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Lorentz Force Equation

  1. Sep 30, 2005 #1
    Hi All,

    Just a thought. The Lorentz force equation as we all know is: F = qE + qvxB. We know that Electrical Field can be written as del(Phi), where Phi is the electrical potential. Also, Force can be written as del(Energy) - correct me on this one. Hence is there a representative term for vxB. Can vxB be written as del(something) where something is a meaningful quantity?

    Just curious,
    Thanks - Michael
     
  2. jcsd
  3. Sep 30, 2005 #2

    Tom Mattson

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    It sure can. Taking [itex]\vec{F}=-\vec{\nabla}U[/itex] and [itex]\vec{E}=-\vec{\nabla}\Phi[/itex], we have:

    [itex]-\vec{\nabla}U=-q\vec{\nabla}\Phi+q\vec{v}\times\vec{B}[/itex].

    Rearranging terms we get:

    [itex]q\vec{v}\times\vec{B}=q\vec{\nabla}\Phi-\vec{\nabla}U[/itex].

    Thanks to the linearity of the [itex]\vec{\nabla}[/itex] operator, we have:

    [itex]q\vec{v}\times\vec{B}=\vec{\nabla}(q\Phi-U)[/itex]
    [itex]\vec{v}\times\vec{B}=\vec{\nabla}(\Phi-\frac{U}{q})[/itex].
     
  4. Sep 30, 2005 #3

    jtbell

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    But a U that satisfies [itex]\vec{F}=-\vec{\nabla}U[/itex] exists only if [itex]\vec F[/itex] is a conservative force. The magnetic force isn't conservative.
     
  5. Sep 30, 2005 #4

    Tom Mattson

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    Duh. :frown:

    Well boys and girls, this is what happens when plug-n-chug runs amuck.

    Is it 5:00 yet?
     
  6. Sep 30, 2005 #5
    Last edited by a moderator: Apr 21, 2017
  7. Oct 1, 2005 #6

    jtbell

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    If a force is conservative, you can define a potential energy function for it. The potential energy of a particle can depend only on its position, so a conservative force can depend only on position. But the magnetic force on a particle depends on the velocity (both magnitude and direction!) of the particle, not just on its position (which determines the magnetic field).
     
  8. May 9, 2011 #7
    how can we get potential from lorentz equations..........?
     
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