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.