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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

    F=q[E+(v[itex]\times[/itex]B)]

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

    ∇[itex]\times[/itex](A[itex]\times[/itex]B)=A(∇[itex]\cdot[/itex]B)-B(∇[itex]\cdot[/itex]A)+(B[itex]\cdot[/itex]∇)A-(A[itex]\cdot[/itex]∇)B

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

    3. The attempt at a solution

    So the magnetic force field is conservative if
    ∇[itex]\times[/itex][E+(v[itex]\times[/itex]B)]=0
    =∇[itex]\times[/itex]E+∇[itex]\times[/itex](v[itex]\times[/itex]B)
    =-∂B/∂t+v(∇[itex]\cdot[/itex]B)-B(∇[itex]\cdot[/itex]v)+(B[itex]\cdot[/itex]∇)v-(v[itex]\cdot[/itex]∇)B

    So from here I know that I need to show:
    v(∇[itex]\cdot[/itex]B)-B(∇[itex]\cdot[/itex]v)+(B[itex]\cdot[/itex]∇)v-(v[itex]\cdot[/itex]∇)B=0

    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

    mfb

    User Avatar
    2016 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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