Zero-Divergence of Rate of Change of Magnetic Flux

[tade]

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

$∇⋅(\frac{∂B}{∂t})=0$ ?

 

[Simon Bridge]

... and what do you get?

 

[tade]

... and what do you get?

that's what I'm wondering myself

 

[Simon Bridge]

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.

 

[tade]

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.

 

[tade]

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?

 

[Simon Bridge]

Short answer: it depends.
https://en.wikipedia.org/wiki/Magnetic_potential#Maxwell.27s_equations_in_terms_of_vector_potential