# B Permittivity and Permeability changes

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1. Jul 10, 2017

### mpolo

I have a question about permeability and permittivity of free space. Theoretically speaking. If either one or both were to increase in value would that make the speed of light be faster or slower. I know they are constants. I am just curious if they did change what would happen.

2. Jul 10, 2017

### NFuller

Hypothetically, yes since the speed of light is related to them by
$$c=\sqrt{\frac{1}{\epsilon\mu}}$$.
In different materials the permittivity usually becomes larger which is why light travels slower through materials like water, glass, plastic, etc.

3. Jul 10, 2017

### Staff: Mentor

In current SI units the permeability and permittivity of free space are exactly defined quantities, as is the speed of light. None of them can change except by action of the BIPM committee. They are not subject to any experimental changes, only changes in the committee's definition.

4. Jul 10, 2017

### Electric to be

So if the speed of light did ever somehow "change", it would be our measure of the meter that would suddenly be off, huh?

5. Jul 11, 2017

### Staff: Mentor

If the fine structure constant changed, then the number of atoms that light passes in one wavelength of a specified radiation would change.

6. Jul 11, 2017

### mpolo

Could someone please show me an example of the equation above presented by NFuller. The units confuse me a little. Plug in the accepted values so I can see them and the result. I want to try the equation out on my calculator. Thanks.

7. Jul 11, 2017

### vanhees71

Using the SI in electromagnetism makes it a pretty complicated subject ;-)). Anyway, let's try

The Coulomb force between two point particles (magnitude) in SI units is
$$F=\frac{q_1 q_2}{4 \pi \epsilon_0 r^2}$$
So the dimension for the conversion factor $\epsilon_0$ is
$$[\epsilon_0]=[Q^2 /(r^2 F)]=\text{C}^2 \text{s}^2/ (\text{kg} \; \text{m}^3).$$
Here it's the force between two pieces of current conducting wire defining
$$F=\mu_0 I_1 I_2 L/d \; \Rightarrow \; [\mu_0]=[F/I^2]=\text{kg} \; \text{m}/\text{s}^2 \cdot \text{s}^2/\text{C}^2=\text{kg} \; \text{m}/\text{C}^2.$$
and thus finally
$$[\epsilon_0 \mu_0]=\text{s}^2/\text{m}^2$$
which is the dimension of an inverse squared speed.

Manipulating the free Maxwell equations a bit you get the wave equation
$$(\mu_0 \epsilon_0 \partial_t^2-\Delta) \vec{E}=0,$$
which shows that the phase velocity is indeed $c=1/\sqrt{\epsilon_0 \mu_0}$.

It's now also very clear that all there is for the vacuum permittivity and permeability is that they are fixed constants defining the system of units: $\mu_0$ is fixed by the definition of the Ampere, i.e., the unit of the electric current. $c$ is fixed through the definition of the unit of length, metre, via the definition of the second, and thus also $\epsilon_0$ is fixed by just the definition of the SI for time, length, mass, and electric current.

Of course, in the medium $\epsilon=\epsilon_0 \epsilon_r$ and $\mu=\mu_0 \mu_r$ are parameters from the constitutive equations of the medium. They are not so fundamental since they are defined from the microscopic theory via linear-response theory, i.e., they are related to the description of the medium for weak em. fields.

8. Jul 11, 2017

### mpolo

impressive piece of work vanhees71. I was just looking for the actual numbers that are used in the equation provided by NFuller. Your full explanation is nice though because it shows the links between different aspects of nature. Thank you.