E=?

[pooria]

Hi my friends,


Why E=MC^2 ?????



Thanks for your answers.

[John L]

Originally posted by From Wikipedia
E = mc2

An equation derived by the twentieth-century physicist Albert Einstein, in which E represents units of energy, m represents units of mass, and c2 is the speed of light squared, or multiplied by itself. (See relativity.)

Because the speed of light is a very large number and is multiplied by itself, this equation points out how a small amount of matter can release a huge amount of energy, as in a nuclear reaction.

A kilogram of mass completely converts into

89,875,517,873,681,764 joules or
24,965,421,632 kilowatt-hours or
21.48076431 megatons of TNT

It is important to note that practical conversions of mass to energy are seldom 100 percent efficient. One theoretically perfect conversion would result from a collision of matter and antimatter; for most cases, byproducts are produced instead of energy, and therefore very little mass is actually converted. In the equation, mass is energy, but for the sake of clarity, the word converted is used.

[Matthew]

Originally posted by Vega@Aug 9 2005, 02:25 AM

Why E=MC^2 ?????

The expression can be derived by a fairly simple thought experiment which looks at a ball in a box. If you're interested I can dig the information up from whereever I've put it.

You need to remember that the "m" in the equation refers to the relativistic mass. To get the relativistic mass of an object you must multiply the rest mass by the lorentz factor.

In the following equation; "m" refers to rest mass, "M" refers to relativistic mass, "v" is the objects velocity relative to you, and "c" is the speed of light.

[rahuldandekar]

This is from the book "Simple Nature" by Ben Crowell, available free of charge on the net at www.lightandmatter.com.

Kinteic Energy = mc^2 (gamma - 1) [gamma = 1/_/(1 - v^2/c^2 ]

Let’s consider what happens if a blob of putty moving at velocity v hits another blob that is initially at rest, sticking to it, and as much kinetic energy as possible is converted into heat. (It is not possible for all the kinetic energy to be converted to heat, because then conservation of momentum would be violated.) The nonrelativistic result is that to obey conservation of momentum the two blobs must
fly off together at v/2.

Relativistically, however, an interesting thing happens. A hot object has more momentum than a cold object! This is because the relativistically correct expression for momentum is p = mv, and the more rapidly moving molecules in the hot object have higher values of gamma.

There is no such effect in nonrelativistic physics, because the velocities of the moving molecules are all in random directions, so the random motion’s contribution to momentum cancels out. In our collision, the final combined blob must therefore be moving a little more slowly than the expected v/2, since otherwise the final momentum would have been a little greater than the initial
momentum. To an observer who believes in conservation of momentum and knows only about the overall motion of the objects and not about their heat content, the low velocity after the collision would seem to be the result of a magical change in the mass, as if the mass of two combined, hot blobs of putty was more than the sum of their individual masses.

Heat energy is equivalent to mass.

Now we know that mass is invariant, and no molecules were created or destroyed, so the masses of all the molecules must be the same as they always were. The change is due to the change in with heating, not to a change in m. But how much does the mass appear to change? On page 561 we prove that the perceived change in mass exactly equals the change in heat energy between two temperatures, i.e. changing the heat energy by an amount E changes the effective
mass of an object by E as well. This looks a bit odd because the natural units of energy and mass are the same. Converting back to ordinary units by our usual shortcut of introducing factors of c, we find that changing the heat energy by an amount E causes the apparent mass to change by m = E/c^2. Rearranging, we have the famous E = mc^2.

All energy is equivalent to mass.

But this whole argument was based on the fact that heat is a form of kinetic energy at the molecular level. Would E = mc^2 apply to other forms of energy as well? Suppose a rocket ship contains some electrical energy stored in a battery. If we believed that E =mc^2 applied to forms of kinetic energy but not to electrical energy, then we would have to believe that the pilot of the rocket could slow
the ship down by using the battery to run a heater! This would not only be strange, but it would violate the principle of relativity, because the result of the experiment would be different depending on whether the ship was at rest or not. The only logical conclusion is that all forms of energy are equivalent to mass. Running the heater then has no effect on the motion of the ship, because the total energy
in the ship was unchanged; one form of energy was simply converted to another.

Sadly, I cannot provide page 561 here, but neglecting the "how"s, this describes the "why" pretty well. The only glitch is that Crowell uses the "natural" system of units, where mass and energy have the same units. But he does convert back to the "unnatural" system and gives E=mc^2.

[pooria]

Hi rahuldandekar,

Could you please explain why the result of the experiment would be different depending on whether the ship was at rest or not?

Thank you.

[rahuldandekar]

Originally posted by Vega@Aug 24 2005, 09:04 AM
Hi rahuldandekar,

Could you please explain why the result of the experiment would be different depending on whether the ship was at rest or not?

Thank you.

Hi Vega,

If the ship was moving, due to the increase in mass, it would slow down. It's mass would increase if only heat was equivalent to mass, and thus it would slow down. If it was at rest, it wouldn't slow down. Thus, you could make out 'rest' and 'moving' frames if only heat was equivalent to mass. This could be repeated with any form of energy.

Thus, all energy is equivalent to mass.

[StarLab]

I think the why can be answered by the values, not the units or dimensions. The question asked is 'why does Energy = mass x speedoflightsquared?' The question intended is 'why c^2?' And the answer to that is because the value of c^2 is simply the conversion factor between joules and kilograms. Specifically, one kilogram of whatever yields c^2 joules of Energy of that exact same whatever. The equation itself does not define the speed of light. That is a common misconception. The conversion value of c^2 is a mere, yet wonderful, coincidence.

The 'why' of the equation as in 'why does it exist?', is that it serves as a very wonderful, incredibly useful conversion factor. That is all. Keep in mind that more than one kind of energy (such as potential, mechanical and kinetic) has the units mv^2. The c^2 is a mere simplicity for one kind of energy.