Is \(\vec{\nabla}^2 \vec{E}\) a Dot or Vector Product?

[kasse]

Is \vec{\nabla^{2}} \vec{E} a dot or a vector product?

 

[gabbagabbahey]

Do you mean \nabla ^2 V?

 

[kasse]

No, I mean \vec{\nabla^{2}} \vec{E}, like here in the wave equations

http://en.wikipedia.org/wiki/Electromagnetic_wave_equation

 

[gabbagabbahey]

Whenever you take the Laplacian of a vector, its shorthand for

\hat{x} \frac{\partial ^2 E_x}{\partial x}+\hat{y} \frac{\partial ^2 E_y}{\partial y}+\hat{z} \frac{\partial ^2 E_z}{\partial z}

in Cartesian coordinates and the definition in curvilinear coordinates is similar.

 

[kasse]

How can I then use Maxwell's equations to derive the wave equations for EM waves?

 

[gabbagabbahey]

There is no need to use the above definition for that...

Just use the rules for vector second derivatives and look at

\vec{\nabla} \times (\vec{\nabla} \times \vec{E})

and

\vec{\nabla} \times (\vec{\nabla} \times \vec{B})...you will need to use maxwell's equations and the continuity equation.

 

[kasse]

\vec{\nabla} \times (\vec{\nabla} \times \vec{E}) = ?

 

[gabbagabbahey]

There is a rule that will help you determine that; it should be in your text somewhere (possibly even inside the front cover!)

 

[kasse]

Thanks!

And one more question:

Why can \vec{\nabla} \times (-\partial B / \partial t) be written (-\partial/ \partial t)\vec{\nabla}\times B

 

[gabbagabbahey]

Thanks!

And one more question:

Why can \vec{\nabla} \times (-\partial B / \partial t) be written (-\partial/ \partial t)\vec{\nabla}\times B

Because \vec{\nabla} represents a spatial derivative, and since space and time are assumed to be independent for classical E&M, it doesn't matter whether you take the time derivative before the spatial derivative, or vice versa.