## How do Maxwell's equations predict the speed of light?

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tsoj
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### How do Maxwell's equations predict the speed of light?

Hello,

I am trying to understand Maxwell's electromagnetism. Why do electromagnetic waves not propagate instantly? My thought process was that at some point a curly magnetic field arises in a specific time interval. During this period according to Maxwell's equations (Maxwell-Faraday equation), an electric field gets produced (spatially offset to the original magnetic field). Because during the specified time period the strength of the spatially offset electric field changed according to the Ampère's circuital law there will be also a new spatially offset magnetic field and its strength will change in the specified time interval. That means it begins again: so at any point in space there will be an electromagnetic field even if its strength is very small.

ijuin
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### Re: How do Maxwell's equations predict the speed of light?

Electromagnetic fields (and therefore, also waves) propagate at finite speed because the permeability and permittivity of vacuum (i.e. empty space’s resistance to the fields) are themselves finite. This is a good thing, because if empty space had zero resistance to electromagnetic fields, then space itself would be a superconducting medium connecting all particles.

There is nothing magically special about the particular speed of approximately 300 million meters per second—the actual value for “c” is dependent on the exact values of the permittivity and permeability of the vacuum, and would be different if the properties of the vacuum were different.

Eebster the Great
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### Re: How do Maxwell's equations predict the speed of light?

Importantly, the permeability and permittivity of free space do not depend on your frame of reference, implying the invariance of the speed of light.

Heimhenge
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### Re: How do Maxwell's equations predict the speed of light?

Eebster the Great wrote:Importantly, the permeability and permittivity of free space do not depend on your frame of reference, implying the invariance of the speed of light.

Good insight, but I'm wondering ... since Maxwell's equations include the permeability and permittivity as a product [ c = (1/με)^0.5 ], might motion through space change one in the direction of motion and the other in the direction perpendicular to motion? Thus the product would be constant, and as you point out that implies the constancy of the speed of light.

I ask because both μ and ε are both measured statically and have dimension of length in their units.

Eebster the Great
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### Re: How do Maxwell's equations predict the speed of light?

I was coming from the perspective that the invariance of μ and ε had already been established and the invariance of c was thus what we wished to establish (which follows from c = (με), as you said). But if we wanted to go the other way around, taking the invariance of c as established, I can't say for sure whether that would be sufficient to prove μ and ε invariant. But I think it would be.