Why is E=MC2 Squared and not Cubed?

In summary, the equation e=mc2 applies through dimensional analysis, where both sides of the equation must have the same physical dimension. This is not related to the number of spatial dimensions in the universe. Energy has physical dimensions of ML2/T2 and mass has physical dimensions of M. The speed of light is squared in the equation to convert its units to match those of energy and mass. The constant k in the equation is determined through physical arguments and in this case, it is equal to 1.
  • #1
Trooper149
14
3
Problem Statement: Trying to understand the principles for the equation e=mc2.
Relevant Equations: E=mc2

So just started at a physics A level and so far loving it. I understand this question has indeed been asked before, however for different reasons. I understand the purpose of e=mc2 and I understand the formula. My question is related more the context in which this equation applies? Also, I understand that it is squared to reflect our dimension, however don't we operate in the 3rd dimension, in which case should it not be cubed? Perhaps I have misinterpreted what is being described?

Appreciate any help.
 
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  • #2
Trooper149 said:
Problem Statement: Trying to understand the principles for the equation e=mc2.
Relevant Equations: E=mc2

Also, I understand that it is squared to reflect our dimension, however don't we operate in the 3rd dimension, in which case should it not be cubed? Perhaps I have misinterpreted what is being described?
Unfortunately, you have misinterpreted what is being described, yes. What is likely being discussed is dimensional analysis. This has nothing to do with the number of spatial dimensions that we happen to live in. Instead, it has to do with the physical dimensions of different physical quantities. In the SI system, there are seven base units, the ones you will encounter most in relativity is length (L), time (T), and mass (M). In order for a physical equality to hold, both sides must have the same physical dimension. For example, an area has physical dimension of length squared (L^2), which means that you must measure areas in units of that dimension (such as m2). It would not make sense to measure areas in units of time (such as seconds).

For ##E = mc^2## it is the same. Energy has physical dimensions of ##\mathsf{M L^2/T^2}## and mass physical dimensions of ##\mathsf M##. The only way the relationship can make sense is if the proportionality constant therefore has a physical dimension of ##\mathsf{L^2/T^2}##. The speed of light is a speed and therefore has dimensions of length/time (L/T) and the only way the relationship can make sense is therefore if you have ##E = kmc^2##, where ##k## is just a number (you have to determine such numbers through physical arguments, not through dimensional analysis). In this case, ##k = 1##.
 
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  • #3
Trooper149 said:
Also, I understand that it is squared to reflect our dimension,
No. The power you raise ##c## to has no connection with the number of dimensions.
 
  • #4
In Newtonian mechanics, in perfectly elastic collisions, mv and 1/2 mv^2 are conserved. The latter is the "energy".

The exponent 2 does not depend on the number of spatial dimensions in the universe. It is the same in a universe which has 100 spatial dimensions.
 
  • #5
Orodruin said:
Unfortunately, you have misinterpreted what is being described, yes. What is likely being discussed is dimensional analysis. This has nothing to do with the number of spatial dimensions that we happen to live in. Instead, it has to do with the physical dimensions of different physical quantities. In the SI system, there are seven base units, the ones you will encounter most in relativity is length (L), time (T), and mass (M). In order for a physical equality to hold, both sides must have the same physical dimension. For example, an area has physical dimension of length squared (L^2), which means that you must measure areas in units of that dimension (such as m2). It would not make sense to measure areas in units of time (such as seconds).

For ##E = mc^2## it is the same. Energy has physical dimensions of ##\mathsf{M L^2/T^2}## and mass physical dimensions of ##\mathsf M##. The only way the relationship can make sense is if the proportionality constant therefore has a physical dimension of ##\mathsf{L^2/T^2}##. The speed of light is a speed and therefore has dimensions of length/time (L/T) and the only way the relationship can make sense is therefore if you have ##E = kmc^2##, where ##k## is just a number (you have to determine such numbers through physical arguments, not through dimensional analysis). In this case, ##k = 1##.

This is excellent. So quite simply, because energy is measured in joules and joules is squared, they square light in e=mc2 to convert the value?
 
  • #6
Trooper149 said:
This is excellent. So quite simply, because energy is measured in joules and joules is squared, they square light in e=mc2 to convert the value?
I do not know what you mean by ”joules is squared”, but if you mean that you must square c to get the appropriate units, then yes, it is the only way in which you can relate these three quantities (up to a multiplicative constant that is dimensionless and must be determined, in this case that constant is one).

Note that this is not saying that such a relationship must exist and be physically meaningful (we had a recent thread with that misconception), just that if it exists, then it must be on that form. This type of argument is a special case of Buckingham’s pi theorem, which is an important result in dimensional analysis.

In general, I find that dimensional analysis - although quite basic and not very difficult or lengthy - is not covered satisfactorily in many physics curricula. I have a section covering the basics in the modelling chapter of my book and there are also many other nice and short texts on the subject. I am not aware of a nice summary on the internet, but admittedly I have not looked very hard for it.
 
  • #7
I agree, Orodruin. I teach students at high school how dimensional analysis eases their lives, with a "what else could the formula be" question central. I find it poorly covered in the textbooks we use at schools.
 
  • #8
Thank you for the informative replies guys and gals
 

1. Why is E=MC2 squared?

According to Albert Einstein's theory of special relativity, the equation E=MC2 represents the relationship between energy (E), mass (M), and the speed of light (C). The squared term in the equation is a result of the mathematical derivation of this relationship, and it shows that energy and mass are equivalent and interchangeable.

2. Can E=MC2 be cubed?

No, E=MC2 cannot be cubed because it is a fundamental equation in physics that has been extensively tested and verified through experiments. The squared term is an essential part of the equation and cannot be changed without altering the fundamental principles it represents.

3. What does the squared term in E=MC2 represent?

The squared term in E=MC2 represents the speed of light (C) squared. This value is approximately 9 x 1016 meters squared per second squared (m2/s2). It is a constant value that is crucial in understanding the relationship between energy and mass.

4. How does E=MC2 relate to the concept of mass-energy equivalence?

E=MC2 is a representation of the famous equation proposed by Einstein, which states that mass (M) and energy (E) are equivalent and can be converted into each other. The squared term in the equation shows that the amount of energy produced is directly proportional to the mass of the object.

5. Is E=MC2 only applicable to objects moving at the speed of light?

No, E=MC2 is applicable to all objects, regardless of their speed. However, at speeds much lower than the speed of light, the effects of the equation are not as significant. At the speed of light, the equation shows that an object's mass would become infinite, and it would require an infinite amount of energy to accelerate it further.

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