Flux integral on a radial vector field

[Raziel2701]

Homework Statement

(The S1 after the double integral is supposed to be underneath them btw, I just can't seem to do it right using LaTeX right now so bear with me please.)

Suppose F is a radial force field, S1 is a sphere of radius 9 centered at the origin, and the flux integral ∫∫S1F⋅dS=4.
Let S2 be a sphere of radius 81 centered at the origin, and consider the flux integral ∫∫S2F⋅dS.

(A) If the magnitude of F is inversely proportional to the square of the distance from the origin,what is the value of ∫∫S2F⋅dS?

(B) If the magnitude of F is inversely proportional to the cube of the distance from the origin, what is the value of ∫∫S2F⋅dS?


2. The attempt at a solution

So the flux integral over S1 has a value of 4. The Flux integral can be evaluated by multiplying the magnitude of F times the surface area of the sphere: 4pi r^2. Solving for the magnitude of F I get that it's (1/pi r^2).

For part A, I just used this magnitude because it matches the fact that it has to be inversely proportional to the square of the distance from the radius. And so I got the right answer which is 4.

For part B however just increasing the proportionality to be (1/pi r^3) doesn't work, so I don't know what I'm missing here.

I'd like to know first, how we know that the flux integral is equal to the magnitude of F times the surface area of whatever given surface there is. I was just told this, and I can't find it on the textbook, so I'd like to first understand what is the methodology used behind solving this problem.

Thank you. Also, does anyone know of a resource similar to Paul's Online Math Notes that would cover Surface Integral more thoroughly?

 

[gabbagabbahey]

The Flux integral can be evaluated by multiplying the magnitude of F times the surface area of the sphere: 4pi r^2.

This is only true if \textbf{F} is both radial (so that it is normal to the surface) and spherically symmetric. Otherwise, \int_{\mathcal{S}_1}\textbf{F}\cdot d\textbf{S}\neq 4\pi r^2F.

 

[lanedance]

the 1/r potential leads to a 1/r^2 radial vector field which is the form of many physical fields (gravity, electric etc.). This is a special field as you've shown as ttal flux is conserved, under certian condition, in this case for any bounded surface containing the origin.

Then as you've found the 1/r^3 magnitude radial field does not have that property...

Do you know about divergence, and the divergence theorem?