Why is Gaussian charge not equivalent to SI charge?

[songoku]

TL;DR Summary

Physical quantities have unit and also dimension, such as unit of mass is kg and the dimension of mass is M

But what is the actual use of dimension? We can do dimension analysis but it can be simply changed into unit analysis and the result will be the same. So why introduce dimension for physical quantities? Why unit is not enough?

Thanks

 

[vanhees71]

In different systems of units the units of quantities with different dimensions can be the same. E.g., in natural units with $c=1$ energy, mass, and momentum have the same units, but these quantities have different dimensions.

 

[Dale]

So why introduce dimension for physical quantities? Why unit is not enough?

How else would you know that you can convert miles into kilometers but not into kilograms?

 

[Arjan82]

You can see dimension as a specific ruler that measures some physics quantity, one ruler for each specific quantity. The units are then the small bars on this ruler that tells you what 1 unit in this dimension exactly is.

So take the dimension of mass, the units can be kg, pounds, etc. Each unit tells you what measure we give to 1 (1 kg, 1 pound, etc). But they all measure the same thing: mass.

 

[kuruman]

How else would you know that you can convert miles into kilometers but not into kilograms?

Or that you can convert $\dfrac{\text{(statcoulombs)}^2}{\text{cm}}$ into Joules?

 

[songoku]

How else would you know that you can convert miles into kilometers but not into kilograms?

Sorry I still do not get this part. I think I can know I can convert miles to km and not to kg by knowing the physical quantities that represented by the units, without knowing the dimension. Miles and km are units of length and kg is unit of mass so direct conversion from length to mass is not possible. Am I missing something?

Or that you can convert $\dfrac{\text{(statcoulombs)}^2}{\text{cm}}$ into Joules?

I can't do this one. Statcoulombs is another unit for electric charge and cm is unit for length, so I tried to convert $\frac{C^2}{\text{m}}$ to $\frac{\text{kg}. \text{m}^2}{\text{s}^2}$

$$\frac{C^2}{\text{m}}=\frac{\text{A}^2 \text{s}^2}{\text{m}}$$

I have changed all the units to base units and I do not know how it can be same as Joules

Thanks

 

[Dale]

Miles and km are units of length and kg is unit of mass so direct conversion from length to mass is not possible. Am I missing something?

Yes. Apparently you are missing that length is the dimension of miles and kilometers and mass is the dimension of kilograms. Your response actually supports my argument exactly.

Were you not aware that length is the dimension of the meter? See p 136 here for the dimensions of all SI units https://www.bipm.org/utils/common/pdf/si-brochure/SI-Brochure-9.pdf

Statcoulombs is another unit for electric charge

The statcoulomb is a unit of charge, but charge is not a base dimension in cgs units. The dimension of the statcoulomb is $L^{3/2} M^{1/2} T^{−1}$

 

[songoku]

Yes. Apparently you are missing that length is the dimension of miles and kilometers and mass is the dimension of kilograms. Your response actually supports my argument exactly.

Were you not aware that length is the dimension of the meter? See p 136 here for the dimensions of all SI units https://www.bipm.org/utils/common/pdf/si-brochure/SI-Brochure-9.pdf

I always think like this:

Physical quantity = length
Unit = meter (m)
Dimension = L

Physical Quantity = mass
Unit = kilogram (kg)
Dimension = M

So no, I am not aware that length is the dimension of meter because I think length is physical quantity, not dimension. That is why I ask why we need dimension when we already have units because I used to think that dimension is redundant since the application of dimension that I am aware is dimensional analysis, which can be done the same way by units analysis.

Is it not correct to think length is physical quantity and not a dimension?

The statcoulomb is a unit of charge, but charge is not a base dimension in cgs units. The dimension of the statcoulomb is $L^{3/2} M^{1/2} T^{−1}$

How can we know the dimension of statcoulomb is $L^{3/2} M^{1/2} T^{−1}$? Is there a way to derive it, like we derive C = A.s?

Thanks

 

[vanhees71]

For the electromagnetic quantities in different systems of units, it's more complicated. Take the usual Gaussian units and the SI units.

In the Gaussian units no extra base unit for electric charge is introduced, while in the SI it is, i.e., the Coulomb or Ampere times second.

The mechanical units are easy to convert, because for all three base units in the SI (s, m, kg) there are also base units in the Gaussian system (s, cm, g).

