hobz Posted March 9, 2011 Share Posted March 9, 2011 How can it be that the electric field is [math]c[/math] times greater than the magnetic field while their energy densities are the same? Link to comment Share on other sites More sharing options...

timo Posted March 9, 2011 Share Posted March 9, 2011 Probably an effect of using units where [math]c \neq 1[/math]. Link to comment Share on other sites More sharing options...

hobz Posted March 9, 2011 Author Share Posted March 9, 2011 (edited) So what you're saying is that because coulomb is defined they way it is, and that causes the introduction of [math]\epsilon_0[/math] (in Coulombs law), and similar for the definition of a second in the Biot-Savart law which requires [math]\mu_0[/math] to appear, and these in turn defines [math]c = \frac{1}{\sqrt{\epsilon_0 \, \mu_0}} \neq 1[/math], then the relation has to be like that, but the energy in the fields are equal? I most physics book I am shown perpendicular E and B "waves" along a direction of propagation of the same magnitude? I suppose this is wrong if [math]E=c\, B[/math]? Edited March 9, 2011 by hobz Link to comment Share on other sites More sharing options...

swansont Posted March 9, 2011 Share Posted March 9, 2011 I most physics book I am shown perpendicular E and B "waves" along a direction of propagation of the same magnitude? I suppose this is wrong if [math]E=c\, B[/math]? Do the diagrams actually show the scaling? Just because the curves have equal height does not mean they have the same magnitude; the value and units are different. Link to comment Share on other sites More sharing options...

DrRocket Posted March 9, 2011 Share Posted March 9, 2011 How can it be that the electric field is [math]c[/math] times greater than the magnetic field while their energy densities are the same? As Swansont noted the units for the E and B fields are different. Therefore you cannot say that the E field is greater than the B field. That would be like saying that 1 kilogram is greater than 1/2 meter, and by the same logic then less than 50 centimeters -- an immediate contradiction. Link to comment Share on other sites More sharing options...

timo Posted March 9, 2011 Share Posted March 9, 2011 (edited) So what you're saying is that because coulomb is defined they way it is, ... That is kind of what I was saying; I was a bit in a rush when I wrote it. Let me give a more elaborated version: What you are asking is how 20 tomatoes can have the same mass as two pumpkins when at the same time tomatoes are larger than pumpkins by a factor of 10 tomatoes/pumpkin. This "larger by" is comparing apples with oranges - or tomatoes with pumpkins. There is a bit more to it in this case. In a relativistic treatment it does make a lot of sense to measure time and distance in the same units, which yields [math]c=1[/math]. With these units, a lot of seemingly-different things turn out to be effectively the same, the most prominent example being the center-of-mass energy of an object and its mass (E=m). To some extend that holds true for electromagnetic fields. If you have a vacuum with no electrical charges around, a possible electric field is [math]\vec E(\vec x) = \vec E_0 \sin(\vec k \vec x - |\vec k| t)[/math] with [math] \vec k \vec E_0 = 0[/math] (in whatever units). In this case, the Maxwell equations demand that [math]\vec B(\vec x) = \vec B_0 \cos(\vec k \vec x - |\vec k| t)[/math] with [math]\vec B_0 \vec k= \vec B_0 \vec E_0 = 0[/math] and [math]|\vec B_0| = |\vec E_0|[/math] (you might want to check for yourself that this is true, btw). So the magnetic amplitude of this electromagnetic wave in "B-direction" and the wave's amplitude in "E-direction" are indeed the same (when c=1). I'm a little short on time and I notice I've possibly run off into the wrong direction. A few comments on what I just said: - This is a very special example of a plane wave in vacuum. For the electric field of a point charge, and no magnetic field around, no such scenario occurs. - Even in this special case, [math]|\vec B (\vec x, t)| = |\vec E (\vec x, t)|[/math] usually does not hold true. The value [math] \vec B \vec B + \vec E \vec E[/math] is a constant, though, and is in fact related to the energy. - The direction I probably should have run into: Under relativistic coordinate transformations, electrical fields and magnetic fields do mix. In this sense, they are indeed very related to some extent. Bottom line: there is indeed some relation between electric fields and magnetic fields that becomes more apparent with c=1. Plainly saying E=cB without giving a context is just wrong. Keep in mind that a lot of these equations that people throw around as if they knew what they mean are only valid within the context a special scenario and/or with a special meaning of the symbols (c.f. "what is the equation for time dilatation with gravitational field and velocity?") hope above makes kind of sense; it's a bit rushed. EDIT: Actually, I think you could add a constant E- or B- field to the solutions I presented above and still fulfill the Maxwell equations. Makes my point about their relation even weaker, I guess. But perhaps to still justify why the example might be relevant: such electromagnetic waves are the standard basis for those ominous "photons" that everyone seems to talk about. Edited March 9, 2011 by timo Link to comment Share on other sites More sharing options...

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