Coupled PDEs - all 4 Maxwell's equations?

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SUMMARY

All four of Maxwell's equations are indeed first-order, coupled partial differential equations (PDEs). This coupling is evident in Faraday's and Ampere's laws, where the electric field (E) and magnetic field (H) are interdependent. Although Gauss' laws appear to relate directly to charge densities, they are still part of the coupled system when expressed through the constituent equations: D = epsilon * E, B = mu * H, and J = sigma * E. Thus, all equations involve unknown fields and demonstrate coupling through their interrelations.

PREREQUISITES
  • Understanding of Maxwell's equations
  • Familiarity with concepts of electric field (E) and magnetic field (H)
  • Knowledge of charge density (rho) and current density (J)
  • Basic grasp of partial differential equations (PDEs)
NEXT STEPS
  • Study the derivation and implications of the constituent equations: D = epsilon * E, B = mu * H, J = sigma * E
  • Explore the mathematical formulation of coupled PDEs in electromagnetic theory
  • Investigate the physical interpretations of Gauss' laws in relation to charge densities
  • Examine the role of magnetic monopoles in the context of Maxwell's equations
USEFUL FOR

This discussion is beneficial for physics students, electrical engineers, and researchers in electromagnetism who seek a deeper understanding of the interrelations within Maxwell's equations.

VictorVictor5
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Greetings all,

Quick question. I know that all 4 Maxwell's equations are said to be first-order, coupled PDEs, where each equation has an unknown field. I see that with Faraday's and Ampere's law, because, E and H appear in each of those equations.

But Gauss' laws, I'm not seeing that, since they're both equal to electric/magnetic charge densities (or 0 in the case of the magnetic law due to there not being magnetic monopoles).

So, are Gauss' laws coupled? And do they still have the unknown fields? Sorry I am not seeing this. I just need a quick "reality check" here.

Thanks!
VV5
 
Yes, all 4 Maxwell's equations are coupled. This may be hard to see since they are usually written with notation using 4 different fields, E, B, D and H, plus the current density J. The coupling become more apparent when considering the constituent equations:
1) D = epsilon * E
2) B = mu * H
3) J = sigma * E

Then all 4 Maxwell's equations can be written in terms of E, H, and the charge density, rho.
 

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