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Maxwell Equation's 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.
    the problem is also in pdf.
     
    Last edited: Aug 1, 2006
  2. jcsd
  3. Aug 2, 2006 #2

    siddharth

    User Avatar
    Homework Helper
    Gold Member

    Notice that in your question you've got terms such as
    [tex] - \in _0 \frac{\partial }{{\partial t}}(\mathop E\limits^ \to \times \mathop B\limits^ \to ) [/tex]

    First use the product rule and then
    [tex] \nabla \times \vec{E} = - \frac{\partial \vec{B}}{\partial t} [/tex]

    You should get your answer from this using some vector identites

    P.S I think you've made a typo while typing your expression for F_v in the first step
     
    Last edited: Aug 2, 2006
  4. Aug 2, 2006 #3
    Ok, thanks for your help siddharth, i found my error and finally solved my problem.
     
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