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Homework Help: Verifying Gauss' Law

  1. Feb 12, 2010 #1
    1. The problem statement, all variables and given/known data
    For the given flux density: [tex]\vec{D}[/tex]=(2y2+z)[tex]\widehat{x}[/tex]+(4xy)[tex]\widehat{y}[/tex]^+(xz)[tex]\widehat{z}[/tex]
    a)Determine the charge density.
    b)Find the total charge enclosed if the surface is 0<x<1, 0<y<1, 0<z<1 (unit cube)
    c)Confirm Gauss’s law by finding the net flux through the surface of the volume.

    2. Relevant equations

    3. The attempt at a solution

    I used divergence to find the [tex]\rho[/tex]v=5x

    To find Qenc I integrated [tex]\int\int\int[/tex]5xdxdydz and came up with
    Qenc=5/2 C

    The last step I know is to verify that [tex]\oint[/tex][tex]\vec{D}[/tex]dS=Qenc.

    From what I gather since the divergence only has an x-component we will integrate only the x-component over the dxdydz, but this gives me 7/2. I'm guessing both are incorrect, but obviously at least one of them is.
     
  2. jcsd
  3. Feb 12, 2010 #2

    ehild

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    The divergence is a scalar, and you have to integrate the flux density for all sides of the cube. Remember that the surface element dS is a vector normal to the surface.


    ehild
     
  4. Feb 12, 2010 #3
    Thanks for the response and the insight.

    So for the top of said cube I would integrate the z-hat coefficient by dydx?
    the bottom by z-hat (-dydx)
    the right by y-hat (dxdz)
    left by y-hat(-dxdz)
    front by x-hat(dydz)
    and back x-hat(-dydz)...
     
  5. Feb 12, 2010 #4

    ehild

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    It looks OK.

    ehild
     
  6. Mar 16, 2010 #5
    I apologize for not saying thanks again. I do find this site a valuable resource in attempting to learn this material instead of just blindly using equations and hoping that things work out.
     
  7. Mar 16, 2010 #6

    ehild

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    You are welcome.

    ehild
     
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