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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