# Homework Help: Dimensions of temperature and charge in terms of M, L and T

1. Jul 4, 2011

### HotMintea

"Most physicists do not recognize temperature, Θ, as a fundamental dimension of physical quantity since it essentially expresses the energy per particle per degree of freedom, which can be expressed in terms of energy (or mass, length, and time). Still others do not recognize electric charge, Q, as a separate fundamental dimension of physical quantity, since it has been expressed in terms of mass, length, and time in unit systems such as the cgs system." (http://en.wikipedia.org/wiki/Dimensional_analysis#Definition)

Since the number of particles and the degree of freedom are dimensionless, it seems to imply that temperature has the same dimension as energy, namely, $M\ L^2\ T^{-2}$. However, I could not find any source for how temperature and charge can be expressed in terms of M, L and T. What are the correct expressions and how are they derived?

Last edited: Jul 4, 2011
2. Jul 4, 2011

### Ranger Mike

they may not but anyone manufacturing high precision parts know about thermal expansion and have to deal with it every day.

3. Jul 4, 2011

Hello HotMintea.I don't believe the claim made in the first sentence you quoted from the Wiki article you referred to.Did you notice that Wiki had added the comment "citation needed"?In the same article the author refers to the c.g.s system of units and if you click on that you may find the answers you need.
Personally I have forgotten cgs units and use SI only.In fact I believe that the majority of scientists and engineers use SI so perhaps you will be well advised to familiarise yourself with these units.In the SI system temperature is taken as a dimension and current rather than charge is taken as a dimension.

4. Jul 4, 2011

### HotMintea

Hello Ranger Mike. I cannot make out how thermal expansion may be relevant to the question. Could you elaborate further?

Hello Dadface. Yes, I noticed the "citation needed."

In the cgs article, there is a section titled "Alternate derivations of CGS units in electromagnetism." However, I cannot find the 'standard' derivation. Moreover, I do not understand how to actually do the "alternate" derivations. (http://en.wikipedia.org/wiki/Cgs#Alternate_derivations_of_CGS_units_in_electromagnetism)

In the discussion page of the CGS article, I found a section called "Is this system still in use?". It seems that it is still used by "people which works [sic] with magnetic fields," and "astronomers." (http://en.wikipedia.org/wiki/Talk:C...ystem_of_units#Is_this_system_still_in_use.3F)

"From the point of view of fundamental physics where one does not distinguish between certain (or any) dimensions, argument based on dimensional analysis seem to be nonsense."(http://en.wikipedia.org/wiki/Talk:D...any_dimensions_to_consider.3F_and.2C_Rigor.3F)

Is it true?

Last edited: Jul 4, 2011
5. Jul 4, 2011

### Mapes

Arguing that temperature and energy are dimensionally equivalent implies that entropy is dimensionless. To that point, you may be interested in Leff's http://www.csupomona.edu/~hsleff/EntropyDimensionless.pdf" [Broken]

Note that dimensions are not the same as units. Saying that the constant c is represented dimensionally by L T-1 doesn't confine you to using m s-1 or inch week-1.

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6. Jul 4, 2011

### HotMintea

The article says, "The Kelvin temperature $T$ is defined to be proportional to $\tau$ and to have the value $T_3\ = \ 273.16 K$ at the triple point of H2O. " This definition is not specific enough for me to find an expression for temperature in terms of M, L and T.

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7. Jul 4, 2011

### Studiot

In classical physics you have to have a thermodynamic quantity in addition to M, L & T.
A lot of effort went into choosing M, L & T in the first place (the choice is not unique) and further effort went into choosingtemperature and its standard symbol, theta, since T was already taken.

The whole point of introducing T is to get away from the need for using a defined substance, which they have not done in your reference. Your system then becomes MLT plus 'water' - just as the standard kg and metre have left Paris!

There is a similar situation with charge/current. You cannot avoid introducing something.

go well

8. Jul 4, 2011

### Mapes

The equation of state is different for different systems. For example, consider $n$ moles of an ideal gas in a cylinder with cross-sectional area $L^2$ and height $L$, pressurized by the weight of a piston with mass $M$. The temperature is $T=PV/nR=MgL/nR$, where $g$ is the acceleration of gravity and $R$ is the gas constant, both of which are relatable in terms of $M$, $L$ and $T$. Therefore, the energy per molecule is proportional to $ML$.

9. Jul 4, 2011

### HotMintea

Do you imply the part of the Wikipedia article is incorrect?

Does it mean there is no definite dimension of temperature in terms of M, L and T, because it depends on the situation? How about charge?

10. Jul 4, 2011

### Mapes

In the sense that (1) temperature is defined by $T=(\partial U/\partial S)_{V,N,q\dots}$, and (2) that you might take entropy to be dimensionless (see the Leff article I linked to above; I don't know the current consensus in this area), then temperature could be related in a definite way to dimensionality $M L^2 T^{-2}$. But this doesn't imply that the equation of state is known at this time for every possible system.

11. Jul 4, 2011

### HotMintea

In (1), the definition uses U and S. Can they be defined without temperature? Moreover, can they be expressed in terms of M, L and T?

In the case of (2), the article calls it tempergy. Hence, at least in the article temperature seems to have the different dimension from energy. Thus, when temperature is defined to have the different dimension from energy, did I understand you correctly that its dimension varies in terms of M, L and T?

Last edited: Jul 4, 2011
12. Jul 4, 2011

### Mapes

I've only seen energy have units of $ML^2T^{-2}$ and generally have seen temperature have units of temperature (e.g., K, °C, °F), except in specific, well-defined systems where it has units of $ML^2T^{-2}$ (see my example above).

13. Jul 4, 2011

### HotMintea

It seems we have the conclusion that temperature and charge CANNOT be expressed in terms of M, L and T, except when temperature is defined to have the same dimension as energy.

14. Jul 4, 2011

### Mapes

I'd buy that for the temperature argument (haven't looked into charge).

15. Jul 4, 2011

### HotMintea

The article "Statcoulomb" says $1\ statC\ = \ g^{1/2}\ cm^{3/2}\ s^{-1}$.
(http://en.wikipedia.org/wiki/Statcoulomb)

However, in Coulomb's law the dimension of charge cancels in the equation (http://en.wikipedia.org/wiki/Coulomb's_law), just as the dimension of temperature cancels in $\tau = kT$ (p.1: http://www.csupomona.edu/~hsleff/EntropyDimensionless.pdf [Broken]) or in $T\ = \frac{ \ m\ v_{rms}^2}{3k_B}$. (http://en.wikipedia.org/wiki/Kinetic_theory#Temperature_and_kinetic_energy)

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16. Jul 4, 2011

### Studiot

So why do schoolmasters spend so much time and effort hammering into kids that heat (=energy) and temperature are very different things?

And which has more energy according to this novel theory, the ocean at 10 deg C or my body at 37 deg C and which way does the energy (heat) flow?

17. Jul 4, 2011

### HotMintea

"An examination of thermal physics books shows that although some authors define entropy as a dimensionless quantity and some define temperature as an energy, ..." This is in the first page of the document Mapes refered to.(http://www.csupomona.edu/~hsleff/EntropyDimensionless.pdf [Broken])

I would not like to speculate, since I hardly understand the document. I wonder why no one has answered my question with clear yes, no, or multiple possible answers.

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