So I am gonna be so bold as to say I think I understand Maxwell's Laws pretty well, it's just that they're not what I expected they would be. They explain perfectly the principal interactions between the electric and magnetic forces. But what confuses me is this: that knowledge of how the two forces interact doesn't at all increase my knowledge of "electromagnetism", because electricity and magnetism are still treated as two completely separate, albeit interacting, fields.
Electromagnetism is one of the four fundamental forces, so at some point, the forces of electricity and magnetism have to be unified into a more general force. So my question is this: Is there a single formula that I can call the "Electromagnetic Law" that defines the electromagnetic field and thereby includes all four of Maxwell's Laws, or is my concept of unification of forces fundamentally flawed? And if so, what does "unification" of forces mean?
And please no one bring up the Lorentz force. What I'm looking for isn't just adding electricity to magnetism, I want to know the more general principle, if there is one.
Fundamental Forces Question
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Re: Fundamental Forces Question
I don't think you quite understand the definition of electromagnetism and why Maxwell's equations describe it so well.

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Re: Fundamental Forces Question
screen317 wrote:I don't think you quite understand the definition of electromagnetism and why Maxwell's equations describe it so well.
No, I probably don't, what am I missing?
 Schrollini
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Re: Fundamental Forces Question
It might help if you can let us know your physics background so far, so we can try to tune our answers to it.
If you've had special relativity with fourvectors, you might be interested to know that you can write an eletromagnetic fourpotential A^{μ}, made out of the (electric) scalar potential and (magnetic) vector potential. You can also write a fourcurrent J^{μ}, made out of the charge density and current. Then all of Maxwell's laws can be written in one equation:
If you're not familiar with fourvectors, don't worry about it. Here's a taste of what this means. Consider a particle moving at velocity v in a magnetic field B. You'll see that it accelerates, since it experiences a Lorentz force v ⨯ B. (I know you didn't want to hear this, but hang on just a second.) But now consider what you'd see if you were traveling along with the particle at velocity v. Since the particle isn't moving in this reference frame, it shouldn't experience a Lorentz force. But we know that the particle accelerates, so what could possibly cause this? The only (electromagnetic) thing that can accelerate a charged particle from rest is an electric field, so this observer must see a nonzero electric field. Indeed, that's exactly what happens. What looks like an electric field to one observer looks like a magnetic field to another, and a mix to a third. So electric and magnetic fields really are the same thing, just looked at from different points of view. By the way, this was one of the insights that helped Einstein develop special relativity, and this is why his paper introducing SR is titled On the Electrodynamics of Moving Bodies.
My (limited) understanding of the unification with the weak and strong forces is that these are somewhat different. These symmetries between the forces only becomes apparent above some energy scale, whereas the symmetry between electric and magnetic fields is always existent.
If you've had special relativity with fourvectors, you might be interested to know that you can write an eletromagnetic fourpotential A^{μ}, made out of the (electric) scalar potential and (magnetic) vector potential. You can also write a fourcurrent J^{μ}, made out of the charge density and current. Then all of Maxwell's laws can be written in one equation:
∂_{ν}∂^{ν} A^{μ} = J^{μ}
which is pretty nifty. (For the record, I'm using the Lorenz gauge, ∂_{μ} A^{μ} = 0, and units where ε_{0} = μ_{0} = 1.)If you're not familiar with fourvectors, don't worry about it. Here's a taste of what this means. Consider a particle moving at velocity v in a magnetic field B. You'll see that it accelerates, since it experiences a Lorentz force v ⨯ B. (I know you didn't want to hear this, but hang on just a second.) But now consider what you'd see if you were traveling along with the particle at velocity v. Since the particle isn't moving in this reference frame, it shouldn't experience a Lorentz force. But we know that the particle accelerates, so what could possibly cause this? The only (electromagnetic) thing that can accelerate a charged particle from rest is an electric field, so this observer must see a nonzero electric field. Indeed, that's exactly what happens. What looks like an electric field to one observer looks like a magnetic field to another, and a mix to a third. So electric and magnetic fields really are the same thing, just looked at from different points of view. By the way, this was one of the insights that helped Einstein develop special relativity, and this is why his paper introducing SR is titled On the Electrodynamics of Moving Bodies.
My (limited) understanding of the unification with the weak and strong forces is that these are somewhat different. These symmetries between the forces only becomes apparent above some energy scale, whereas the symmetry between electric and magnetic fields is always existent.
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Re: Fundamental Forces Question
Awesome, thanks Schrollini! So Maxwell didn't unify them, special relativity did! There's a pretty good Stanford set of lectures I just found that'll help, thanks!
Re: Fundamental Forces Question
panther991 wrote:Awesome, thanks Schrollini! So Maxwell didn't unify them, special relativity did! There's a pretty good Stanford set of lectures I just found that'll help, thanks!
Well. Not quite. Take a look at how the electromagnetic wave equations are derived. While both wave equations seem to include only the magnetic or electric field, their derivation depends crucially on the fact that Maxwell's equations relate the two. In short, think of a propagating electromagnetic wave as follows: You've got an electric field changing over time. That creates a magnetic field, which in turn creates an electric field, which in turn creates a magnetic field, ...
Mathematically speaking, wave equations are always second order in derivatives. Maxwells equations are of first order however. Yet since Maxwells equations relate E and B fields, you can combine them to get the second order wave equations.
The effect of special relativity is a bit different: Consider an electron at rest. Clearly there is only a static electric field. A moving observer however can claim (all frames being equivalent) that it's the electron in fact that is moving. A moving charge creates a magnetic field. So here it becomes really clear that the two fields are different sides of the same coin. The Lorentz transformations of special relativity (which you use to go from one frame to another) map electric to magnetic fields and vice versa.
If I remember correctly, the Lorentz transformations in electromagnetism where actually known before the formulation of special relativity. (Maybe Pais' book "subtle is the lord" mentions this).
Re: Fundamental Forces Question
The analogy there is not quite accurate because it suggests a pi/2 phase difference between the E and M components when they're actually in phase.
It's also not even SR that allows you to say the charge is moving, that's Galilean.
Yup, Maxwell's equations were known to be Lorentz invariant (but not Galilean invariant) before Einstein and, in fact, this observation is often credited as being one of the things which led to the discovery of SR.
It's also not even SR that allows you to say the charge is moving, that's Galilean.
Yup, Maxwell's equations were known to be Lorentz invariant (but not Galilean invariant) before Einstein and, in fact, this observation is often credited as being one of the things which led to the discovery of SR.
my pronouns are they
Magnanimous wrote:(fuck the macrons)
 Schrollini
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Re: Fundamental Forces Question
panther991 wrote:So Maxwell didn't unify them, special relativity did!
Don't sell Maxwell short. He introduced the displacement current, essentially by inspection, that gives the Maxwell equations the symmetry necessary for both electromagnetic waves and being written in the covariant form I give above.
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Re: Fundamental Forces Question
this tutorioal on EM teaching issues may help http://johnwarthur.com/The%20Fundamenta ... 80820a.pdf
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