Exploring Spin Force and Curl in Electromagnetic Waves

[jtleafs33]

Homework Statement

I put this in the math forum because although it's for my EM waves class, it's a math question.

Show that the spin force can be written as:

[itex]F_{spin}=\frac{-1}{2}Im(\alpha)Im(E\cdot\nabla E^{*})=\nabla\times L_s[/itex]

Find [itex]L_s[/itex].

Where [itex]\alpha[/itex] is complex. I'm using [itex]E^{*}[/itex] to denote the complex conjugate of [itex]E[/itex]. Also, since these are all vectors, I'm omitting the arrow notation atop the vector quantities.

Homework Equations

[itex]Im(z)=\frac{1}{2i}(z-z^{*})[/itex]

The Attempt at a Solution

From the relevant equations:
[itex]Im(\alpha)=\frac{1}{2i}[\alpha-\alpha^{*}][/itex]
[itex]Im(E\cdot\nabla E^{*})=\frac{1}{2i}[E\cdot\nabla E^{*}-(E\cdot\nabla E^{*})^{*}][/itex]

Substituting in,
[itex]F_{spin}=\frac{1}{8}[\alpha-\alpha^{*}][E\cdot\nabla E^{*}-(E\cdot\nabla E^{*})^{*}]=\nabla\times L_s[/itex]

Here, in order to make a curl appear, I'd like to apply the identity:
[itex]\nabla\times(A\times B)=A(\nabla\cdot B)-B(\nabla\cdot A)+(B\cdot\nabla)A-(A\cdot\nabla)B[/itex]

However, I'm not sure what the quantity [itex][(E\cdot\nabla E^{*})^{*}][/itex] looks like... I don't know how to conjugate this and I'm stuck here for the moment.

 

[haruspex]

If E is a vector then I'm not sure what ∇E means. ∇.E would be a scalar, making E.(∇.E) problematic. Do you mean ∇×E?

 

[jtleafs33]

E is the electric field vector. it is a function of position and time, so that's just its gradient vector, also a function of position and time.

 

[haruspex]

E is the electric field vector. it is a function of position and time, so that's just its gradient vector, also a function of position and time.

If you have a scalar field f, then ∇f = grad f is the gradient vector. But here you say E is a vector, so I'm at a loss to understand what ∇E represents. See item e) at http://www.math.ucla.edu/~ronmiech/Calculus_Problems/32B/chap14/section5/930d31/930_31.html