass
of 1.95 grams in motion.

> The amount of energy required for that particle accelerator is
> approximately 21 megaGigaJoules (still huge).

The particle accelerator must supply energy to the electrons and
positrons to get them moving which becomes their kinetic energy:

   K.E. = (m - m_o) c^2 = 261 kilo-gigajoules

So far, the total energy investment is (90 + 261) kilo-gigajoules =
351 kilo-gigajoules.

> However, the increased mass in the collision would (approximately)
> create 351 mega Giga Joules.

This should be 351 kilo-gigajoules, the same as the energy required
to create and accelerate the electrons and positrons.  Energy is

Incidentally, S.I. prefixes such as "giga" are not capitalized, and
S.I. units like "joule" are also lower case when they are written out.
The abbreviations are capitalized: gigajoule = GJ.

--Randall J. Scalise

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