Gauss Law for Magnetic fields?

You probably know Gauss Law very well. It states that if you take any closed surface, the electric flux through this surface is proportional to the total charge enclosed. Mathematically: \[ \Phi_{E}:=\oint \vec{E} \cdot d\vec{A}=\frac{Q_{enc}}{\epsilon_{0}}.\] What about the magnetic flux? It turns out that \[ \Phi_{M}:=\oint \vec{B} \cdot d\vec{A}=0 \quad (\textrm{always!}).\] This is a Law of Nature, equivalent to one of Maxwell's equations and it reflects the experimental fact that there are no magnetic charges . In particular, \( \Phi_{M}=0\) implies that not every magnetic field configuration can be realized in nature. For example, one can show that it is impossible to have a magnetic field that increases along the z-axis having only a z-component.

Consider an axially symmetric field with z-component (the field is symmetric about the z-axis) given by \[ B_{z}=B_{0}+ b z \] where \(B_{0}= 2~ \mu \mbox{T}\) and \( b=1 ~\mu \mbox{T/m}\).

Show that in addition to the z-component, this field must have a radial component \(B_{r}\). Find \(|B_{r}|\) in Teslas at a point located \(50~\textrm{cm}\) away from the z-axis.

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