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Maxwell Equations Problem

  1. Aug 1, 2006 #1
    Hi, i am solving a problem about Maxwell Equation that invoves a lot of Vector Calculus, i have a partial solution for it but i have a few terms missing :cry: , i appreciate any help in this. Thanks

    The problem is the following

    Starting with the expression of the force by volume unit over a free space region with charges and currents:
    [itex]
    F_{v}=\rho E+J\timesB
    [/itex]
    and using Maxwell's Equations:
    [itex]

    \begin{array}{l}
    \nabla \cdot \mathop E\limits^ \to = \frac{\rho }{{ \in _0 }} \\
    \nabla \times \mathop E\limits^ \to = - \frac{{\partial \mathop B\limits^ \to }}{{\partial t}} \\
    \nabla \cdot \mathop B\limits^ \to = 0 \\
    \nabla \times \mathop B\limits^ \to = \mu _0 \mathop J\limits^ \to + \in _0 \mu _0 \frac{{\partial \mathop E\limits^ \to }}{{\partial t}} \\
    \end{array}


    [/itex]
    and the following vectorial identity:
    [itex]

    \mathop B\limits^ \to \times \nabla \times \mathop B\limits^ \to = \nabla ({\textstyle{1 \over 2}}B^2 ) - (\mathop B\limits^ \to \cdot \nabla )\mathop B\limits^ \to
    [/itex],

    Show that:
    [itex]

    \begin{array}{l}
    \mathop {F_v }\limits^ \to = - \in _0 \frac{\partial }{{\partial t}}(\mathop E\limits^ \to \times \mathop B\limits^ \to ) + \in _0 \mathop E\limits^ \to \nabla \cdot \mathop E\limits^ \to - \frac{1}{2} \in _0 \nabla (E^2 ) + \in _0 (\mathop E\limits^ \to \cdot \nabla )\mathop E\limits^ \to \\
    {\rm{ + }}\frac{1}{{\mu _0 }}\mathop B\limits^ \to \nabla \cdot \mathop B\limits^ \to - \frac{1}{{2\mu _0 }}\nabla \mathop {(B^2 ) + }\limits^{} \frac{1}{{\mu _0 }}(\mathop B\limits^ \to \cdot \nabla )\mathop B\limits^ \to \\
    \end{array}
    [/itex]

    Now, let me show you my partial solution:

    First, by Maxwell Equations, i get J:
    [itex]

    \mathop J\limits^ \to = \frac{1}{{\mu _0 }}(\nabla \times \mathop B\limits^ \to ) - \in _0 \frac{{\partial \mathop E\limits^ \to }}{{\partial t}}
    [/itex]

    And i replace it in the first equation for [itex]F_{v}[/itex] to get:
    [itex]
    \[
    \begin{array}{l}
    \mathop F\limits^ \to _v = \rho \mathop E\limits^ \to + \left( {\frac{1}{{\mu _0 }}(\nabla \times \mathop B\limits^ \to ) - \in _0 \frac{{\partial \mathop E\limits^ \to }}{{\partial t}}} \right) \times \mathop B\limits^ \to \\
    {\rm{ }} = \rho \mathop E\limits^ \to + \frac{1}{{\mu _0 }}(\nabla \times \mathop B\limits^ \to ) \times \mathop B\limits^ \to - \in _0 (\frac{{\partial \mathop E\limits^ \to }}{{\partial t}} \times \mathop B\limits^ \to ) \\
    \end{array}
    \]

    [/itex]
    Changing the order of the cross product (the sign changes), then
    [itex]
    \[
    \mathop F\limits^ \to _v = \rho \mathop E\limits^ \to - \frac{1}{{\mu _0 }}(\mathop B\limits^ \to \times \nabla \times \mathop B\limits^ \to ) - \in _0 (\frac{{\partial \mathop E\limits^ \to }}{{\partial t}} \times \mathop B\limits^ \to {\rm{)}}
    \]

    [/itex]
    Now i can use the vectorial identity, thus:
    [itex]
    \[
    = \rho \mathop E\limits^ \to - \frac{1}{{\mu _0 }}(\nabla ({\textstyle{1 \over 2}}B^2 ) - (\mathop B\limits^ \to \cdot \nabla )\mathop B\limits^ \to ) - \in _0 (\frac{{\partial \mathop E\limits^ \to }}{{\partial t}} \times \mathop B\limits^ \to {\rm{) }}
    \]

    [/itex]

    Also, from Maxwell equations:
    [itex]
    \rho = (\nabla \cdot \mathop E\limits^ \to ) \in _0
    [/itex]
    and replacing it into the last equation for [itex]F_{v}[/itex], i get:
    [itex]
    \mathop F\limits^ \to _v = \mathop E\limits^ \to (\nabla \cdot \mathop E\limits^ \to ) \in _0 - \frac{1}{{\mu _0 }}(\nabla ({\textstyle{1 \over 2}}B^2 ) - (\mathop B\limits^ \to \cdot \nabla )\mathop B\limits^ \to ) - \in _0 (\frac{{\partial \mathop E\limits^ \to }}{{\partial t}} \times \mathop B\limits^ \to {\rm{) }}
    [/itex]
    Some terms of what i should get can be seen already, but i have another terms missing. My last step is the following, what you think i should do to get the missing terms?
    [itex]
    \mathop {F_v }\limits^ \to = - \in _0 (\frac{{\partial \mathop E\limits^ \to }}{{\partial t}} \times \mathop B\limits^ \to ) + \in _0 (\mathop E\limits^ \to \cdot \nabla )\mathop E\limits^ \to - \frac{1}{{2\mu _0 }}\nabla \mathop {(B^2 ) + }\limits^{} \frac{1}{{\mu _0 }}(\mathop B\limits^ \to \cdot \nabla )\mathop B\limits^ \to
    [/itex]

    ¿What should I do to get the missing terms?
    I think I must develop the first term in the last equation, but I don’t know how, can you help me?

    I appreciate any help. Thanks a lot.
     
    Last edited: Aug 1, 2006
  2. jcsd
  3. Aug 1, 2006 #2

    siddharth

    User Avatar
    Homework Helper
    Gold Member

    Hi gjfelix2006, welcome to PhysicsForums.

    First of all, please don't double post. I see you've made a duplicate thread in the other subforum as well. Also, to make it easier for those who wish to help, you might want to read this thread which explains how to use LaTeX mathematical typesetting. That way you needn't wait till your attachment is approved.
     
  4. Aug 1, 2006 #3
    I am sorry for my double post, i am new here and i dont know how to erase one of them, i made a double post because i dont know if my problem is a basic problem or an advanced problem. So sorry, if you can tell me how to erase it, i'll appreciate. Bye
     
  5. Aug 2, 2006 #4
    I haven't followed your math thru but it looks like that you are missing
    3 terms. The one with del dot B is obviously zero from Maxwell's
    equations. Try writing a one dimensional version of the other two terms
    and I think that they will also cancel out to zero!!!
     
  6. Aug 2, 2006 #5
    Thanks a lot for your help. I have solved the problem already. Thanks J Hann.
     
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