Electromagnetic - finding the complete field by knowing either E or B

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SUMMARY

The discussion focuses on solving for the electric field (E) given a magnetic field (B) defined as B = xt in the z direction. The challenge lies in satisfying Maxwell's equations while accounting for charge density (ρ) and conduction current (J). The participant attempted to apply Ampere's law but encountered difficulties due to the presence of two unknowns, J and dE/dt. A backward approach using Faraday's law was also considered but proved ineffective, highlighting the complexity of the problem.

PREREQUISITES
  • Understanding of Maxwell's equations
  • Familiarity with vector calculus
  • Knowledge of electromagnetic theory, specifically Faraday's law and Ampere's law
  • Basic concepts of charge density (ρ) and conduction current (J)
NEXT STEPS
  • Study the application of Maxwell's equations in electromagnetic fields
  • Learn about vector calculus techniques relevant to electromagnetic theory
  • Explore the implications of charge density and conduction current in electromagnetic fields
  • Investigate advanced methods for solving coupled differential equations in electromagnetism
USEFUL FOR

Students and professionals in physics and engineering, particularly those specializing in electromagnetism and seeking to deepen their understanding of Maxwell's equations and their applications.

sixers0130
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Homework Statement



B field = xt in z direction in Cartesian Coordinate, t is time.

Homework Equations



obtaining the remaining field expression together with charge densty "rho" and conduction current J, so that all Maxwell's equations are satisfied. There is no unique solution.

The Attempt at a Solution



I tried to apply Ampere's law on the given B field, but the right side will have two unknown variables to solve (J and dE/dt). I know E filed will only have y component by taking curl of B, so I try to do it backward on Faraday's law... and I'm stucked. Can someone give me a basic method to approach?
 
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forgot to mention all quantities are vectors here
 

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