What happens if you increase μ0 and decrease ϵ0? Or vice versa

[em3ry]

I have been thinking about the physical significance of Planck's constant.

The effect of increasing Planck's constant on blackbody radiation is the red line below. Y-axis is frequency

Apparently if Planck's constant were infinite then there would be no blackbody radiation. But we know that as long as the speed of light is finite then accelerating charges will emit light.

Wikipedia says:

The physical meaning of the Planck constant could suggest some basic features of our physical world. These basic features include the properties of the vacuum constants μ0 and ϵ0.

So that got me thinking. The speed of light is $$\sqrt{\frac{1}{μ0 ϵ0}}$$. So that means that if you increase μ0 and decrease ϵ0 by the same amount then the speed of light would be the same but surely something in the universe would change.

So what would change if you increased μ0 and decreased ϵ0 by the same amount so that the speed of light was the same as before? Surely this would have some effect on the universe.

My question is about μ0 and ϵ0 but if you have any insight into Planck's constant then I will be glad to hear it too.

edit: I see that the Larmor formula depends on ϵ0 and c but not μ0. Decreasing ϵ0 increases the energy radiated by an accelerating charge.

$$P = {2 \over 3} \frac{q^2 a^2}{ 4 \pi \varepsilon_0 c^3}$$

edit2: Dale mentioned the Fine structure constant in post 1 below. It is:
$$\alpha = \frac{1}{4 \pi \varepsilon_0} \frac{e^2}{\hbar c} = \frac{\mu_0}{4 \pi} \frac{e^2 c}{\hbar}$$

edit3: The force between two separated electric charges with spherical symmetry (in the vacuum of classical electromagnetism) is given by Coulomb's law:
$$F_\text{C} = \frac{1} {4 \pi \varepsilon_0} \frac{q_1 q_2} {r^2}$$

comparing to the equation for the Fine structure constant we get:
$$F_\text{C} r^2 = constant = \alpha \hbar c$$

edit4: As Dale explains in post 65 his point about alpha is that given an equation like 1=abcxyz if you change z then you must change at least one other variable but more importantly the equation doesn't tell you which one to change. You must decide that on your own. Dale's point went under my radar and I apologize.

 

[Dale]

So what would change if you increased μ0 and decreased ϵ0 by the same amount so that the speed of light was the same as before?

To get actual physical changes you need to change the dimensionless constants like the fine structure constant. Merely changing the dimensionful constants simply alters your system of units.

In SI units the fine structure constant, $\alpha$, is proportional to $\mu_0$

 

[Paul Colby]

Not certain this counts for anything but the impedance of free space ($\sqrt{\frac{\mu_o}{\epsilon_o}} =$ 377ohms) would change. Antennas, for example, would all work differently?

 

[Delta2]

Increasing $\mu_0$ will have an effect of how strong are the magnetic fields. For example the magnetic field from a long thin conductor carrying current I is $$\mathbf{B}=\frac{\mu_0}{2\pi}\frac{I}{r}$$. We would have stronger magnetic field for the same current or for the same magnets.
Also because $\epsilon_0$ appears in Gauss's Law $$\nabla\cdot\mathbf{E}=\frac{\rho}{\epsilon_0}$$
decreasing it will also make stronger the electric field for the same charge distribution. So we would have stronger electrostatic interactions.

 

[em3ry]

The energy in the field is $$u_{EM} = \frac{\varepsilon}{2} |\mathbf{E}|^2 + \frac{1}{2\mu} |\mathbf{B}|^2$$

It looks like increasing e0 would increase the energy but the definition of E already has 1/e0 in it. The net result is that increasing e0 decreases the energy in the field. (Just like increasing the spring constant decreases the energy in the spring under any given load and increases the speed of waves in the spring)

I suspect that the definition of E shouldn't have 1/e0 in it.

 

[Delta2]

The energy in the field is $$u_{EM} = \frac{\varepsilon}{2} |\mathbf{E}|^2 + \frac{1}{2\mu} |\mathbf{B}|^2$$

It looks like increasing e0 would increase the energy but the definition of E already has 1/e0 in it. The net result is that increasing e0 decreases the energy in the field. (Just like increasing the spring constant decreases the energy in the spring under any given load and increases the speed of waves in the spring)

I suspect that the definition of E shouldn't have 1/e0 in it.

