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Integrating a vector (Electromagnetism)

  1. Jan 7, 2014 #1
    1. The problem statement, all variables and given/known data

    Given [itex]\textbf{E}(z,t) = E_{0}cos(kz+ωt)\textbf{i}[/itex]
    Find B

    2. Relevant equations

    ∇ x E = -[itex]\frac{\partial\textbf{B}}{\partial t}[/itex]


    3. The attempt at a solution

    Taking the curl of [itex]\textbf{E}[/itex] gives [itex](0, -ksin(kz+\omega t), 0)[/itex]
    so
    [itex]\frac{\partial\textbf{B}}{\partial t} = (0,ksin(kz+\omega t),0)[/itex]

    I'm not too confident integrating this, I got
    [itex]\textbf{B} = (f(z),-\frac{k}{\omega}cos(kz+\omega t), g(z)) + \textbf{c}[/itex]
    where c is a constant of integration.

    Is this right? The next part of the question asks for the poynting vector and it seems like a lot of work calculating [itex]\textbf{E} \times \textbf{B}[/itex] , would i be allowed to set [itex]f = g = 0[/itex]?
     
  2. jcsd
  3. Jan 7, 2014 #2

    HallsofIvy

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    Where did you get this "f(z)" and "g(z)"? The "constant" of integration is the vector c you have added.
     
  4. Jan 7, 2014 #3
    Yeah thats what i'm confused about.
    My reasoning is that I have [itex]\frac{\partial \textbf{B}}{\partial t}[/itex] in terms of [itex]\textbf{E}[/itex], and since [itex]\textbf{E}[/itex] is a function of z and t I get the functions of z from partially integrating wrt t.
    Should they be 0?
     
  5. Jan 7, 2014 #4
    Oh wait I think i just got it.
    I can put those functions of z into the arbitrary constant vector c can't I?

    Thanks!
     
  6. Jan 7, 2014 #5

    vanhees71

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    From the Faraday Law alone, we can in fact only conclude that
    [tex]\vec{B}(t,\vec{r})=-E_0 \frac{k}{\omega} \cos(\omega t+k z)+\vec{B}_0(\vec{r})[/tex]
    where [itex]\vec{B}_0[/itex] is an arbitrary static magnetic field.

    From Gauss's Law for the magentic field we find
    [tex]\vec{\nabla} \cdot \vec{B}=\vec{\nabla} \cdot \vec{B}_0=0.[/tex]

    From the Maxwell-Ampere Law, assuming that there are no currents, we get
    [tex]\partial_t \vec{E}=\vec{\nabla} \times \vec{B} \; \Rightarrow \; \vec{\nabla} \times \vec{B}_0=0.[/tex]
    Thus static magnetic field [itex]\vec{B}_0[/itex] is both source and vortex free. Thus if it should vanish at inifinity, it must be 0.

    From these additional assumptions we get the usual plane-wave solution.
     
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