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

  1. Nov 22, 2012 #1
    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.
  2. jcsd
  3. Nov 22, 2012 #2
    In component form ∇.B=0 reads,
    ∂Bx/∂x+∂By/∂y+∂Bz/∂z=0,there is no vector here because the product is scalar as implied by ∇.B
  4. Nov 22, 2012 #3


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

    [tex]\nabla \cdot B = \frac{\partial B_x}{\partial x} + \frac{\partial B_y}{\partial y} + \frac{\partial B_z}{\partial z} = 0[/tex]

    Any of these can individually be non-zero. So long as the sum is zero.
  5. Nov 22, 2012 #4
    Thank you, andrien and K^2. I completely overlooked that ∇ ⋅ B is regarded as a dot product.
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