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Plane EM wave in a vacuum, quick identity question

  1. Feb 16, 2014 #1
    Okay the question is, given a plane electromagnetic wave in a vacuum given by E=(Ex,Ey,Ez)exp[itex]^{(i(k_{x}x+k_{y}y+k_{z}z-wt)}[/itex] and B=(Bx,By,Bz)exp[itex]^{(i(k_{x}x+k_{y}y+k_{z}z-wt)}[/itex] ,
    where k = (kx,ky,kz),
    to show that kXE=wB.

    So I'm mainly fine with the method. I can see the maxwell's equaion ∇XE=-dB/dt, is the equation required.
    -dB/dt=iwB.
    And using ∇XE=ikXE, [1], the result follows.
    My question is identifying equation [1]. How do you deduce this? Is it supposed to be obvious in any way. (I've done a check on the LHS and RHS so I can see its true), but should this be obvious?

    Thanks in advance for your assistance .
     
  2. jcsd
  3. Feb 16, 2014 #2

    BvU

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    Deduce, deduce? How would you write out ∇xE if ∇= ##( {d\over dx}, {d\over dy}, {d\over dz}) ## and ##\vec E = \Bigl( E_x \exp ({i(k_{x}x+k_{y}y+k_{z}z-wt)}) , E_y \exp(i(k_{x}x+k_{y}y+k_{z}z-wt)) , E_y \exp(i(k_{x}x+k_{y}y+k_{z}z-wt))\Bigr)## where ##\vec E =(E_x, E_y, E_z)## is a constant vector.
    You don't even have to carry out the cross product. Only the diferentiation.
     
  4. Feb 23, 2014 #3
    I'm not seeing it to be honest. I'm writing out ∇xE in determinant form.
     
  5. Feb 23, 2014 #4

    BvU

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    Determinant form has $$\left \vert \matrix{\hat \imath&\hat \jmath&\hat k\cr {d\over dx}&{d\over dy}&{d\over dz}\cr E_x &E_y &E_z }\right \vert $$ if I remember well.
    Write out e.g the x component and compare with the x-component of ##\vec k\times\vec E##, etc.
     
  6. Feb 28, 2014 #5
    In the bottom line of my first post - I've done this. But it's still not obvious to me until I do this?
     
  7. Feb 28, 2014 #6

    BvU

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    Ah, I see. You did it, you see it's correct by inspection, but apparently you need something more to be really convinced ?

    Looks to me as if you are interpreting "required" somewhat unusual.

    My interpretation of the exercise is more like: here we have an expression for ##\vec E(\vec x,t)## and ##\vec B(\vec x,t)## that we propose as a solution for Faraday's law (i.e. one of the Maxwell equations). And you are asked to make a few steps towards confirming it satisfies this equation -- under certain conditions for (Ex,Ey,Ez), (Bx,By,Bz), ##\vec k## and ##\omega##.
     
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