Finding the average energy density of an EM wave in a magnetic field

[Matt S]

Hi, I've not posted on here before but I'm trying to keep on top of work over the summer and I'm having some real problems with this question

Homework Statement

Consider the plane polarised EM wave in a source free vacuum with magnetic field B = (1,1,0)B₀cos(kz-wt) where B₀ = 0.001T. Find the direction of propagation of the wave, the average energy density of the wave and the fraction of this energy which would pass through a perfect polaroid sheet at normal incidence if the angle between B and the transmission axis of the polariser is 30^o.

Homework Equations

For the second part, I'm pretty sure you have to use U=B²/u₀ where U is the energy density and u₀ is the permeability of free space.

The Attempt at a Solution

The first part seems relatively simple, the wave travels perpendicularly to the magnetic field, so it moves in the (0,0,1) direction.

The second part asks for the average energy density but since B varies with respect to both time and position I'm not sure how to find it, would I have to integrate the expression for B with respect to t and then use the equation for U?

Obviously without an answer to part two I can't complete the third section. However, I'm pretty sure I have to find the fraction of the wave that passes through the polariser and hence find its energy density and then express that as a fraction of the incident energy density. I'm quite new to polarisation so I don't really know where to start and looking online has just confused me so far. Suggestions or links to relevant material would be really appreciated.

Thanks a lot.

 

[merryjman]

The E field and B field are perpendicular, and both of these are perpendicular to the direction of propagation of the wave. If you can find the E field expression, the Poynting vector (cross product of E and B) will give you the direction of propagation and the energy flux. The gradient of the Poynting vector will give you the energy density.

As far as the polarizer part, I'm pretty sure when they give you transmission axis, that's referring to the electric field and not the magnetic field. So again you need to find which way the E field points and go from there.

 

[Matt S]

Okay, so I used the relationship E=cB to find that the electric field acts in the (1,-1,0) direction and hence that the wave does indeed propagate in the (0,0,1) direction. I then used U=ε₀E² to find that the energy density is U=2c²B₀²ε₀

I'm still confused about the polariser, did you mean that the angle between the magnetic field and electric field is 30^o?