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Math Help - flux through each side of cube

  1. #1
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    flux through each side of cube

    Consider the vector field  \vec{F} = 10\vec{i} + y\vec{j} + 5\vec{k} .

    Find the flux of the given vector field through a cube in the first octant with edge length c, one corner at the origin and edges along the axes.


    I need to find the flux at each sides of the cube, can someone help me how I can do this?
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    Quote Originally Posted by TheRekz View Post
    Consider the vector field  \vec{F} = 10\vec{i} + y\vec{j} + 5\vec{k} .

    Find the flux of the given vector field through a cube in the first octant with edge length c, one corner at the origin and edges along the axes.


    I need to find the flux at each sides of the cube, can someone help me how I can do this?
    I hope you know Gauss' theorem otherwise you will need to evaluate 6 surface integrals.

    \iiint_V \nabla \cdot F dV

    \int_{0}^{c}\int_{0}^{c}\int_{0}^{c}1dV=c^3
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    MHF Contributor Calculus26's Avatar
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    Use the divergence thm --beats 6 flux integrals

    Flux = volume integral( divF) in this case divF =1

    so you have simply c^3 for the flux

    If you absolutely have to do 6 Flux calculations

    since the x and z components are constant the flux through the bottom
    cancels the flux through the top. Similarly for the front and back since the x component is 0

    On the left side y= 0 so the flux is 0

    the right face then is the only contribution to the flux

    where y =c and you integrate over the square with sides c in the xzplane
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    Quote Originally Posted by Calculus26 View Post
    Use the divergence thm --beats 6 flux integrals

    Flux = volume integral( divF) in this case divF =1

    so you have simply c^3 for the flux

    If you absolutely have to do 6 Flux calculations

    since the x and z components are constant the flux through the bottom
    cancels the flux through the top. Similarly for the front and back since the x component is 0

    On the left side y= 0 so the flux is 0

    the right face then is the only contribution to the flux

    where y =c and you integrate over the square with sides c in the xzplane
    yes.. I actually need to find the flux through each surface.. how can I do that?
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    Quote Originally Posted by TheRekz View Post
    yes.. I actually need to find the flux through each surface.. how can I do that?
    The definiton of Flux is
    \int F \cdot \vec n dS

    Since each of the sides of the cube are parallel to a coordinate axisis the normal vectors will be the unit vectors i, j k.

    I will set one up for you

    So on the top of the cube the normal vector point up in the Z direction.

    \int_{0}^{c}\int_{0}^{c} (10 \vec i +y \vec y + 5 \vec k)\cdot \vec k dxdy=\int_{0}^{c}\int_{0}^{c}5dxdy=5c^2

    You will need to do this with the other 5 sides and then add all of them up.
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    Quote Originally Posted by TheEmptySet View Post
    The definiton of Flux is
    \int F \cdot \vec n dS

    Since each of the sides of the cube are parallel to a coordinate axisis the normal vectors will be the unit vectors i, j k.

    I will set one up for you

    So on the top of the cube the normal vector point up in the Z direction.

    \int_{0}^{c}\int_{0}^{c} (10 \vec i +y \vec y + 5 \vec k)\cdot \vec k dxdy=\int_{0}^{c}\int_{0}^{c}5dxdy=5c^2

    You will need to do this with the other 5 sides and then add all of them up.
    for the bottom would it be just the inverse of the top?
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    Quote Originally Posted by TheRekz View Post
    for the bottom would it be just the inverse of the top?
    yes becuase your unit vector is -\vec k
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    MHF Contributor Calculus26's Avatar
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    the negative not the inverse -- yes since n = -k (outward normal)
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    I am just confuse on how to get the left side of the cube... can someone enlighten me

    so far I have:

     \int_{0}^{c}\int_{0}^{c} y dx dz
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  10. #10
    MHF Contributor Calculus26's Avatar
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    First n = -j also y = 0 on the left face so F*n = 0 --on the right face
    y = c
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    Quote Originally Posted by Calculus26 View Post
    First n = -j also y = 0 on the left face so F*n = 0 --on the right face
    y = c
    sorry I meant to ask for the right face as the left face is 0
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    when y = c, so what do I set the integral as?
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  13. #13
    MHF Contributor Calculus26's Avatar
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    For

    simply put y= c in above integral
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    thanks!
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