Maxwell's equations outside electrodynamics?

In summary: Your Name]In summary, the conservation of charge is a fundamental principle in electromagnetism and is directly linked to the inhomogeneous Maxwell equations. While other conserved quantities may also lead to the creation of fields, they may not necessarily result in the same equations. This principle is related to Noether's theorem and plays a crucial role in understanding the laws of physics.
  • #1
jjustinn
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So I've been reading Hehl's Foundations of Classical Electrodynamics - which builds up Electrodynamics from a six of axioms - and their proof that the conservation of charge alone is sufficient to derive the inhomogenous Maxwell equations got me thinking - why don't these extremrly basic equations show up everywhere?

The example from the book basically showed that since charge is conserved (in the language of differential forms, dJ = 0, J being the charge/current 3-form), it is expressible (at least locally, via Poincare's theorem, and they go on to show how it holds globally) as the exterior derivative of a 2-form H: e.g. dH = J -- which is the inhomogenous Maxwell equations, as desired.

So -- from just that, it would appear that *any* conserved quantity Q (eg dQ = 0) should at least lead to the creation of fields that follow those equations, right? I got to thinking of conserved quantities to try to find a counterexample, but I hit a wall...energy/momentum and mass/energy *are* the source of fields (and IIRC they follow dF=8∏T) , and so is color-charge, but that's cheating because it's just a generalization of electrodynamics. But...what else is there?

So...what am I missing here? The fact that it deals with conserved quantities makes it seem like it could be somehow related to Noether's theorem, but it seems like if conserved quantities always lead to fields, that fact would be more well-known.
 
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Dear fellow scientist,

Thank you for bringing up this interesting topic. The inhomogeneous Maxwell equations, as you mentioned, can be derived from the conservation of charge. This is a fundamental principle in electromagnetism and plays a crucial role in understanding the behavior of electric and magnetic fields.

It is true that other conserved quantities can also lead to the creation of fields, but the specific form of the equations may vary depending on the nature of the quantity. For example, the conservation of energy and momentum leads to the equations of motion in classical mechanics, which are different from the Maxwell equations.

In terms of Noether's theorem, it is indeed related to the conservation laws in physics. The theorem states that for every continuous symmetry in a physical system, there exists a corresponding conserved quantity. This is a powerful tool in understanding the underlying principles behind the laws of physics.

In terms of other conserved quantities, there are many in nature, such as angular momentum, lepton number, and baryon number. However, not all of these quantities can be directly linked to the creation of fields. For example, angular momentum is conserved in rotational systems, but it does not directly lead to the creation of fields.

Overall, the conservation of charge is a unique and fundamental principle in electromagnetism, and its connection to the inhomogeneous Maxwell equations is well-established. While other conserved quantities may also play a role in the creation of fields, they may not necessarily lead to the same equations. I hope this clarifies your question.
 

1. What are Maxwell's equations outside of electrodynamics?

Maxwell's equations are a set of four fundamental equations in physics that describe the behavior of electric and magnetic fields. They are named after James Clerk Maxwell, who first developed them in the mid-19th century. These equations are used to describe the propagation of electromagnetic waves and the behavior of electric and magnetic fields in various mediums.

2. What is the significance of Maxwell's equations in other fields of science?

Maxwell's equations have a wide range of applications in various fields of science, including optics, quantum mechanics, and fluid dynamics. They are also used in the study of plasma physics, which is important in understanding the behavior of stars and other celestial bodies. Additionally, these equations have been instrumental in the development of modern technologies such as radio, television, and wireless communication.

3. Can Maxwell's equations be applied to non-electromagnetic phenomena?

While Maxwell's equations were originally developed to explain the behavior of electric and magnetic fields, they have also been used in other areas of physics. For example, they have been applied to the study of gravitational waves and the behavior of subatomic particles. However, some modifications may be necessary to account for the differences in these phenomena.

4. Are Maxwell's equations still relevant in modern science?

Yes, Maxwell's equations are still considered to be one of the cornerstones of modern physics. They have been confirmed by numerous experiments and have been integrated into the standard model of particle physics. They continue to be studied and applied in various fields of science, and their predictions have been shown to be incredibly accurate.

5. What are some real-world applications of Maxwell's equations?

Maxwell's equations have numerous practical applications in our daily lives. For example, they are used to design and optimize electronic devices such as cell phones and computers. They are also used in medical imaging technologies, such as MRI machines, and in the development of renewable energy sources like solar panels. Additionally, these equations are used in the study of weather patterns and the behavior of the Earth's magnetic field.

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