What is the physical interpretation of zero divergence?

[aaaa202]

When a vector field representing a physical quantity (e.g. B) has ∇$\cdot$B = 0 what is then the physical interpretation of this? Some people have said that the field doesn't diverge away from anything, but as far as I can tell magnetic field can easily get weaker and weaker the further you go away.
I also have some trouble understanding exactly what my book means by the fact that ∇$\cdot$B = 0 reflects the fact that there exists no magnetic charge. I know in the electrical case ∇$\cdot$E = ρ/ε₀ but that just comes from coulombs law and how electric fields behave.
Who says magnetic charge can't be different producing fields with zero divergence (like that of the Biot-Savart law).

 

[SammyS]

When a vector field representing a physical quantity (e.g. B) has ∇$\cdot$B = 0 what is then the physical interpretation of this? Some people have said that the field doesn't diverge away from anything, but as far as I can tell magnetic field can easily get weaker and weaker the further you go away.
I also have some trouble understanding exactly what my book means by the fact that ∇$\cdot$B = 0 reflects the fact that there exists no magnetic charge. I know in the electrical case ∇$\cdot$E = ρ/ε₀ but that just comes from coulombs law and how electric fields behave.
Who says magnetic charge can't be different producing fields with zero divergence (like that of the Biot-Savart law).

If $\displaystyle \vec{\nabla}\cdot\vec{F}=0\,,$ for some vector field$,\ \vec{F}\,,\$ then the "field-lines" have no "sources" or "sinks".

This is unlike the electric field, for which the field-lines originate on positive charges and terminate on negative charges.

Since for the magnetic field$,\ \vec{B}\,,\$ we have $\displaystyle \vec{\nabla}\cdot\vec{B}=0\,,$ each field-line forms a closed loop, with no "source" or "sink" !

 

[PrakashPhy]

To physically interpret what zero gradiance of vector field is; you first have to be able to physically interpret what gradience of a vector field actually is.

A vector point function assigns every point in a coordinate space with a vector. So consider such a vector field; the velocity of flowing water in pipe which assigns every point inside the pipe with a vector equal to the velocity of water. Now consider any point in the pipe and construct an infinitesimal volume with dimensions dx ,dy, dz respectively in your x , y and z axes. The gradiance of the vector field at that point gives the total amount of water ( flux i.e field * area) exiting from that volume minus the total amount of water ( flux )entering that volume. [ you can do the mathematical workout if needed i will provide]

If there is is a source of water inside the infinitesimally chosen volume then all water must flow out of the volume, which would provide +ve divergence. Oppositively if there was a sink that somehow consumed, more water would flow inward than that exit out of the volume and hence the gradience would be negative.

Now see what does zero divergence mean> It simply means the total flux entering any infinitesimal volume in the filed of that vector field is equal to the total flux exiting, since divergence is the difference of flux exiting and flux entering.

Now ∇.B)= 0 means fore any arbitrarily taken volume in the magnetic field, there is always equal amount of flux entering and exiting. Since the filed of N pole is outward directed (conventionally) and of S pole is inward directed. Their difference is always zero means equal amount of flux enter and exit out of a arbitrarily chosen infinitesimally volume; meaning where there is N pole to produce outward flux there must be S pole to produce inward flux. Clearly MAGNETIC MONOPOLE DO NOT EXIST.

Hope this helped