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Thread: Help with scalar equations.

  1. #1
    Aug 2007

    Help with scalar equations.

    I'm trying to find all the scalar equations in Maxwell's equations and I think I just need to get to grips with all the definitions.

    $\displaystyle \nabla \cdot E=\frac{\rho}{\epsilon_{0}}$

    $\displaystyle \nabla \cdot B=0$

    $\displaystyle \nabla \times E=-\frac{\partial B}{\partial t}$

    $\displaystyle \nabla \times B=\mu_{0} J + \mu_{0} \epsilon_{0} \frac{\partial E}{\partial t}$

    where $\displaystyle E = (E_{1},E_{2},E_{3})$, $\displaystyle B = (B_{1},B_{2},B_{3})$ and $\displaystyle J = (J_{1},J_{2},J_{3})$

    What is a scalar equation exactly? Is it an equation without vectors like ax + by +cz = 0?

    Will my answer have to do with the divergence/curling operators? Or is it something to do with the partial derivatives on vectors?
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  2. #2
    Aug 2007
    are there are 8 equations? I tried to break it into scalar bits and make new equations. The first two can be changed to make 1 scalar equation each and the second two equations break into 3 each using the definition of the curl operator.


    $\displaystyle \frac{\partial E_{1}}{\partial x}+\frac{\partial E_{2}}{\partial y}+\frac{\partial E_{3}}{\partial z}=\frac{\rho}{\epsilon_{0}}$


    $\displaystyle \frac{\partial B_{1}}{\partial x}+\frac{\partial B_{2}}{\partial y}+\frac{\partial B_{3}}{\partial z}=0$


    $\displaystyle \frac{\partial E_{3}}{\partial y}-\frac{\partial E_{2}}{\partial z}=\frac{\partial B_{1}}{\partial t}$

    $\displaystyle \frac{\partial E_{1}}{\partial z}-\frac{\partial E_{3}}{\partial x}=\frac{\partial B_{2}}{\partial t}$

    $\displaystyle \frac{\partial E_{2}}{\partial x}-\frac{\partial E_{1}}{\partial y}=\frac{\partial B_{3}}{\partial t}$


    $\displaystyle \frac{\partial B_{3}}{\partial y}-\frac{\partial B_{2}}{\partial z}=\mu_{0} J_{1} + \mu_{0} \epsilon_{0} \frac{\partial E_{1}}{\partial t}$

    $\displaystyle \frac{\partial B_{1}}{\partial z}-\frac{\partial B_{3}}{\partial x}=\mu_{0} J_{2} + \mu_{0} \epsilon_{0}\frac{\partial E_{2}}{\partial t}$

    $\displaystyle \frac{\partial B_{2}}{\partial x}-\frac{\partial B_{1}}{\partial y}=\mu_{0} J_{3} + \mu_{0} \epsilon_{0}\frac{\partial E_{3}}{\partial t}$

    Is this okay at a glance? Are all the terms scalars (if that's what a scalar equation is)?
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