Ronald C. Lasky, director of the Cook Engineering Design Center at Dartmouth College, explains the significance behind this hallowed equation:
It is the most famous equation in the world. Many can recite it—and attribute it to Albert Einstein—but few know its significance.
It tells us that mass and energy are related, and, in those rare instances where mass is converted totally into energy, how much energy that will be. The elegance with which it ties together three disparate parts of nature—energy, the speed of light and mass—is profound.
Here is where the equation of all equations comes from:
It was known for some time before Einstein's insights that electromagnetic radiation (light, for example) possessed momentum. This quality of radiation is small in magnitude—after all, you needn't worry about being knocked over by sunlight—but easily measurable. Applying an understanding of light's momentum within a little thought experiment, it is possible to see how E = mc2 comes about.
Consider a cubic hollow box at rest in space with sides of length D and a mass of M. This box is also symmetrical in its mass distribution. One of the faces inside the box is coated with a fluorescing material, and, at a given moment, a photon (i.e., a particle of light) is emitted from that material, perpendicular to its surface. The momentum of this photon causes the box to move in the opposite direction as the photon, and it continues to move until the photon hits the opposite wall. During this time the box moves a very small distance, Δx.
 Image: RONALD LASKY |
Newton's laws of mechanics tell us that the center of mass cannot move, because the box has not been acted upon by an outside force. However, in order to keep the center of mass constant, since the box has moved, some mass must have been transferred from the fluorescing side of the box to the absorbing side in the process of generating the photon and its striking the opposite side. Therefore the photon must have a mass, m.
So the photon, which also possesses energy E, is emitted from the fluorescing side of the box. Its momentum, Pphoton, is equal to its energy divided by the speed of light: Pphoton = E / c. The photon will impart this momentum to the box, causing the box to move a small distance, Δx, during the time, t, in which the photon travels to the opposite side of the box. The momentum of the box, Pbox, is also equal to its mass, M, times the velocity, vbox, at which it moves before the photon strikes its target. (Note: The box loses the photon's mass, m, during this process, but this slight loss can be neglected here.) Hence:
Pphoton = Pbox = E / c = Mvbox
Then vbox = E / cM (1)
We can also determine the time it takes for the photon to travel across the box: it is equal to the length, between parallel faces, of the box (which is D), minus the amount the box moved in the opposite direction (Δx), divided by the speed of light, c. (The target will essentially have moved a slight distance closer, meaning the photon did not have to travel the full distance D.):
t = (D - Δx) / c
But, since Δx is a minute fraction of D, we essentially get:
t = D / c (2)
Now, since vbox = Δx / t , using equation 2, vbox can be restated as:
vbox = Δxc / D (3)
Substituting equation 3 for the term vbox in equation 1:
Δxc / D = E / cM
Next, we rearrange the terms to get:
ΔxM = ED / c2 (4)
Assuming the center of the box is initially at x = 0, this position is also the center of mass, xm. After the photon event, the box moves Δx to the left, as shown in the figure below, and the equivalent mass of the photon, m, is deposited on the opposite side. As mentioned above, we recall from Newton's Laws that the center of mass must not change, because the box is not acted upon by an outside force. This concept is expressed in the center of mass equation below. The center of mass is at x = 0 in the left half of the equation and it is still at 0 after the photon strikes the opposite wall as described in the right half of the equation.
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24 Comments
Add CommentPlease Explain the meaning of the Inertial clock paradox and the real meaning of the Lorentz transformations written by Mr. Frank Baird
Reply | Report Abuse | Link to thiswhen the photon strike the opposite wall , the box will try to move back to the orignal position due to the momentum imparted on the absorbing side.
Reply | Report Abuse | Link to thisIn the para beginning with "so the photon" it appears that when you equate the momentum of the photon (p=E/c) that you have allready assumed that E=MC2. Pphoton = E/c comes from Pphoton = Mmassphoton * c . rearranging E=MC2 for m=E/C2 and then substituting E/C2 in the equation for momentum; Pphoton = Mphoton * C= E/C?
Reply | Report Abuse | Link to thisI hate to tell you this. But your so-called expert has make a huge mistake. First of all, photons are massless particles so m=0 and therefore acording to your expert E=0, which is obviously wrong. Secondly, all of relativistic energy formulas and in fact all relativistic dynamics can be derived from Einstein's mass-enengy formula, which your expert fails to mention (the questioner had asked about the significance of the formula). Furthermore, the fact that p=E/c cannot be assumed in proving Einstein's relation as it is a consequence of the relation. In other words, to assume that p=E/c is to assume that E=mc2 since momentum is mass times velocity so that p=mc. Anyways, as a person who teaches relativity at university I find your expert's answer completely misguided and unfortunate. I used to read Scientific American as a graduate student and had the highest regard for your magazine. It saddens me to see how over the years you have grown quantitavely and shrunck qualitatively.