You are right here about the effect of $\epsilon_0$ on energy density of the EM field. However it is not the definition of E that has the $\frac{1}{\epsilon_0}$, it rather comes from Maxwell's equations and their general solution Jefimenko's Equations Jefimenko's equations - Wikipedia . Similar arguments hold for B and $\mu_0$.

 

[Vanadium 50]

I see everybody ignored poor Dale.

ε₀ and μ₀ define the Coulomb and the Ampere. (And by all that is holy, please can we not turn this into Yet Another Thread About the SI Redefinition? ) You can't switch between "regular" and "alternative" definitions of ε₀ and μ₀ without redefining the Coulomb and the Ampere in a consistent fashion. You most certainly cannot take equations and redefine quantities willy-nilly and expect anything beyond an inconsistent mess.

Also, I see the OP is changing his message after getting replies. That can only increase confusion not clarity.

 

[em3ry]

Planck's constant has units of

charge^2 * ohms

or

charge^2 * impedance of free space

charge * ohms = magnetic flux

https://en.wikipedia.org/wiki/Dipole_antenna#Radiation_resistance

 

[em3ry]

Now we are getting somewhere
https://en.wikipedia.org/wiki/Radiation_resistance
Radiation resistance * average current^2 = power radiated
(Impedance/Impedance of free space) * charge^2 / time^2 = energy / time
Impedance * charge^2 = energy * time = energy per frequency
energy per frequency = Planck's constant

 

[em3ry]

The energy in the field is $$u_{EM} = \frac{\varepsilon}{2} |\mathbf{E}|^2 + \frac{1}{2\mu} |\mathbf{B}|^2$$

It looks like increasing e0 would increase the energy but the definition of E already has 1/e0 in it. The net result is that increasing e0 decreases the energy in the field. (Just like increasing the spring constant decreases the energy in the spring under any given load and increases the speed of waves in the spring)

I suspect that the definition of E shouldn't have 1/e0 in it.

ε₀ and μ₀ define the Coulomb and the Ampere. (And by all that is holy, please can we not turn this into Yet Another Thread About the SI Redefinition? ) You can't switch between "regular" and "alternative" definitions of ε₀ and μ₀ without redefining the Coulomb and the Ampere in a consistent fashion. You most certainly cannot take equations and redefine quantities willy-nilly and expect anything beyond an inconsistent mess.

B = μ₀ H
So μ₀ is part of the definition of B but is not part of the definition of H (unless H already has 1/μ₀ in it)

D = ε₀ E (But E already has 1/ε₀ in it)
So ε₀ is part of the definition of E but is not part of the definition of D

 

[Dale]

Also, I see the OP is changing his message after getting replies. That can only increase confusion not clarity.

@em3ry please do not edit your posts after they have received a reply. Edits afterwards can cause confusion. Instead, simply write a new reply with the additional information.

 

[Vanadium 50]

Now we are getting somewhere

That's one of us who thinks so.

To me, this looks like more willy-nilly. Pleasew reread what @Dale said.

 

[em3ry]

To me, this looks like more willy-nilly.

Sigh. Its dimensional analysis. I am simply looking for some sort of electromagnetic phenomenon that has the right units. It not only has the right units it has the right effect too

 

[Dale]

Sigh. Its dimensional analysis. I am simply looking for some sort of electromagnetic phenomenon that has the right units. It not only has the right units it has the right effect too

The dimensional analysis will not tell you about any physical changes. You need to look at the dimensionless constants, in this case the fine structure constant.

As the fine structure constant changes the effects are pretty profound. For example, an increase in the fine structure constant essentially increases the strength of the electromagnetic interaction relative to the strength of other interactions. For example, as nuclear protons repel more the maximum size of stable nuclei becomes smaller.

 

[Vanadium 50]

I am simply looking for some sort of electromagnetic phenomenon that has the right units. It not only has the right units it has the right effect too

It's more willy-nilly, I'm afraid. The path you are on will tell you energy and toque are the same thing.

 

[em3ry]

Thats why its so difficult. Thats also why I don't think that the Planck's constant in blackbody radiation is the same as the Planck's constant in the Bohr model. One has units of angular momentum and the other is an electromagnetic phenomenon involving energy emitted per frequency.

