Gauss's Law of Magnetism shows us that the divergence of Magnetic field is 0, ▽ ⋅ B → = 0Then how do you derive that statement by showing the divergence of a magnetic field upon an axis of a current carrying coil where radius is much smaller that distance so that we can use, B z = μ o I 2 z 3 z ^ ∴ ▽ ⋅ B ≡ ∂ ∂ z ⋅ u o I 2 z 3 z ^ ≠ 0This doesn't equal zero? What am I missing?

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Gauss&#039;s Law of Magnetism shows us that the divergence of Magnetic field is 0,   ▽  ⋅            B      →        =  0Then how do you derive that statement by showing the divergence of a magnetic field upon an axis of a current carrying coil where radius is much smaller that distance so that we can use,      B    z    =                    μ        o            I              2              z        3                        z      ^        ∴  ▽  ⋅  B  ≡      ∂          ∂      z        ⋅                    u        o            I              2              z        3                        z      ^        ≠  0This doesn&#039;t equal zero? What am I missing?

Darion Sexton · Accepted Answer

When you compute the divergence, it&#039;s not enough to know the field at a given point or on an axis. As you need to compute all possible directional derivatives, you need to know the field in a neighbour of the point where you are calculating the divergence*. In your case, you only know the field on the axis, so no derivatives on the orthogonal plane. Your formula for divergence only have       ∂    z        B    z  : surely the other components (that you are not able to calculate) will take care of bringing the divergence to zero.*Actually, you just need to be able to perform some derivatives: in Cartesian coordinates, you need to know       B    x   in a 1D neighbour of the point along the x^ axis, and analogously for       B    y   and       B    z   substituting the versor appropriately.

Gauss's Law of Magnetism shows us that the divergence of Magnetic field is 0, ▽

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