Reply | Report Abuse | Link to thisThat is a marvelously ingenious mind experiment. Thank you.
Reply | Report Abuse | Link to thisHere, I believe, is another one. Same box, same photon emitted from the flourescing surface. Energy imparted to photon is (1/2)(mc^2). Reaction energy to box is same. Total energy required to get the photon moving at speed of light is mc^2. When the photon is stopped at the opposing inside surface, then mc^2 energy is returned.
Proposition: Any mass moving a velocity c required energy mc^2 to accelerate it to that speed from rest.
Corollary: Any particle of mass m which gives up mc^2 energy when 'stopped' (decelerated to rest, or anahilated) was travelling at velocity c before it was 'stopped'.
Therefore all mass in our vicinity of the universe is travelling at velocity c.
When the author says "The momentum of this photon causes the box to move in the opposite direction as the photon, and... ...Therefore the photon must have a mass, m.", where is the demonstration? In the very beginning of the explanation of why the box moves, the momentum of the photon is used... and this would allow to conclude the photon has a mass (and therefore a momentum). The snake is eating its tail...
Reply | Report Abuse | Link to thisFrom the author:
Reply | Report Abuse | Link to thisMotomonty is correct, a photon does not have a rest mass. I should have said the photon's "equivalent" mass. Equivalent in the respect that energy (which a photon has) and mass have an equivalence through E = mc2. This is also sometimes referred to as inertial mass. The transfer of mass from one side of the box to the other is real. The photon leaving the fluorescing material is a loss of mass to that side. Whereas, the photon being absorbed on the other side of the box is a gain of mass.
I disagree with Motomonty Re the momentum of a photon = E/c not being separable from relativity theory. The momentum of a light (p=E/c) was shown experimentally before 1900 and perfected by Nichols and Hull in 1901 (http://en.wikipedia.org/wiki/Nichols_radiometer), before Einstein's Relativity Theory circa 1905.
The purpose of this exercise is to show that E=mc2 can be derived without the more challenging math of relativity theory.
Ron Lasky
When ‘Meaning’ is asked for, the request is usually intended to mean something like: ‘How does the item affect the human race’, or, even more specifically: ‘How does the item affect or relate to Me’.
Reply | Report Abuse | Link to thisThe equation is making a statement about Reality, of which we constitute a part. Basically, the things that are in our spatial universe we perceive or classify as either forms of Energy or forms of Matter as we note that things express the nature and reactions of an unstable fluidity that we call Energy or the nature of a more stable density which we call Matter. The equation states that Matter is simply another form of Energy, a compression of energy.
Therefore, although we observe a variety of forms in the universe, actually there is only one thing that makes up everything. This meaning should be weighed up by humanity as it has consequences for philosophy, politics, religion and the future existence of mankind.
Philosophy, politics and religion rule and divide the world because they tend to focus on differences. Each culture or sub-culture group selects for itself ideas and opinions with which to identify itself, to differentiate itself from others. Each group tends to fear, detest or oppose any opinion or idea that differs from those that it has previously chosen and held dear to its bosom. Ego, self-interest and self-preservation rule. In this war between persons, groups and nations, humanity suffers and may eventually self-destruct.
When it is considered that we are all of the one thing, as it were; all of the same clay simply made into different shapes and densities; all the same basic atoms or energies built up like Lego into highly complex chemical and opinionated structures; it can be seen that individual selves (though real) are an illusion in the sense that our normal defining of ourselves has so far required that we ignore or contest whatever is outside of ourselves. What is outside, or persons not identifiable with ourselves, has been disregarded or resented. However, in the view of Physics, it appears that we are all the same thing or the one Self in our material being. The hordes of competing, conflicting humanity are like an Argos of myriad eyes, each eye looking and glaring at the other eyes. When we hurt another we are hurting another part of us, damaging another part of the immense sculpture that is all mankind, all life, all matter. What we do unto others we do unto ourselves.
Mathematics and Physics are tools or windows that reflect or provide insights into Reality. So far as the Meaning of the equation is concerned, there is more that can be learned from it than is ordinarily dreamt of by just the mathematical part of a mind. Meaning is asking the greater or outside thing to relate to the smaller subject at hand or to the Enquirer. On the other side of the equation or question, the answer can be used the assist in knowing how the Questioner can relate, in turn, to the greater Whole.
As in quantum states, it is the observer that interferes or decides what the outcome will be.
Balan Oakenring
what i conclude ,it relates matter & how much energy contained.
Reply | Report Abuse | Link to thisbefor this equation mass was considerd as a constant whereas this equation considerd as variable
rcdohare25@yahoo.co.in
What noone seems to publish is the units associated with the formula. What are the units?
Reply | Report Abuse | Link to thisI am by no means any sort of scientist but one day I thought I would figure out the number of foot pounds of energy in a pound of matter - if the pound of matter were converted entirely to energy.