 

[em3ry]

The dimensional analysis will not tell you about any physical changes. You need to look at the dimensionless constants, in this case the fine structure constant.

As the fine structure constant changes the effects are pretty profound. For example, an increase in the fine structure constant essentially increases the strength of the electromagnetic interaction relative to the strength of other interactions. For example, as nuclear protons repel more the maximum size of stable nuclei becomes smaller.

All I know about the fine structure constant is that it is the velocity of the electron in the Bohr model of hydrogen. To change it you would have to change the strength of the electrical attraction or change the angular momentum of the electron

 

[Vanadium 50]

Thats also why I don't think that the Planck's constant in blackbody radiation is the same as the Planck's constant in the Bohr model.

Good heavens.

If you think PF is the place to explore personal theories that run contrary to established science, you should re-read the PF Rules.

 

[em3ry]

what do you mean contrary? They have the same value. But they don't represent the same underlying phenomenon. There is no conflict with established science at all.

I feel like you're trying to pick a fight with me here. I have been going to great lengths to fit in here

You said yourself that just because something has the same unit doesn't mean that it represents the same thing.

if I am wrong then just explain to me what the angular momentum of the electron has to do with black body radiation

 

[Delta2]

ε0 and μ0 define the Coulomb and the Ampere. (And by all that is holy, please can we not turn this into Yet Another Thread About the SI Redefinition? ) You can't switch between "regular" and "alternative" definitions of ε0 and μ0 without redefining the Coulomb and the Ampere in a consistent fashion

The Ampere is defined as $\frac{Coulomb}{sec}$. Can you explain why changing $\epsilon_0$ or $\mu_0$ redefines the coulomb?

 

[Vanadium 50]

They convert forces to charges and currents. (Or the other way around if you like)

 

[em3ry]

it sounds like I've stepped into some pre-existing arguments between you people. I want nothing to do with your arguing.

I would prefer it if we could stick to the subject of this thread which is the physical significance of the Planck's constant.

usually you can just look at the units and it will tell you what it represents but in this case it doesn't seem to be that easy

It seems to represent an aspect of electromagnetic interactions that is not immediately obvious. I think the impedance of free space is a good place to start

 

[Delta2]

They convert forces to charges and currents. (Or the other way around if you like)

Ok if i understand you well, you mean that the coulomb is defined from Newton, meter and $\epsilon_0$ via the electrostatic force law (Coulomb law) $$F=\frac{1}{4\pi\epsilon_0}\frac{q_1q_2}{r^2}$$. Is that it?

 

[Dale]

They have the same value. But they don't represent the same underlying phenomenon.

They most certainly are exactly the same thing. That is why it is such an important quantity that we now use it to define our system of units that we use to measure all of these phenomena.

I would prefer it if we could stick to the subject of this thread which is the physical significance of the Planck's constant.

Hmm, I thought it was $\mu_0$ you were investigating.

The same thing can be said about Planck’s constant as I said in post 2. The dimensionful constants just tell you about your system of units. Currently, the SI uses Planck’s constant to define the kilogram. To see actual physical effects, not just unit changes, you have to consider changes to the dimensionless constants.

However, in the current SI system Planck’s constant is fixed, so any change to Planck’s constant would require a different system of units. I would therefore recommend sticking with $\mu_0$ (proportional to $\alpha$) as you started since then you could keep the SI units as currently defined.

 

[em3ry]

Whoops you are right. I was asking about $\mu_0$

I am thinking about Planck's constant but I was asking about $\mu_0$

 

[Dale]

Well, $\mu_0$ is the better one since in the current SI it is a measured quantity instead of a defined quantity. $$\mu_0=\alpha \frac{4\pi \hbar}{e^2 c}$$Everything else in that expression is defined.

 

[em3ry]

The dimensionful constants just tell you about your system of units. Currently, the SI uses Planck’s constant to define the kilogram. To see actual physical effects, not just unit changes, you have to consider changes to the dimensionless constants.

I understand you better than you think I do but we seem to be talking past one another here. If a car is going 100 mph and I change it so that it is moving 50 mph then that is not just a change of units. I didnt redefine 100 mph to be 50 mph. I actually changed the cars motion.

