- #1
gjfelix2006
- 11
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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:
<br /> F_{v}=\rho E+J\timesB<br />
and using Maxwell's Equations:
<br /> <br /> \begin{array}{l}<br /> \nabla \cdot \mathop E\limits^ \to = \frac{\rho }{{ \in _0 }} \\ <br /> \nabla \times \mathop E\limits^ \to = - \frac{{\partial \mathop B\limits^ \to }}{{\partial t}} \\ <br /> \nabla \cdot \mathop B\limits^ \to = 0 \\ <br /> \nabla \times \mathop B\limits^ \to = \mu _0 \mathop J\limits^ \to + \in _0 \mu _0 \frac{{\partial \mathop E\limits^ \to }}{{\partial t}} \\ <br /> \end{array}<br /> <br /> <br />
and the following vectorial identity:
<br /> <br /> \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 <br />,
Show that:
<br /> <br /> \begin{array}{l}<br /> \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 \\ <br /> {\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 \\ <br /> \end{array}<br />
Now, let me show you my partial solution:
First, by Maxwell Equations, i get J:
<br /> <br /> \mathop J\limits^ \to = \frac{1}{{\mu _0 }}(\nabla \times \mathop B\limits^ \to ) - \in _0 \frac{{\partial \mathop E\limits^ \to }}{{\partial t}}<br />
And i replace it in the first equation for F_{v} to get:
<br /> \[<br /> \begin{array}{l}<br /> \mathop F\limits^ \to _v = \rho \mathop E\limits^ \to + \left( {\frac{1}{{\mu _0 }}(\nabla \times \mathop B\limits^ \to ) - \in _0 \frac{{\partial \mathop E\limits^ \to }}{{\partial t}}} \right) \times \mathop B\limits^ \to \\ <br /> {\rm{ }} = \rho \mathop E\limits^ \to + \frac{1}{{\mu _0 }}(\nabla \times \mathop B\limits^ \to ) \times \mathop B\limits^ \to - \in _0 (\frac{{\partial \mathop E\limits^ \to }}{{\partial t}} \times \mathop B\limits^ \to ) \\ <br /> \end{array}<br /> \]<br /> <br />
Changing the order of the cross product (the sign changes), then
<br /> \[<br /> \mathop F\limits^ \to _v = \rho \mathop E\limits^ \to - \frac{1}{{\mu _0 }}(\mathop B\limits^ \to \times \nabla \times \mathop B\limits^ \to ) - \in _0 (\frac{{\partial \mathop E\limits^ \to }}{{\partial t}} \times \mathop B\limits^ \to {\rm{)}}<br /> \]<br /> <br />
Now i can use the vectorial identity, thus:
<br /> \[<br /> = \rho \mathop E\limits^ \to - \frac{1}{{\mu _0 }}(\nabla ({\textstyle{1 \over 2}}B^2 ) - (\mathop B\limits^ \to \cdot \nabla )\mathop B\limits^ \to ) - \in _0 (\frac{{\partial \mathop E\limits^ \to }}{{\partial t}} \times \mathop B\limits^ \to {\rm{) }}<br /> \]<br /> <br />
Also, from Maxwell equations:
<br /> \rho = (\nabla \cdot \mathop E\limits^ \to ) \in _0 <br />
and replacing it into the last equation for F_{v}, i get:
<br /> \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{) }}<br />
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?
<br /> \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 <br />
¿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.
, i appreciate any help in this. ThanksThe problem is the following
Starting with the expression of the force by volume unit over a free space region with charges and currents:
<br /> F_{v}=\rho E+J\timesB<br />
and using Maxwell's Equations:
<br /> <br /> \begin{array}{l}<br /> \nabla \cdot \mathop E\limits^ \to = \frac{\rho }{{ \in _0 }} \\ <br /> \nabla \times \mathop E\limits^ \to = - \frac{{\partial \mathop B\limits^ \to }}{{\partial t}} \\ <br /> \nabla \cdot \mathop B\limits^ \to = 0 \\ <br /> \nabla \times \mathop B\limits^ \to = \mu _0 \mathop J\limits^ \to + \in _0 \mu _0 \frac{{\partial \mathop E\limits^ \to }}{{\partial t}} \\ <br /> \end{array}<br /> <br /> <br />
and the following vectorial identity:
<br /> <br /> \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 <br />,
Show that:
<br /> <br /> \begin{array}{l}<br /> \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 \\ <br /> {\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 \\ <br /> \end{array}<br />
Now, let me show you my partial solution:
First, by Maxwell Equations, i get J:
<br /> <br /> \mathop J\limits^ \to = \frac{1}{{\mu _0 }}(\nabla \times \mathop B\limits^ \to ) - \in _0 \frac{{\partial \mathop E\limits^ \to }}{{\partial t}}<br />
And i replace it in the first equation for F_{v} to get:
<br /> \[<br /> \begin{array}{l}<br /> \mathop F\limits^ \to _v = \rho \mathop E\limits^ \to + \left( {\frac{1}{{\mu _0 }}(\nabla \times \mathop B\limits^ \to ) - \in _0 \frac{{\partial \mathop E\limits^ \to }}{{\partial t}}} \right) \times \mathop B\limits^ \to \\ <br /> {\rm{ }} = \rho \mathop E\limits^ \to + \frac{1}{{\mu _0 }}(\nabla \times \mathop B\limits^ \to ) \times \mathop B\limits^ \to - \in _0 (\frac{{\partial \mathop E\limits^ \to }}{{\partial t}} \times \mathop B\limits^ \to ) \\ <br /> \end{array}<br /> \]<br /> <br />
Changing the order of the cross product (the sign changes), then
<br /> \[<br /> \mathop F\limits^ \to _v = \rho \mathop E\limits^ \to - \frac{1}{{\mu _0 }}(\mathop B\limits^ \to \times \nabla \times \mathop B\limits^ \to ) - \in _0 (\frac{{\partial \mathop E\limits^ \to }}{{\partial t}} \times \mathop B\limits^ \to {\rm{)}}<br /> \]<br /> <br />
Now i can use the vectorial identity, thus:
<br /> \[<br /> = \rho \mathop E\limits^ \to - \frac{1}{{\mu _0 }}(\nabla ({\textstyle{1 \over 2}}B^2 ) - (\mathop B\limits^ \to \cdot \nabla )\mathop B\limits^ \to ) - \in _0 (\frac{{\partial \mathop E\limits^ \to }}{{\partial t}} \times \mathop B\limits^ \to {\rm{) }}<br /> \]<br /> <br />
Also, from Maxwell equations:
<br /> \rho = (\nabla \cdot \mathop E\limits^ \to ) \in _0 <br />
and replacing it into the last equation for F_{v}, i get:
<br /> \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{) }}<br />
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?
<br /> \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 <br />
¿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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