Astrophysics
  • Astrophysics
Flux
Mathematics
  • Stacey Warren - Expert brainly.com
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schrodinger
  • schrodinger
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
Astrophysics
  • Astrophysics
Find a flux of the vector field \[\vec F(x,y,z) = \frac{ m \vec r }{ |\vec r|^3 }\], where m = constant. \[\vec r = x \vec i + y \vec j + z \vec k\] out of a sphere od radius 1 centred at origin.
Astrophysics
  • Astrophysics
Don't really remember how to do this, so thought I'd ask for some help, I believe if F is a continuous vector field defined on an oriented surface S with a normal vector n, then the surface integral of F over S is \[\int\limits \int\limits_S \vec F \cdot d \vec S = \int\limits \int\limits \vec F \cdot \vec n dS\]
Astrophysics
  • Astrophysics
@Empty @ganeshie8

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Astrophysics
  • Astrophysics
So I guess \[x^2+y^2+z^2=1\] then we can use a parametric representation?
Astrophysics
  • Astrophysics
I think the troubling part is when we cross multiply, not really sure how to visualize it and what way it should be
Astrophysics
  • Astrophysics
|dw:1441476251414:dw|
IrishBoy123
  • IrishBoy123
use a Gaussian surface as it's totally symmetrical or stuff it into spherical if you want a slightly harder time
Astrophysics
  • Astrophysics
Hmm interesting, never really worked with gaussian surfaces, I will look into it, thanks @IrishBoy123
hwyl
  • hwyl
that's what you do anyway start from cube then use the same for curved
hwyl
  • hwyl
|dw:1441476650300:dw|
IrishBoy123
  • IrishBoy123
attached is the spherical coords way you have an inverse square law here and you can use spherical and gaussian surfaces for all that stuff, even for gravity!! Lewin's lectures on this are epic
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Astrophysics
  • Astrophysics
Thanks @hwyl Awesome, this is one reason I want to take E&M haha, thank you very much @IrishBoy123
hwyl
  • hwyl
|dw:1441476754530:dw|
hwyl
  • hwyl
apply gauss law
hwyl
  • hwyl
ty for your time
beginnersmind
  • beginnersmind
You can use the divergence theorem:|dw:1441477056233:dw| div F is particularly simple in this example
Astrophysics
  • Astrophysics
Thanks everyone, nice @beginnersmind I will try your approach as well! :)
IrishBoy123
  • IrishBoy123
well divergence theorem sounds good in theory, but in spherical is it really any easier than cranking out the surface integral in rectangular, yuck.
beginnersmind
  • beginnersmind
Yeah, scratch that. Probably easiest to use the fact that F is normal to the surface and equal on the whole surface. So I guess the solution is just |F| at the surface times the surface area of the sphere?
IrishBoy123
  • IrishBoy123
@beginnersmind that is just so neat! simplifies down all the other Gaussian ideas. i think i may have let my enthusiasm get the better of me - again!
beginnersmind
  • beginnersmind
I mean |F| is equal on the whole surface. It has different directions (always perpendicular to the surface) but the same magnitude.
anonymous
  • anonymous
bae is here XD
Jhannybean
  • Jhannybean
*
anonymous
  • anonymous
who is "bae"
Jhannybean
  • Jhannybean
Ohh is this a conservative vector field?
beginnersmind
  • beginnersmind
Well, it's like the electric field of a positive charge so it better be :)
Jhannybean
  • Jhannybean
This might help? http://ltcconline.net/greenl/courses/202/vectorIntegration/vectorFields.htm
dan815
  • dan815
divergence theorem cleans it up
dan815
  • dan815
|dw:1441507539703:dw|
dan815
  • dan815
|dw:1441507619509:dw|
dan815
  • dan815
you can do surface integral if u must practice
beginnersmind
  • beginnersmind
Why is div F = 3 though?
dan815
  • dan815
oh oops that r looks like an f to me
beginnersmind
  • beginnersmind
@Astrophysics do you still need this, or have you figured it out?
dan815
  • dan815
hmm im not sure how well F dot N ds really simplifies it doesnt seem liek it would simplify that well but
beginnersmind
  • beginnersmind
dan: F is m in the direction of the normal. So F dot dn is m. You integrate that over that over the whole surface, you get m times the surface area. So 4m*pi
dan815
  • dan815
ya i noticed it simplfied after i tried it earlier xD
dan815
  • dan815
i took F=m*r_, since mag of r is just 1 everywhere on surface of sphere we can cancel out the |r_|^3 unit vector n = r_/|mag r_| F dot n ds m*(x^2+y^2+z^2) ds since its on surface thats just m =integral m ds
Astrophysics
  • Astrophysics
I got it, thanks everyone :)!

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