Find \nabla x curl E: Solve Using Properties of Vector Calculus

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In summary, by using the equations given and some properties of curl and divergence, you can find the value of \nabla \times (\nabla \times E) and simplify it using the original fields. You could also take the time derivative outside the curl, but it is recommended to confirm this before using it. Alternatively, you can use the equality \nabla \times (\nabla \times E) = \nabla (\nabla \cdot E) - \nabla^2 E to simplify the expression.
  • #1
magnifik
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given curl E = -1/c*([tex]\partial[/tex]H/[tex]\partial[/tex]t)
div E = 0
div H = 0
curl H = 1/c*([tex]\partial[/tex]E/[tex]\partial[/tex]t), find

[tex]\nabla[/tex] x ([tex]\nabla[/tex] x E)

how do i take [tex]\nabla[/tex] x curl E? i tried to do it by determinants, but I'm not sure which values correspond to the i, j, and k. so my next assumption is that there is some property that i can take advantage of to solve the problem. please help. thanks.
 
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  • #2
[tex]
\nabla \times E = -\frac{1}{c} \frac{\partial H}{\partial t}
[/tex]


you could do it directly
[tex]
\nabla \times (\nabla \times E)
=\nabla \times ( -\frac{1}{c} \frac{\partial H}{\partial t})
=-\frac{1}{c}(\nabla \times \frac{\partial H}{\partial t})
[/tex]

and i think it should be ok to take the time derivative outside the curl, though you may want to confirm that...
[tex]
-\frac{1}{c}(\nabla \times \frac{\partial H}{\partial t})=-\frac{1}{c}\frac{\partial}{\partial t}(\nabla \times H)
[/tex]

and it should follow, otherwise, if you know the original field you could make use of the equality
[tex]
\nabla \times (\nabla \times E)
=\nabla (\nabla \cdot E) - \nabla^2 E
[/tex]

which should simplify with some of the other info
 
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