Origin of the equation c^2=1/epsilon_0*mu_0

[Sunfire]

Hello,

Could someone describe the origin of the equation c²=1/ε₀μ₀

Is it derived from fundamental reasons?

Thanks.

 

[Vanadium 50]

Start with \nabla \times \nabla \times E and apply Maxwell's Equations.

 

[Sunfire]

I see, because ε₀ and μ₀ appear in the Maxwell's (SI) equations. Then, - how do ε₀ and μ₀ get to appear in the Maxwell's equations?

ε₀ is the coefficient in the Coulomb's force (2 static charges)
μ₀ is the coefficient in the Ampere's force (2 parallel wires)

I am guessing via F=qE, E gets to be expressed via ε₀ and then ε₀ gets to appear in Maxwell's equations because E is there. In the end, I am trying to understand how the deliberate choice (inserted "by hand", e.g. artificially) of the value of μ₀ determines the value of ε₀.

The shortest connection between ε₀ and μ₀ seems to be the relation c² = 1/ε₀μ₀. But it seems that the derivation of this relation is not short, but rather long and winded.

Is there a SHORTER way of seeing how the choice of μ₀ determines the value of ε₀?

 

[Nugatory]

The shortest connection between ε₀ and μ₀ seems to be the relation c² = 1/ε₀μ₀. But it seems that the derivation of this relation is not short, but rather long and winded.

Is there a SHORTER way of seeing how the choice of μ₀ determines the value of ε₀?

Not in classical electrodynamics. We have three values, c, ε₀, and μ₀ any two of which determine the value of the third through c² = 1/ε₀μ₀. All three have been measured experimentally, so we know that the relationship holds, but without experimental data we would be free to vary any two of them to determine the value of the third.

[edit: Historically, there was a leap of imagination involved: Maxwell's equations predict electromagnetic radiation traveling at a speed given by the square root of 1/ε₀μ₀. This value is equal to the already known speed of light, leading to the suggestion that light was the electromagnetic radiation predicted by Maxwell's equations.]

 

[jtbell]

I see, because ε₀ and μ₀ appear in the Maxwell's (SI) equations. Then, - how do ε₀ and μ₀ get to appear in the Maxwell's equations?

ε₀ enters because we define Coulomb's Law (which in turn leads to Gauss's Law) in such a way that we can use the pre-existing units of coulombs and Newtons. μ₀ enters similarly via Ampere's Law. They're basically unit-conversion factors, and disappear when we use Gaussian units.

In the case of Ampere's Law and μ₀, the logic is a bit convoluted, because we actually use Ampere's Law to define the ampere, by defining $\mu_0 = 4 \pi \times 10^{-7}$ exactly, in SI units. But the choice isn't arbitrary, because we want to also have 1 ampere = 1 coulomb/second.

 

[rbj]

my response to the original question is to take Maxwell's Eqs and derive the wave propagation equation (say in one dimension, to make your life easier). figure out what the speed of propagation is.

 

[Sunfire]

When one reads Wikipedia (http://en.wikipedia.org/wiki/SI_base_unit), it says plainly that the Ampere is a base unit in SI. The Coulomb is thus a derived unit.

I asked the above question in trying to understand the logical flow:

Start from the Ampere, a base unit, then move on to the definition of μ₀ -->... (missing logic here!) ... -->then end at the definition of ε₀ and its value.

It seems the relation c^2=1/epsilon_0*mu_0 should be part of the missing logic

 

[Sunfire]

This is the full logic as I understand it:

Decide to choose (e.g., pick) the Ampere as a base unit in SI.

Then, pick the Ampere to have a value such, that it is convenient for household and engineering purposes in the measurement of electric current (no one in everyday life can use numbers like 10^-8 etc. , because most people are uncomfortable with the concept of negative numbers, let alone powers of 10). This seems to be the most "fundamental" reason for the value of μ₀ and as we shall see, of ε₀ also.

Pick the Ampere force law as a defining equation (force between 2 parallel conductors). The Newton and the meter already have fixed definitions, thus we need to adjust the ampere to fit them in the defining equation. But we don`t want to do that, because we have already picked a household-specific value for the ampere. Then, we insert by hand a hand-picked value of a conversion factor, which we call μ₀, into the defining equation. We also pick a factor of 4π and insert it into the definition of μ₀. It will come in handy later, when we deal with spheres with surface areas of 4πr².

But, recall that I=\frac{dQ}{dt}, e.g. the electrical current is a `charge flux``, much like the fluid mass flux in a pipe. Then, because time has an established unit of measurement, our hand-picked definition of the Ampere imposes the unit of measurement of the electric charge. This means something has to be adjusted in the Coulomb`s law, where the electric charge is related to force and distance.

Moving on to Coulomb`s law. Force, time and charge now have their defined units of measurement. To reconcile them, insert a factor of ε₀ (and throw in a factor of 4π).

Then, because the Coulomb`s law and Ampere force law determine the form of Maxwell`s equations, the new hand-picked factors ε₀ and μ₀ get to appear in the Maxwell`s equations, where one can derive the relation

c²=\frac{1}{\epsilon_{0}\mu_{0}}.

 

[DiracPool]

This is the full logic as I understand it:

Huh? The answer to your question is the same as asking why is the fine structure constant 1/137, or asking why plank's length is 10^-35. It's just a fundamental constant(s) relationship.