I am not talking about changing the definition of Planck's constant. I am asking about what would happen if we could magically change the actual value of Planck's constant itself. How would the world change? I ask because I want to know what the constant represents so I can understand the equations better. Of course we can't actually change Planck's constant but its just a thought experiment.

 

[Delta2]

The dimensionful constants just tell you about your system of units

To see physical effects you have to consider changes to the dimensionless constants.

Sorry @Dale can you elaborate a bit more on these claims, why changes to dimensionful constants are all about which system of units we use and we need changes to dimensionless constants to see physical effects.
Thanks.

 

[em3ry]

Because those are the things that don't depend on your system of units

 

[Delta2]

For example if I say that I could change the speed of light to be $300\frac{m}{sec}$ you will again tell me that this would redefine meter and second? I don't think meter and second are defined through the speed of light in SI.

 

[Dale]

If a car is going 100 mph and I change it so that it is moving 50 mph then that is not just a change of units. I didnt redefine 100 mph to be 50 mph. I actually changed the cars motion.

Yes, but let’s think a little more carefully about what this statement means. You have a reference speed 1 mph, and the dimensionless ratio of the car’s speed to the reference speed is changing from 100 to 50. Since that is a dimensionless change it can have physical meaning.

This is as opposed to changing from 100 mph to 45 m/s. There we have changed our dimensionful quantities, but nothing dimensionless has changed. We changed both our number and our reference speed by the same proportion, so all the dimensionless quantities are the same.

I am not talking about changing the definition of Planck's constant. I am asking about what would happen if we could magically change the actual value of Planck's constant itself.

You cannot do that without changing your units. So now all of your reference values have changed. Specifically your unit for mass and all units that depend on the kg have changed.

By doing this you have unavoidably done the equivalent of changing from mph to m/s, except that our new units are unknown. So we can no longer disentangle changes due to the value and changes due to the units.

How would the world change? I ask because I want to know what the constant represents so I can understand the equations better. Of course we can't actually change Planck's constant but its just a thought experiment.

I am not objecting to the thought experiment at all, but I am trying to explain that your question, as posed, is under-specified. There is not enough information to discuss it.

Because $$\mu_0=\alpha\frac{4\pi \hbar}{e^2 c}$$ it is not possible to change only $\hbar$. You must also specify how the other quantities change to keep this equation true. Once you have done so, the changes to $\alpha$ completely determine the physical changes and the changes to the other quantities determine the changes to your units.

 

[Dale]

For example if I say that I could change the speed of light to be $300\frac{m}{sec}$ you will again tell me that this would redefine meter and second? I don't think meter and second are defined through the speed of light in SI.

Yes. You can look up the definition for the meter on the official BIPM website:

“ The metre, symbol m, is the SI unit of length. It is defined by taking the fixed numerical value of the speed of light in vacuum c to be 299 792 458 when expressed in the unit m s–1, where the second is defined in terms of the caesium frequency DeltanuCs.”

https://www.bipm.org/en/measurement-units/base-units.html

The new SI system is really clean and consistent. But to understand it you have to understand the fact that the dimensionful constants are arbitrary. So much so that the BIPM, a self appointed committee, can fix their values by decree.

 

[em3ry]

I get the impression that there's been a lot of arguing lately about changing systems of units.

 

[em3ry]

...the subject of this thread which is the physical significance of the Planck's constant.

usually you can just look at the units and it will tell you what it represents but in this case it doesn't seem to be that easy

It seems to represent an aspect of electromagnetic interactions that is not immediately obvious. I think the impedance of free space is a good place to start

On the subject of obscure electromagnetic phenomena, I think the emission of electromagnetic waves by oscillating charges is fairly well understood (Larmor formula) but that the reverse process of absorbing electromagnetic radiation, especially blackbody radiation, is less well understood. Perhaps this is why the physical significance of Plancks constant is obscure. It has something to do with absorbance

There doesn't seem to be a Larmor formula for absorbance

 

[Dale]

I get the impression that there's been a lot of arguing lately about changing systems of units.

The BIPM changed the SI quite substantially in 2019, but a lot of people are still not aware of the change. So it comes up often, usually in the context of questions like yours. If you are unaware of the SI change usually you are unaware of the difficulties I mentioned above regarding this type of question.