# Why can the speed of light be found from magnetic and electrical permeability?

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#### thebrain13

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##### Why can the speed of light be found from magnetic and electrical permeability?
« on: 01/04/2012 20:18:54 »
It's possible to calculate the speed of light by knowing the electric permittivity and the magnetic permittivity. the equation is c= 1/sqrt(magnetic permeability*electric permeability) It's neat that one can calculate the speed of light by knowing those two numbers, but I have no idea how it works. I was wondering if somebody has a good conceptual explanation about how those two things control the speed of an electromagnetic wave.
« Last Edit: 02/04/2012 20:43:06 by chris »

#### Phractality

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##### Re: calculating the speed of light
« Reply #1 on: 01/04/2012 21:22:33 »
If you look at the definitions of permittivity and permeability of vacuum, you'll see that they are defined in terms of the speed of light in the vacuum. Calculating the speed of light from permittivity and permeability is nothing more than turning the definitions around. It proves absolutely nothing.
Imagination is more important than knowledge. Einstein

#### thebrain13

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##### Re: calculating the speed of light
« Reply #2 on: 01/04/2012 21:25:37 »
interesting, so you are saying it is circular logic?

#### Phractality

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##### Re: calculating the speed of light
« Reply #3 on: 01/04/2012 21:40:12 »
interesting, so you are saying it is circular logic?
Absolutely!
Imagination is more important than knowledge. Einstein

#### imatfaal

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##### Re: calculating the speed of light
« Reply #4 on: 02/04/2012 12:46:51 »
Not sure that is completely correct.

The speed of light can be defined as follows
$$c^2 = \frac{1}{\mu_0\epsilon_0}$$

As c is fundamental $$\mu_0$$ the vacuum permeability and $$\epsilon_0$$ the vacuum permittivity are now defined in terms of each other and the speed of light.  BUT - this is not the only way they can be defined and the other methods involve other fundamental units that are not dependant (I think) on the speed of light

$$F_m = \frac{I^2\mu_0}{2\pi r}$$

Force between two wires based on the separation and the current flowing - this is Ampere's Law

and

$$F_c = \frac{1}{4\pi \epsilon_0}\frac{q_1q_2}{r^2}$$

Force between charged particles based on charge magnitude and separation - this is Coulomb's Law

Thus both $$\mu_0$$ the vacuum permeability and $$\epsilon_0$$ the vacuum permittivity can be determined
independently from speed of light.
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#### Phractality

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##### Re: calculating the speed of light
« Reply #5 on: 02/04/2012 19:35:51 »
Not sure that is completely correct.

The speed of light can be defined as follows
$$c^2 = \frac{1}{\mu_0\epsilon_0}$$

As c is fundamental $$\mu_0$$ the vacuum permeability and $$\epsilon_0$$ the vacuum permittivity are now defined in terms of each other and the speed of light.  BUT - this is not the only way they can be defined and the other methods involve other fundamental units that are not dependant (I think) on the speed of light

$$F_m = \frac{I^2\mu_0}{2\pi r}$$

Force between two wires based on the separation and the current flowing - this is Ampere's Law

and

$$F_c = \frac{1}{4\pi \epsilon_0}\frac{q_1q_2}{r^2}$$

Force between charged particles based on charge magnitude and separation - this is Coulomb's Law

Thus both $$\mu_0$$ the vacuum permeability and $$\epsilon_0$$ the vacuum permittivity can be determined
independently from speed of light.
I haven't fully analyzed the connection yet, but I think those formulas amount to definitions of the Coulomb in terms of $$\mu_0$$ and $$\epsilon_0$$. Thus Coulomb's law is based on the speed of light.
Imagination is more important than knowledge. Einstein

#### JP

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##### Re: calculating the speed of light
« Reply #6 on: 02/04/2012 20:22:31 »
In addition to the quantities $$\mu_0$$ and $$\epsilon_0$$, the only dimensionful units in the equations Matthew posted (and in Maxwell's equations + Lorentz force, which describe more fully how the electric and magnetic forces interact with charges) have units of: charge, distance and time.

You certainly can redefine your unit of charge to eliminate either $$\mu_0$$ or$$\epsilon_0$$.  To eliminate both, you need to redefine your measure of space or time, which is simply choosing a longer/shorter measuring stick or choosing to measure units of time that are longer/shorter than 1 second.  Doing so doesn't change the speed of light, it just changes the units you're using to measure it.
« Last Edit: 02/04/2012 20:28:12 by JP »

#### Pmb

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• Physicist
##### Re: Why can the speed of light be found from magnetic and electrical permeability?
« Reply #7 on: 02/04/2012 20:54:57 »
It's possible to calculate the speed of light by knowing the electric permittivity and the magnetic permittivity. the equation is c= 1/sqrt(magnetic permeability*electric permeability) It's neat that one can calculate the speed of light by knowing those two numbers, but I have no idea how it works. I was wondering if somebody has a good conceptual explanation about how those two things control the speed of an electromagnetic wave.
The laws of physics, when expressed as a set of differential equations, are called Maxwell's Equations. Those equations can be manipulated resulting in wave equations, one equation for each component of the electric and magnetic fields. The speed of the waves in those equations is calculated to be c = 1/sqrt(magnetic permeability*electric permeability). The wave equation has a certain form and that form has the speed of the wave expressley written in the form of the equation.

How well do you understand vector calculus? If you're familiar with it then I can be more precise.

Pete
« Last Edit: 02/04/2012 20:59:45 by Pmb »

#### Phractality

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##### Re: Why can the speed of light be found from magnetic and electrical permeability?
« Reply #8 on: 03/04/2012 02:22:55 »
An altermative definition of Coulomb is Avogadro's number of electron charges. But Avogadro's number is derived in a roundabout way from Coulomb's law, so we still have circular logic when deriving the speed of light from constants that are defined in terms of the speed of light.
Imagination is more important than knowledge. Einstein

#### JP

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• 3366
##### Re: Why can the speed of light be found from magnetic and electrical permeability?
« Reply #9 on: 03/04/2012 02:44:53 »
You're mistaking definitions for circular logic here.  You can measure units of charge in terms of the electron charge and Maxwell's equations will still hold.  The speed of light will still take on exactly the same physical value as a result of the two measured constants, no matter what units you decide to use to measure charge, distance or time.

In fact it's a standard problem in undergraduate physics courses to take Maxwell's equations, which have been derived from experimental knowledge of how magnetic and electric fields interact with charges and show that they're also describing electric and magnetic fields that propagate as waves which move at the speed $$1/\sqrt{\mu_0 \epsilon_0}.$$  The fact that any reference to the motion of the observer drops out means that this speed is constant and reference frame independent.  There is no circular reasoning in there.
« Last Edit: 03/04/2012 02:47:50 by JP »

#### graham.d

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##### Re: Why can the speed of light be found from magnetic and electrical permeability?
« Reply #10 on: 03/04/2012 11:23:53 »
There are good historic and practical reasons why we measure Force in Newtons, distances in metres, time in seconds etc. If we choose different units, some of the constants we use in equations disappear. It is quite common, for example, to set c=1 when dealing with General Relativity because it simplifies the equations and saves a lot of writing/typing and reveals more about the fundamental concepts.