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Flux integral and Gauss's theorem

  1. Oct 2, 2014 #1
    1. The problem statement, all variables and given/known data
    Please help me to solve this assignment.

    a) A vector field is given by v(x,y,z) = (y, x, z-x).
    Calculate the flux from this field out of the unit cube, given by x,y,z = [0,1].

    b) Use Gauss's theorem to calculate the same flux. Check that you get the same result.


    2. Relevant equations
    Electric Flux: Φ = E. A .cosθ
    Electric Flux: Φ = E. dA = Qencl / ε

    3. The attempt at a solution
    For part a) one can calculate ∫∫ v . n dxdy, where n is the unit normalvector for the surface A, for each cube faces and add them together.
     
  2. jcsd
  3. Oct 2, 2014 #2

    BvU

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    Well ?
     
  4. Oct 2, 2014 #3
    I just don't know how to start!
     
  5. Oct 2, 2014 #4
     
  6. Oct 2, 2014 #5
     
  7. Oct 2, 2014 #6

    BvU

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    Take one of the faces of the unit cube and write out ##\vec v \cdot \hat n## then integrate over that face. Proceed to the next face, etc. Repeat until done !
     
  8. Oct 3, 2014 #7
     
  9. Oct 3, 2014 #8
    The flux through the face parallell to the z-axis, which I call for A1 and A2, is
    Φ = ∫∫v.k dA1, where k is the normal vector in z-direction. But, what is V and dA1 now?!
    Is v the z-component of the vector field?

    I'm hoping someone can help walk me through this problem!
     
  10. Oct 3, 2014 #9

    BvU

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    Do one face at the time. Both the face with x=1 and the face with y=1 are parallel to th z axis...
    Or do you mean the face with z = 1 ? That is perpendicular to the z axis.
    The normal in the z direction is the surface vector for the face with z = 1. In other words, (0,0,1)
    A point on the surface is characterized by (x,y,z) = (x,y,1)

    Can you now write out the integral for that face ?
     
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