Zero-Divergence of Rate of Change of Magnetic Flux

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SUMMARY

The discussion centers on the mathematical implications of Maxwell's equations, specifically the condition that the divergence of the time derivative of the magnetic field must equal zero, represented as ∇⋅(∂B/∂t) = 0. Participants explore the relationship between charge density ρ(x,y,z,t) and current density J(x,y,z,t), and how these fields interact under Gauss' Law for Magnetic Flux. The conversation also touches on the conditions required for the existence of a vector field B defined as B = ∇×A, questioning whether the divergence of A must equal zero for B to exist.

PREREQUISITES
  • Understanding of Maxwell's equations
  • Familiarity with vector calculus
  • Knowledge of Gauss' Law for Magnetic Flux
  • Concept of vector fields and their properties
NEXT STEPS
  • Study the implications of Gauss' Law for Magnetic Flux in electromagnetic theory
  • Learn about the mathematical properties of divergence and curl in vector calculus
  • Explore the relationship between charge density and current density in electromagnetic fields
  • Investigate the conditions for the existence of vector potentials in electromagnetism
USEFUL FOR

Physicists, electrical engineers, and students studying electromagnetism who seek to deepen their understanding of magnetic fields and their mathematical foundations.

tade
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Let's introduce a time-varying scalar field ρ(x,y,z,t) [charge density] and vector field J(x,y,z,t) [current density]

Assuming the system follows Maxwell's equations, what must both fields satisfy such that

##∇⋅(\frac{∂B}{∂t})=0## ?
 
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... and what do you get?
 
Simon Bridge said:
... and what do you get?
that's what I'm wondering myself
 
Well do the maths and see ... i.e. can you change the order between the time-partial and the div in that last equation?

If I don't see how you are attempting the problem I don't know how it is a problem for you so I don't know how to help you.
 
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Simon Bridge said:
Well do the maths and see ... i.e. can you change the order between the time-partial and the div in that last equation?

If I don't see how you are attempting the problem I don't know how it is a problem for you so I don't know how to help you.

Oh I get it! Gauss' Law for Magnetic Flux! Right under my nose the whole time.
 
This leads to the next part of my question, if you don't mind.

Let's say we are given a vector field ##A##.

Vector field ##B## is defined as ##B = ∇×A##

Must ##∇⋅A=0## in order for ##B## to exist?
 

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