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Flux Integral

  • #1
798
1

Homework Statement


Find the flux of [itex]\vec{F}=(x, y, z)[/itex] outward across the sphere [itex]x^2+y^2+z^2=a^2[/itex].


I am able to get it to this point:
[tex]\int\int_Cadxdy[/tex] and I then convert it to polar coordinates, and integrate rdr from 0 to a, and theta from zero to 2pi. However, this does not give me the correct result, as the answer is 4a^3*pi, and Im getting a^3*pi.
 

Answers and Replies

  • #2
tiny-tim
Science Advisor
Homework Helper
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hi sandy.bridge! :smile:
I am able to get it to this point:
[tex]\int\int_Cadxdy[/tex] and I then convert it to polar coordinates, and integrate rdr from 0 to a, and theta from zero to 2pi. However, this does not give me the correct result, as the answer is 4a^3*pi, and Im getting a^3*pi.
it's the surface of a sphere …

what does r have to do with it? :confused:

(and why are you integrating? surely you know the surface area of a sphere? :wink:)
 
  • #3
798
1
Okay, well I know that once I get it to this point, it's right:
[tex]\int\int_CadS=\int\int_Ca(1)dxdy[/tex]

The projection of the sphere on the xy-plane is a circle, no? So why can I not use
[tex]dxdy=rdrd\theta[/tex]?
 
  • #4
tiny-tim
Science Advisor
Homework Helper
25,832
251
The projection of the sphere on the xy-plane is a circle, no?
projection? are you treating F as if it was a parallel field along one of the axes? :confused:

in that case, yes, the projection perpendicular to the field would be a circle

but the given F is radial ((x,y,z) = t), and constant in magnitude over the sphere,

so you just need the amount of surface it cuts through, which is 4πa2
 

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