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 , i appreciate any help in this. Thanks(adsbygoogle = window.adsbygoogle || []).push({});

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.

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# Homework Help: Maxwell Equations Problem

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