Why not a magnetic field local maximum?

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Solution 1

For a stationary point to be a local maximum or local minimum, then the eigenvalues of the Hessian matrix must either be all negative or all positive respectively.

However, a neat trick is that the trace of the Hessian is equal to the sum of the eigenvalues and is also the Laplacian.

So, to answer the question we ask what the Laplacian of the magnetic field magnitude looks like, or more conveniently, the Laplacian of $B^2$.

$$\nabla^2 B^2 = \nabla^2(B_{x}^2 + B_{y}^2 + B_{z}^2)$$ $$\nabla^2 B^2 = \nabla \cdot \nabla(B_{x}^2 + B_{y}^2 + B_{z}^2)$$ $$\nabla^2 B^2 = 2\nabla \cdot (B_x \nabla B_x + B_y \nabla B_y + B_z \nabla B_z)$$ $$\nabla^2 B^2 = 2( [\nabla B_x]^2 + [\nabla B_y]^2 + [\nabla B_z]^2 + B_x \nabla^2 B_x + B_y \nabla^2 B_y + B_z \nabla^2 B_z)\ \ \ (1)$$

For a time-independent, current free situation we know that $\nabla \times {\bf B} =0$. If we take the curl of both sides $$\nabla \times \nabla \times {\bf B} = -\nabla^2 {\bf B} + \nabla(\nabla \cdot {\bf B}) = 0$$ and therefore, because $\nabla \cdot {\bf B} = 0$, we can say $\nabla^2 {\bf B} = 0$ and we can also say that the individual components $\nabla^2 B_x = \nabla^2 B_y = \nabla^2 B_z = 0$. Using this in equation (1), we get $$\nabla^2 B^2 = 2( [\nabla B_x]^2 + [\nabla B_y]^2 + [\nabla B_z]^2),$$ but because the square of the individual gradients must always be $\geq 0$, we can say that $$\nabla^2 B^2 \geq 0$$.

From the discussion of the Hessian and its relationship with the eigenvalues and the Laplacian at the beginning we can thus say that whilst a local minimum in the magnitude of the magnetic field is possible because the trace of the Hessian can be $>0$, it is impossible to have a local field magnitude maximum, because it cannot be $<0$.

Solution 2

Static magnetic field is a harmonic function (its curl and divergence vanish, you can then derive that its Laplacian vanishes), so it does not have local minima or maxima.

EDIT (10/3/2015): I am grateful to the authors of the comments for their criticism. What I wrote seems to be applicable to the components of static magnetic field (their Laplacian is zero, say, if there are no charges/currents in the area). As for the magnitude of magnetic field, it looks like it can have local minimums, but not maximums (https://books.google.com/books?id=iULpBKHIbeoC&pg=PA77&lpg=PA77&dq=magnetic+field+local+maximum&source=bl&ots=N8ubx1w1QS&sig=FOyZyUjQNdPz92X3bmVlGyrDORY&hl=en&sa=X&ved=0CD8Q6AEwBGoVChMIw6Kqnc-nyAIVhIYNCh08JQAg#v=onepage&q=magnetic%20field%20local%20maximum&f=false)

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Updated on October 05, 2020

Comments

  • Klopmint
    Klopmint about 3 years

    Reading a brief on magnetic traps I've read that you can't have a magnetic field local maximum. I believe this is linked to the divergence of B being 0 in vacuum, but I don't see why one can have a local minimum but not a maximum.

    • Klopmint
      Klopmint about 8 years
      Well I can't justify what I've read... I mean... 0 divergence should give a stationary point locally, am I wrong?
  • Floris
    Floris about 8 years
    Are you sure about this? I can think of simple situations where magnetic field has either a minimum or a maximum.
  • hyportnex
    hyportnex about 8 years
    @akhmeteli the lack of local extrema of the potential does not carry over to the vector intesity
  • Klopmint
    Klopmint about 8 years
    I see what theorem you are referring to. But wasn't there some additional hypothesis ? I must admit I can't remember that much of analysis as I've studied it some time ago
  • Klopmint
    Klopmint about 8 years
    Maybe these conditions are what I'm missing... Even of what I've read seemed so linear. Anyway I see your point. I'm actually a bit confused
  • Floris
    Floris about 8 years
    Can you give a reference to the thing you read? Maybe copy it into your question?
  • Klopmint
    Klopmint about 8 years
    The first time I've read this loca-maximum thing is on Adams' article on laser cooling; the same statement can be found on wikipedia's page about magnetic traps (I don't know anyway how wikipedia can be attendible).
  • Klopmint
    Klopmint about 8 years
    I perfectly agree with you, that's why I don't understand why local minimum points can be created. I'm insisting this much on this point beacuse the fact that minimum points can be created appears to me as the reason magnetic traps for atoms work.
  • ProfRob
    ProfRob about 8 years
    @Klopmint Sorted. You just need to consider the Laplacian of $B^2$ to see why the field magnitude cannot have a local maximum.