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Math Help - tough tough gauss law

  1. #1
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    tough tough gauss law

    1)The electric field at apoint P(x,y,z) due to a point charge q located at the origin is given by the inverse square field
    E=qr/||r||^3
    where r=xi+yj+zk
    (a)Suppose S is a closed surface,Sa is a sphere x^2+y^2+z^2=a^2 lying completely within s, and D is the region bounded between S and Sa.Show that the outward flux of E for the region D is zero.
    (b)Use the result of part (a) to prove Gauss's Law:
    int int (E.n)dS=4*pi*q
    that is, the outward flux of the electric field E through any closed surface (for which the divergence theorem applies) contaning the origin is 4*pi*q.

    2)int int y^2/x dA,where R isthe region bounded by the graphs y=x^2,y=x^2/2,x=y^2,x=y^2/2;u=x^2/y,v=y^2/x.

    In the question above,"int int" means double integral.Any idea about the question.Too tough for me.
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    Quote Originally Posted by prescott2006 View Post
    1)The electric field at apoint P(x,y,z) due to a point charge q located at the origin is given by the inverse square field
    E=qr/||r||^3
    where r=xi+yj+zk
    (a)Suppose S is a closed surface,Sa is a sphere x^2+y^2+z^2=a^2 lying completely within s, and D is the region bounded between S and Sa.Show that the outward flux of E for the region D is zero.
    You can use a qualitative argument to do this, at least if you make the assumption that the electric field has no "psychotic" features (such as entering the region D and then looping around the interior forever.) Unless there is a charge inside the region any field lines entering D must also leave it, meaning that the incoming flux is equal to the outgoing flux, which means that the net flux through the surface is 0.

    As far as the E field being non-psychotic, I can't think of a way to prove that. After all, certain situations with the magnetic field can warp that into different shapes (though Gauss' Law still holds for magnetic fields as well for any finite sized surface.)

    Quote Originally Posted by prescott2006 View Post
    (b)Use the result of part (a) to prove Gauss's Law:
    int int (E.n)dS=4*pi*q
    that is, the outward flux of the electric field E through any closed surface (for which the divergence theorem applies) contaning the origin is 4*pi*q.
    A pretty good derivation of Gauss' Law may be found here.

    -Dan
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