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Nabla calculus and conservative forces

  1. May 1, 2013 #1
    1. The problem statement

    I'm trying to show that the magnetic force is only conservative if dB/dt=0

    2. Relevant equations


    Conservative if ∇[itex]\times[/itex]F=0


    Maxwells equation: ∇[itex]\times[/itex]E=-∂B/∂t

    3. The attempt at a solution

    So the magnetic force field is conservative if

    So from here I know that I need to show:

    But when I write it out in all its components I get lost in the algebra. And then it got me thinking, if that equals zero, why doesn't ∇[itex]\times[/itex](A[itex]\times[/itex]B) always equal zero.
    I know there's a simple answer to this as we did it in lectures but I can't seem to find it in my notes anywhere.
    Last edited: May 1, 2013
  2. jcsd
  3. May 1, 2013 #2


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    2017 Award

    Staff: Mentor

    You cannot assign a scalar potential to the magnetic field. You can show that the magnetic field does not influence the speed (and therefore the energy) of the particle, however. This is related to ##\vec{v} \cdot \vec{F}##.
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