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

by Sunfire
Tags: epsilon, permeability, permitivity, speed of light
 P: 177 Hello, Could someone describe the origin of the equation c2=1/ε0μ0 Is it derived from fundamental reasons? Thanks.
 Mentor P: 15,625 Start with $\nabla \times \nabla \times E$ and apply Maxwell's Equations.
 P: 177 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?
Thanks
P: 2,972

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

 Quote by Sunfire 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 travelling 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.]
Mentor
P: 11,255
 Quote by 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 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.
 P: 2,265 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.
 P: 177 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
 P: 177 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 dont 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 Coulombs law, where the electric charge is related to force and distance. Moving on to Coulombs 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 Coulombs law and Ampere force law determine the form of Maxwells equations, the new hand-picked factors ε0 and μ0 get to appear in the Maxwell`s equations, where one can derive the relation c2=$\frac{1}{\epsilon_{0}\mu_{0}}$.
P: 492
 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.

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