So to see, how statcoulombs are converted in SI Coulombs, you have to use the definition via Coulomb's force law for two charges of the same magnitude.
$$F=\frac{Q_G^2}{r^2}.$$
The unit of charge is
$$1 \text{statC}=1\text{Fr}=1 \sqrt{\text{dyn}} \cdot \text{cm}=1 \text{g}^{1/2} \text{cm}^{3/2} \text{s}^{1/2}.$$
I.e., $1\text{statC}$ is defined such that for two charges of $1 \text{statC}$ at a distance of 1 cm you get a force of 1 dyn.

To get this charge in SI Coulombs just consider two charges at a distance of $1 \; \text{cm}=10^{-2} \text{m}$ leading to a force of $1 \; \text{dyn}$. Now $1 \text{dyn}=1 \text{g} \; \text{cm}/\text{s}^2=10^{-5} \text{kg} \; \text{m}/\text{s}^2=10^{-5} \text{N}.$
The charge corresponding to 1 dyn force between two equal charges at the distance of 1 cm in the SI then leads to
$$1 \; \text{statC} \hat{=} \sqrt{4 \pi \epsilon_0} (10^{-5} \text{N})^{1/2} 10^{-2} \text{m} \simeq 3.336 \cdot 10^{-10} \text{C}.$$
The confusing aspect of the em. units is that the quantities have not only different units but also different dimensions! That's why I can't write an equality sign between a charge in statC and a charge measured in SI-C but only a "refers-to sign".

 

[Arjan82]

Sorry I still do not get this part. I think I can know I can convert miles to km and not to kg by knowing the physical quantities that represented by the units, without knowing the dimension.

What you call physical quantity IS the dimension. It is exactly the same thing.

Physical Quantity = mass
Unit = kilogram (kg)
Dimension = M

Here, again, the dimension is mass, the letter 'M' is just shorthand thereof. And thus, what you call Physical Quantity is called 'Dimension' in Physics.

 

[Orthoceras]

The confusing aspect of the em. units is that the quantities have not only different units but also different dimensions! That's why I can't write an equality sign between a charge in statC and a charge measured in SI-C but only a "refers-to sign".

Wouldn't it be less confusing if it was explicitly stated that Gaussian charge is not equal to SI charge? For example, wikipedia states: $$Q_{Gaussian}=\frac{1}{\sqrt{4 \pi \epsilon_0}} Q_{SI}$$

 

[Dale]

I always think like this:

Physical quantity = length
Unit = meter (m)
Dimension = L

Physical Quantity = mass
Unit = kilogram (kg)
Dimension = M

So no, I am not aware that length is the dimension of meter because I think length is physical quantity, not dimension.

If you read the link that I posted earlier you will see that it says "Physical quantities can be organized in a system of dimensions, where the system used is decided by convention. Each of the seven base quantities used in the SI is regarded as having its own dimension. The symbols used for the base quantities and the symbols used to denote their dimension are shown in Table 3"

So according to this the dimensions are how we organize the physical quantities. Thus, length is a physical quantity and in order to organize physical quantities SI assigns length its own dimension, also called length. The symbol L is not the dimension of length, it is just a symbol used to denote length. The dimension of length is length.

Charge is also a physical quantity. As described above, the organization of charge into a system of dimensions depends on the convention used. In SI, charge is assigned the dimension of electric current times time, $I T$. In cgs, charge is assigned the dimension of length to the 3/2 times mass to the 1/2 times time to the -1, $L^{3/2} M^{1/2} T^{−1}$.

The dimension is not the physical quantity, and it is not the symbol, and it is not the unit. It is a convention for organizing physical quantities. I don't think that any of them are redundant with each other. So going back to your OP you cannot discard dimension.

 

[songoku]

Thank you very much for all the help and explanation vanhees71, Dale, Arjan82, kuruman, Orthoceras

 

[symbolipoint]

What kind of physical quality or kind of quantity: Dimension.

@Arjan82 gave the most direct best response for what is dimension.

 

[vanhees71]

Wouldn't it be less confusing if it was explicitly stated that Gaussian charge is not equal to SI charge? For example, wikipedia states: $$Q_{Gaussian}=\frac{1}{\sqrt{4 \pi \epsilon_0}} Q_{SI}$$

Yes, that should be explicitly stated. Neither many textbooks nor Wikipedia are very accurate on this.

That's why I would write the $\hat{=}$ symbol, not an equality sign in the quoted formula. Also see

https://en.wikipedia.org/wiki/Centimetre–gram–second_system_of_units