# Zero-Divergence of Rate of Change of Magnetic Flux

In summary, both the scalar field ρ(x,y,z,t) (charge density) and the vector field J(x,y,z,t) (current density) must satisfy Maxwell's equations in order for the equation ∇⋅(∂B/∂t)=0 to hold. This leads to the conclusion that Gauss' Law for Magnetic Flux is applicable. Additionally, whether or not the vector field B exists depends on the vector field A and whether or not ∇⋅A=0.

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## ?

... 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.

berkeman
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?

## 1. What is zero-divergence of rate of change of magnetic flux?

The zero-divergence of rate of change of magnetic flux is a physical law that states that the net flow of magnetic field lines into any closed surface is zero. This means that the amount of magnetic flux entering a surface is equal to the amount of magnetic flux leaving the surface.

## 2. Why is zero-divergence of rate of change of magnetic flux important?

This law is important because it is a fundamental principle in electromagnetism and is essential for understanding and predicting the behavior of magnetic fields. It also has practical applications in various fields, such as in the design of electronic devices and magnetic sensors.

## 3. How does zero-divergence of rate of change of magnetic flux relate to Gauss's law?

The zero-divergence of rate of change of magnetic flux is essentially the magnetic analogue of Gauss's law, which states that the net electric flux through a closed surface is equal to the enclosed electric charge. Both laws involve the concept of flux and show that the net flow of a certain quantity through a closed surface is zero.

## 4. Can zero-divergence of rate of change of magnetic flux be violated?

No, zero-divergence of rate of change of magnetic flux is a fundamental law of electromagnetism and cannot be violated. In fact, it has been extensively tested and found to hold true in all observed cases.

## 5. How is zero-divergence of rate of change of magnetic flux mathematically expressed?

This law can be expressed mathematically using the divergence theorem, which states that the flux of a vector field through a closed surface is equal to the volume integral of the field's divergence over the volume enclosed by the surface. In the case of magnetic flux, this integral will always equal zero, hence the law of zero-divergence.