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Astrophysics
 one year ago
Flux
Astrophysics
 one year ago
Flux

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Astrophysics
 one year ago
Best ResponseYou've already chosen the best response.4Find 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
 one year ago
Best ResponseYou've already chosen the best response.4Don'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
 one year ago
Best ResponseYou've already chosen the best response.4@Empty @ganeshie8

Astrophysics
 one year ago
Best ResponseYou've already chosen the best response.4So I guess \[x^2+y^2+z^2=1\] then we can use a parametric representation?

Astrophysics
 one year ago
Best ResponseYou've already chosen the best response.4I think the troubling part is when we cross multiply, not really sure how to visualize it and what way it should be

Astrophysics
 one year ago
Best ResponseYou've already chosen the best response.4dw:1441476251414:dw

IrishBoy123
 one year ago
Best ResponseYou've already chosen the best response.4use a Gaussian surface as it's totally symmetrical or stuff it into spherical if you want a slightly harder time

Astrophysics
 one year ago
Best ResponseYou've already chosen the best response.4Hmm interesting, never really worked with gaussian surfaces, I will look into it, thanks @IrishBoy123

hwyl
 one year ago
Best ResponseYou've already chosen the best response.1that's what you do anyway start from cube then use the same for curved

IrishBoy123
 one year ago
Best ResponseYou've already chosen the best response.4attached 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

Astrophysics
 one year ago
Best ResponseYou've already chosen the best response.4Thanks @hwyl Awesome, this is one reason I want to take E&M haha, thank you very much @IrishBoy123

beginnersmind
 one year ago
Best ResponseYou've already chosen the best response.1You can use the divergence theorem:dw:1441477056233:dw div F is particularly simple in this example

Astrophysics
 one year ago
Best ResponseYou've already chosen the best response.4Thanks everyone, nice @beginnersmind I will try your approach as well! :)

IrishBoy123
 one year ago
Best ResponseYou've already chosen the best response.4well divergence theorem sounds good in theory, but in spherical is it really any easier than cranking out the surface integral in rectangular, yuck.

beginnersmind
 one year ago
Best ResponseYou've already chosen the best response.1Yeah, 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
 one year ago
Best ResponseYou've already chosen the best response.4@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
 one year ago
Best ResponseYou've already chosen the best response.1I mean F is equal on the whole surface. It has different directions (always perpendicular to the surface) but the same magnitude.

Jhannybean
 one year ago
Best ResponseYou've already chosen the best response.0Ohh is this a conservative vector field?

beginnersmind
 one year ago
Best ResponseYou've already chosen the best response.1Well, it's like the electric field of a positive charge so it better be :)

Jhannybean
 one year ago
Best ResponseYou've already chosen the best response.0This might help? http://ltcconline.net/greenl/courses/202/vectorIntegration/vectorFields.htm

dan815
 one year ago
Best ResponseYou've already chosen the best response.0divergence theorem cleans it up

dan815
 one year ago
Best ResponseYou've already chosen the best response.0you can do surface integral if u must practice

beginnersmind
 one year ago
Best ResponseYou've already chosen the best response.1Why is div F = 3 though?

dan815
 one year ago
Best ResponseYou've already chosen the best response.0oh oops that r looks like an f to me

beginnersmind
 one year ago
Best ResponseYou've already chosen the best response.1@Astrophysics do you still need this, or have you figured it out?

dan815
 one year ago
Best ResponseYou've already chosen the best response.0hmm im not sure how well F dot N ds really simplifies it doesnt seem liek it would simplify that well but

beginnersmind
 one year ago
Best ResponseYou've already chosen the best response.1dan: 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
 one year ago
Best ResponseYou've already chosen the best response.0ya i noticed it simplfied after i tried it earlier xD

dan815
 one year ago
Best ResponseYou've already chosen the best response.0i 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
 one year ago
Best ResponseYou've already chosen the best response.4I got it, thanks everyone :)!
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