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Cleonis
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Lately I'm thinking about the question of how to convey how Maxwell was able to derive from first principles how fast electromagetic waves propagate. How is that possible at all?In Maxwell's time it was already well known, of course, that the electric field and the magnetic field are very much interconnected.
In retrospect we see that Maxwell contributed what amounts to a principle of equivalence. Let me call it Maxwell's principle .*
Maxwell hypothesized that changing of an electric field has the same physical effects as current has, and he suggested a thought experiment to arrive at a quantitive relation. Imagine a wire that is conducting a current, with at some point a capacitor in the circuit. As we know a magnetic field will surround that wire. With the capacitor we have that as the current proceeds an electric field builds up between the capacitor plates. That changing electric field will be surrounded by a magnetic field that is equivalent to the magnetic field that surrounds the current carrying wire.
In all, a changing electric field induces a magnetic field, a changing magnetic field induces an electric field. This mutual effect gives rise to a wave phenomenon. As all textbooks mention: Maxwell pointed out that the speed of this electromagnetic wave coincides with the speed of light.
http://www.phys.virginia.edu/People/Personal.asp?UID=mf1i"
Maxwell's equations imply the following relation:
[tex] c^2 = \frac{1}{\mu_0 \epsilon_0} [/tex]
Amazing. How come the electric constant and the magnetic constant relate to the speed of light?
As I understand it the following two factors are key:
- Maxwell's equations involve rate of change: rate of change of one field induces the other field.
- Electric field and magnetic field are in a certain ratio to each other. (Relativistic physics allows us to describe magnetism as a relativistic side effect of the Coulomb force. (See Daniel Schroeder's: http://physics.weber.edu/schroeder/mrr/MRRtalk.html" ) The relativistic nature is why the ratio of electric field and magnetic field has a factor 'c' in it.)
Let's say that at a signal emitter an electric field changes 1 volt in 1 nanosecond. Assume that change propagates away from that emitter at a velocity 'v'. That change is not an instantaneous jump, it's like an incline. The faster the propagation, the steeper the incline. When 'v' equals c the rate of change is just right, that is how Maxwell was able to conclude that the speed of electromagnetic waves coincides with the speed of light.* (I googled, there is already a 'Maxwell's principle', but it's used by just a few authors.)
In retrospect we see that Maxwell contributed what amounts to a principle of equivalence. Let me call it Maxwell's principle .*
Maxwell hypothesized that changing of an electric field has the same physical effects as current has, and he suggested a thought experiment to arrive at a quantitive relation. Imagine a wire that is conducting a current, with at some point a capacitor in the circuit. As we know a magnetic field will surround that wire. With the capacitor we have that as the current proceeds an electric field builds up between the capacitor plates. That changing electric field will be surrounded by a magnetic field that is equivalent to the magnetic field that surrounds the current carrying wire.
In all, a changing electric field induces a magnetic field, a changing magnetic field induces an electric field. This mutual effect gives rise to a wave phenomenon. As all textbooks mention: Maxwell pointed out that the speed of this electromagnetic wave coincides with the speed of light.
http://www.phys.virginia.edu/People/Personal.asp?UID=mf1i"
Maxwell's equations imply the following relation:
[tex] c^2 = \frac{1}{\mu_0 \epsilon_0} [/tex]
Amazing. How come the electric constant and the magnetic constant relate to the speed of light?
As I understand it the following two factors are key:
- Maxwell's equations involve rate of change: rate of change of one field induces the other field.
- Electric field and magnetic field are in a certain ratio to each other. (Relativistic physics allows us to describe magnetism as a relativistic side effect of the Coulomb force. (See Daniel Schroeder's: http://physics.weber.edu/schroeder/mrr/MRRtalk.html" ) The relativistic nature is why the ratio of electric field and magnetic field has a factor 'c' in it.)
Let's say that at a signal emitter an electric field changes 1 volt in 1 nanosecond. Assume that change propagates away from that emitter at a velocity 'v'. That change is not an instantaneous jump, it's like an incline. The faster the propagation, the steeper the incline. When 'v' equals c the rate of change is just right, that is how Maxwell was able to conclude that the speed of electromagnetic waves coincides with the speed of light.* (I googled, there is already a 'Maxwell's principle', but it's used by just a few authors.)
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