Zero-Divergence of Rate of Change of Magnetic Flux

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Discussion Overview

The discussion revolves around the mathematical implications of the zero-divergence condition of the rate of change of magnetic flux, particularly in the context of Maxwell's equations. Participants explore the relationships between scalar and vector fields, specifically charge density and current density, as well as the conditions under which certain vector fields can be defined.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant introduces a time-varying scalar field ρ and a vector field J, questioning the conditions under which the divergence of the time derivative of the magnetic field B is zero.
  • Several participants express uncertainty about the implications of the mathematical formulation, asking for clarification on the results of the equations presented.
  • Another participant suggests examining whether the order of operations between the time-partial derivative and the divergence operator can be interchanged in the equation.
  • A later reply references Gauss' Law for Magnetic Flux as a potential connection to the problem being discussed.
  • One participant raises a follow-up question about the necessity of the divergence of vector field A being zero for the existence of vector field B defined as the curl of A.
  • A participant provides a brief response indicating that the answer to the previous question is conditional, suggesting that it depends on specific circumstances.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the mathematical implications of the zero-divergence condition or the conditions necessary for the existence of vector field B. Multiple competing views and uncertainties remain present throughout the discussion.

Contextual Notes

The discussion includes unresolved mathematical steps and assumptions regarding the interchangeability of derivatives and the conditions for vector fields, which are not fully clarified.

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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