Reply | Report Abuse | Link to thisIts really quite simple math.
c = 983571056 feet per second.
c^2 = 9.674120222 x 10^17 feet^2 per sec^2.
m being 1 pound, mc^2 is 9.674120222 x 10^17 feet^2 pounds per sec^2.
That's not yet foot pounds. Foot pounds are derived from lifting a pound mass in earth's gravitational field which means we now have to divide by the acceleration of gravity at the earth's surface, which is
32.174048 feet^2 per sec^2.
the desired quotion comes out
3.0068085 x 10^16 foot pounds. Aproximately 30 quadrillion foot pounds.
Easy math. Of course, I used a calculator.
This interpretation of the meaning of E=mc2 is highly misleading...
Reply | Report Abuse | Link to thisThis interpretation of E=mc2 is highly misleading...
Reply | Report Abuse | Link to thisAh, you probably have in mind that the energy of a pound of matter, if obtained, would probably not manifest itself in foot pounds, a unit applicable to mechanical work. It would probably come out as heat.
Reply | Report Abuse | Link to thisOne can convert foot pounds to other units according to established relationships:
A calorie will raise the temperature of 1 gm water by 1 degree C
A Calorie (used by nutritionists) = 1000 calories = 1 kilocalorie
1 BTU (British thermal unit) will raise the temperature of 1 lb water 1 degree F
1 BTU = 252 calories
1 calorie = 4.186 joules =3.087 foot pounds
1 J (joule) = 0.2388 cal = .7375 foot pounds
1 foot-pound = 1.356 joules =.3239 calories
1 kilowatt-hour = 3414 BTUs = 860,300 calories = 2,655,000 foot pounds
(your electric bill uses the kilowatt hour unit)
1 barrel of oil is equivalent to 1.46 million kilocalories or 6.1 billion joules
What would you say to someone who claims that the formula E = mc2 explains the equivalence of matter and energy; that it proves that tiny amounts of mass could be converted into huge amounts of energy; that Einstein's discovery proves that all things broken down to their most basic form consist of one and the same kind of energy; that we all - including our thoughts - are pure energy, according to Einstein's famous formula"?
Reply | Report Abuse | Link to thisSimply put, everything that exist whether air, oxygen, emotions, your favorite bag, your lovable dog, your house, your bed, your body, the moon, your five senses, your computer and everything around you, is made up of one kind of energy vibrating at different levels of frequency. So are your THOUGHTS !
What would you say to someone who claims that the formula E = mc2 explains the equivalence of matter and energy; that it proves that tiny amounts of mass could be converted into huge amounts of energy; that Einstein's discovery proves that all things broken down to their most basic form consist of one and the same kind of energy; that we all - including our thoughts - are pure energy, according to Einstein's famous formula"?
Reply | Report Abuse | Link to thisIf E=MC2 then wat does time have to do with energy?
Reply | Report Abuse | Link to thisIf E=MC2 then wat does time have to do with energy?
Reply | Report Abuse | Link to thisE=MC2 M is mass, not matter. Matter has mass/inertia. Mass/inertia is a measurement of the energy contained in matter in motion or it's resistance to change in motion. The formula is also misleading in other areas. It is a nice and simple piece of mathematic short hand but does not truly represent reality. The speed of light squared is also an incorrect concept that makes the math work out, but gives the wrong view point. When mass/inertia is converted to energy it changes it's volumn 4/3*pi*r3*(10to6) at the speed of light. This is the change of a neutron into a hydrogen atom, the basis of all atomic energy, Mass/inertia conversion to energy event. Not matter conversion.
Reply | Report Abuse | Link to thisI am upset that in this great thought experiment, the assumption that (delta)x is minute compared to D is necessary. Afterall, light does not travel at infinite speed. Is there any way around it so as to make this proof completely flawless? I am not saying that this proof is wrong.. I enjoyed it very much. I just wish it did not have to cross out a minor detail however minor it may be...
Reply | Report Abuse | Link to thisThis derivation of Einstein's, which I have seen before, relies on an (excellent) approximation, whereas E = mc2 is an exact formula. Thus, it is somewhat unsatisfying.
Reply | Report Abuse | Link to thisI don't do math, but I have to question what the c^2 represents in E = mc^2.
Reply | Report Abuse | Link to thisWhat precisely does c^2 represent, other than some arbitrary large number?
I know it is often referred to the conversion factor - does this mean it has no meaningful relationship with the speed of light, or perhaps more generally, velocity applied to mass?
Despite the fact of rearrangement from m = E / c2 to E= mc2,
Reply | Report Abuse | Link to thismy father said:
Uncle Al made a small mistake, instead of symbol = it should be <.
When applied it makes sense and it is true.
Details and further explanations in Sequel Box, Sequel 1 at: http://www.visutech.net/peace365/index.asp?pageID=86
;~)
Yes, ABQJON, it is somewhat unsatisfying.
Reply | Report Abuse | Link to thisAnother point of view on post 23.
Please check the link - very much explanatory.
;~)