Solving Differential Equations Involving Vector Fields

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SUMMARY

This discussion focuses on solving differential equations involving vector fields, specifically when the divergence is zero. The problem is contextualized within electromagnetism, where the static magnetic field is induced by a steady-state electric current density J(x). Key concepts include the relationship between the curl of the magnetic field (B field) and the current density, as well as the application of Maxwell's equations. The Biot-Savart Law is referenced for calculating the B field, although its use is limited in scenarios involving point particles due to the changing nature of current density.

PREREQUISITES
  • Understanding of vector calculus, particularly curl and divergence
  • Familiarity with Maxwell's equations in electromagnetism
  • Knowledge of Biot-Savart Law for magnetic field calculations
  • Concept of current density in physics
NEXT STEPS
  • Study the derivation and applications of Maxwell's equations
  • Learn about the Biot-Savart Law and its limitations in dynamic systems
  • Explore advanced vector calculus techniques for solving differential equations
  • Investigate the behavior of magnetic fields generated by time-varying current densities
USEFUL FOR

Physicists, electrical engineers, and students studying electromagnetism or vector calculus who are interested in solving complex differential equations involving vector fields.

Savant13
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Given the curl and divergence of a vector field, how would one solve for that vector field?

In the particular case I would like to solve, divergence is zero at all coordinates.
 
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The problem you describe comes up in electromagnetism. The case of zero divergence would be modeled by finding the static magnetic field induced by a steady state electric current density J(x). The curl of the B field is proportional to the current density.

Look up the Biot-Savart's Law which gives the B field as an integral involving the current.
 
The application here is maxwell's equations. However, my use of point particles precludes the Biot-Savart Law (as the current density is constantly changing)
 

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