Transverse Wave Incident on Absorbing Surface: Pressure & Energy Density

[ptabor]

Suppose a light wave is traveling in the z-direction.
Then the Electric field is in the x direction, and the B field is in the y direction.

My understanding is that these E and B fields are independent of the x and y directions, respectively. This is to say, that if I take the partial derivative of E with respect to x, I get zero - Likewise for B and y.

I'm trying to show that for a transverse plane wave incident normally on a perfectly abosrbing surface, starting from the conservation of linear momentum, that the pressure exerted on the screen is equal to the energy density per unit volume.

Since we're in vacuum there is no charge or current density, and I only have to consider the momentum density of the wave itself (no mechanical momentum density for the charges). I reduce the expression to something that is the energy density plus the scalar product of B and nabla (or del, whichever you prefer) acting on B - likewise for E. I need to show these are zero.

 

[nrqed]

Suppose a light wave is traveling in the z-direction.
Then the Electric field is in the x direction, and the B field is in the y direction.

My understanding is that these E and B fields are independent of the x and y directions, respectively. This is to say, that if I take the partial derivative of E with respect to x, I get zero - Likewise for B and y.

I'm trying to show that for a transverse plane wave incident normally on a perfectly abosrbing surface, starting from the conservation of linear momentum, that the pressure exerted on the screen is equal to the energy density per unit volume.

Since we're in vacuum there is no charge or current density, and I only have to consider the momentum density of the wave itself (no mechanical momentum density for the charges). I reduce the expression to something that is the energy density plus the scalar product of B and nabla (or del, whichever you prefer) acting on B - likewise for E. I need to show these are zero.

I am probably asking a stupid question...but what do you mean by nabla of B? You are not talking about ${\vector \nabla} \cdot {\vec B}$ since this is a scalar. So you mean what exactly? (${\partial B_x \over \partial x } + \ldots$? But that's not nabla applied to B)

:-( The tex editor does not accept my symbol \nabla

 

[robphy]

${\vector \nabla} \cdot {\vec B}$
Hmm...
$$\nabla$$
$\nabla$ ${\nabla}$
$\vec\nabla$