What are the missing terms in the equation for F_{v}?

[gjfelix2006]

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

The problem is the following

Starting with the expression of the force by volume unit over a free space region with charges and currents:
$F_{v}=\rho E+J\timesB$
and using Maxwell's Equations:
$\begin{array}{l} \nabla \cdot \mathop E\limits^ \to = \frac{\rho }{{ \in _0 }} \\ \nabla \times \mathop E\limits^ \to = - \frac{{\partial \mathop B\limits^ \to }}{{\partial t}} \\ \nabla \cdot \mathop B\limits^ \to = 0 \\ \nabla \times \mathop B\limits^ \to = \mu _0 \mathop J\limits^ \to + \in _0 \mu _0 \frac{{\partial \mathop E\limits^ \to }}{{\partial t}} \\ \end{array}$
and the following vectorial identity:
$\mathop B\limits^ \to \times \nabla \times \mathop B\limits^ \to = \nabla ({\textstyle{1 \over 2}}B^2 ) - (\mathop B\limits^ \to \cdot \nabla )\mathop B\limits^ \to$,

Show that:
$\begin{array}{l} \mathop {F_v }\limits^ \to = - \in _0 \frac{\partial }{{\partial t}}(\mathop E\limits^ \to \times \mathop B\limits^ \to ) + \in _0 \mathop E\limits^ \to \nabla \cdot \mathop E\limits^ \to - \frac{1}{2} \in _0 \nabla (E^2 ) + \in _0 (\mathop E\limits^ \to \cdot \nabla )\mathop E\limits^ \to \\ {\rm{ + }}\frac{1}{{\mu _0 }}\mathop B\limits^ \to \nabla \cdot \mathop B\limits^ \to - \frac{1}{{2\mu _0 }}\nabla \mathop {(B^2 ) + }\limits^{} \frac{1}{{\mu _0 }}(\mathop B\limits^ \to \cdot \nabla )\mathop B\limits^ \to \\ \end{array}$

Now, let me show you my partial solution:

First, by Maxwell Equations, i get J:
$\mathop J\limits^ \to = \frac{1}{{\mu _0 }}(\nabla \times \mathop B\limits^ \to ) - \in _0 \frac{{\partial \mathop E\limits^ \to }}{{\partial t}}$

And i replace it in the first equation for $F_{v}$ to get:
$\[ \begin{array}{l} \mathop F\limits^ \to _v = \rho \mathop E\limits^ \to + \left( {\frac{1}{{\mu _0 }}(\nabla \times \mathop B\limits^ \to ) - \in _0 \frac{{\partial \mathop E\limits^ \to }}{{\partial t}}} \right) \times \mathop B\limits^ \to \\ {\rm{ }} = \rho \mathop E\limits^ \to + \frac{1}{{\mu _0 }}(\nabla \times \mathop B\limits^ \to ) \times \mathop B\limits^ \to - \in _0 (\frac{{\partial \mathop E\limits^ \to }}{{\partial t}} \times \mathop B\limits^ \to ) \\ \end{array} \]$
Changing the order of the cross product (the sign changes), then
$\[ \mathop F\limits^ \to _v = \rho \mathop E\limits^ \to - \frac{1}{{\mu _0 }}(\mathop B\limits^ \to \times \nabla \times \mathop B\limits^ \to ) - \in _0 (\frac{{\partial \mathop E\limits^ \to }}{{\partial t}} \times \mathop B\limits^ \to {\rm{)}} \]$
Now i can use the vectorial identity, thus:
$\[ = \rho \mathop E\limits^ \to - \frac{1}{{\mu _0 }}(\nabla ({\textstyle{1 \over 2}}B^2 ) - (\mathop B\limits^ \to \cdot \nabla )\mathop B\limits^ \to ) - \in _0 (\frac{{\partial \mathop E\limits^ \to }}{{\partial t}} \times \mathop B\limits^ \to {\rm{) }} \]$

Also, from Maxwell equations:
$\rho = (\nabla \cdot \mathop E\limits^ \to ) \in _0$
and replacing it into the last equation for $F_{v}$, i get:
$\mathop F\limits^ \to _v = \mathop E\limits^ \to (\nabla \cdot \mathop E\limits^ \to ) \in _0 - \frac{1}{{\mu _0 }}(\nabla ({\textstyle{1 \over 2}}B^2 ) - (\mathop B\limits^ \to \cdot \nabla )\mathop B\limits^ \to ) - \in _0 (\frac{{\partial \mathop E\limits^ \to }}{{\partial t}} \times \mathop B\limits^ \to {\rm{) }}$
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?
$\mathop {F_v }\limits^ \to = - \in _0 (\frac{{\partial \mathop E\limits^ \to }}{{\partial t}} \times \mathop B\limits^ \to ) + \in _0 (\mathop E\limits^ \to \cdot \nabla )\mathop E\limits^ \to - \frac{1}{{2\mu _0 }}\nabla \mathop {(B^2 ) + }\limits^{} \frac{1}{{\mu _0 }}(\mathop B\limits^ \to \cdot \nabla )\mathop B\limits^ \to$

¿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.

 

[siddharth]

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

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

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

 

[gjfelix2006]

Ok, thanks for your help siddharth, i found my error and finally solved my problem.