Show wave equations for E & B consistent with potentials

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SUMMARY

The discussion focuses on deriving wave equations for the electric field (E) and magnetic field (B) consistent with potentials in the context of Maxwell's equations. Specifically, it addresses how to separate these equations under the condition of a time-independent and divergence-free current density. The user successfully derived a homogeneous wave equation for B and an inhomogeneous wave equation for E, influenced by the term ∇(ρ/ε). The challenge lies in substituting the vector potential (A) and electrostatic potential (φ) into these equations while applying the Coulomb Gauge and Poisson's equations.

PREREQUISITES
  • Understanding of Maxwell's equations
  • Familiarity with wave equations in electromagnetism
  • Knowledge of vector potentials and electrostatic potentials
  • Concept of the Coulomb Gauge
NEXT STEPS
  • Study the derivation of wave equations from Maxwell's equations
  • Learn about the application of the Coulomb Gauge in electromagnetic theory
  • Explore Poisson's equation and its implications in electrostatics
  • Investigate techniques for simplifying complex equations in vector calculus
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This discussion is beneficial for physics students, particularly those studying electromagnetism, as well as researchers and educators looking to deepen their understanding of wave equations and potentials in the context of Maxwell's equations.

insynC
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Homework Statement



There are a couple of introductory questions that aren't relevant to the last parts then it says:

(d) If the current density is time independent and divergence free, show that the Maxwell Equations separate into independent equations for E and B.

(e) Express E in terms of the electrostatic potential φ, and B in terms of the vector potential A. Show that the equations in question (d) are satisfied when (in Coulomb Gauge) φ and A satisfy Poisson equations. Write down those equations.

The Attempt at a Solution



I have (with some help) figured out (d) and I got a homogeneous wave equation for B and an inhomogeneous wave equation for E (the inhomogeneity created by a ∇(ρ/ε) term).

However subbing in A and φ for B and E leads only to mess, I have no idea how to simplify it. I looked through a list of identities, but they were no help. I need to find a way to rearrange it so that I can make use of properties like the Coulomb Gauge and Poisson's equation, but I'm not sure how.

Any help would be great, thanks.
 
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