How to explain that ∇ ⋅ B = 0 but ∂Bz/∂z can be non zero?

[Dyon]

It is known that Gauss's law for magnetism is ∇ ⋅ B = 0.
If we write this in component form it becomes (∂Bx/∂x)i + (∂By/∂y)j + (∂Bz/∂z)k = 0, where i, j, k are unit vectors in a cartesian coordinate system and Bx, By, Bz are the components of the magnetic field on these axes.
It would follow then that all the partial derivatives must be zero: (∂Bx/∂x) = 0, (∂By/∂y) = 0 and (∂Bz/∂z) = 0 for this equation [ (∂Bx/∂x)i + (∂By/∂y)j + (∂Bz/∂z)k = 0 ] to obtain.
But we know that there are magnetic fields with spatial gradients as, for example, in Stern-Gerlach experiment, where the magnetic force on a dipole of magnetic moment F is m⋅(∂Bz/∂z).
How to reconcile mathematically ∇ ⋅ B = 0 with the fact that ∂Bz/∂z can be non-zero?
Thank you.

 

[andrien]

In component form ∇.B=0 reads,
∂B_x/∂x+∂B_y/∂y+∂B_z/∂z=0,there is no vector here because the product is scalar as implied by ∇.B

 

[K^2]

It's a dot product. That should tell you right away that the answer needs to be a scalar.

$$\nabla \cdot B = \frac{\partial B_x}{\partial x} + \frac{\partial B_y}{\partial y} + \frac{\partial B_z}{\partial z} = 0$$

Any of these can individually be non-zero. So long as the sum is zero.

 

[Dyon]

Thank you, andrien and K^2. I completely overlooked that ∇ ⋅ B is regarded as a dot product.

 

[pmacias]

The reason for this apparent contradiction lies in the fact that ∇ ⋅ B = 0 is a statement about the overall behavior of the magnetic field, while ∂Bz/∂z is a measure of the local change in the magnetic field in the z-direction. In other words, ∇ ⋅ B = 0 tells us that there is no net flow or accumulation of magnetic field at a given point, while ∂Bz/∂z measures the rate of change of the magnetic field in the z-direction at that point.

To reconcile this mathematically, we can think of ∇ ⋅ B as a measure of the divergence of the magnetic field, while ∂Bz/∂z is a measure of the gradient of the magnetic field in the z-direction. These are two different mathematical concepts that are related but not equivalent. Just like how a river can have a uniform flow (divergence) but also have areas of faster or slower flow (gradient), the magnetic field can have a zero overall divergence but still have local variations in its gradient.

In terms of the Stern-Gerlach experiment, the fact that ∂Bz/∂z can be non-zero means that there is a spatial gradient in the magnetic field, which is necessary for the experiment to work. However, the overall divergence of the magnetic field is still zero, as seen by the null result of ∇ ⋅ B = 0. This is because the variations in the gradient of the magnetic field cancel out when considering the overall behavior of the field.

In summary, ∇ ⋅ B = 0 and ∂Bz/∂z are two different measures of the magnetic field that can coexist without contradiction. It is important to understand the distinction between the overall behavior of the field and its local variations in order to fully grasp the concept of ∇ ⋅ B = 0.