Calculating Flux Through Cylinder w/ Vector Field

[-EquinoX-]

Homework Statement

Find the flux of the vector field through the surface of the closed cylinder of radius c and height c, centered on the z-axis with base on the xy-plane.

Homework Equations

The Attempt at a Solution

Can I just use the divergence theorem here? Find the divergence of the vector field and then multiply it by the volume of the cylinder?

 

[HallsofIvy]

Yes, you can use the divergence theorem but the is not "find the divergence of the vector field and then multiply it by the volume of the cylinder" unless the divergence is a constant. By the divergence theorem, the integral over the surface is the integral of the divergence over the cylinder.

 

[-EquinoX-]

so if it's not a constant then I should integrate it right

 

[HallsofIvy]

Yes!

 

[-EquinoX-]

if I have:

$$\vec{F} = 6x\vec{i} + 6y\vec{j}$$

then according to greens theorem this is 0 right? so therefore the flux is 0 as well?

 

[-EquinoX-]

bump!

 

[dx]

No, it's not zero. You don't use green's theorem in this problem. The theorem you have to use is Gauss' divergence theorem. First find the divergence of F. What is the divergence of (6x, 6y)?

 

[-EquinoX-]

it's just 12... and then multiply it by the volume right, which is $$\pi c^3$$ so it's $$12\pi c^3$$

 

[dx]

That's right. Now just use the divergence theorem. The flux out of the cylinder is ∫(div F)dV over the volume of the cylinder.

 

[-EquinoX-]

what if I am asked to find the limit? is it just 12? or no the flux density is 12 right.. what if I am asked this:

http://img245.imageshack.us/img245/751/limw.th.jpg

 

[dx]

What limit?

 

[-EquinoX-]

read my attachment above.. I don't understand the question it self

 

[dx]

Yes, the answer is 12.

 

[-EquinoX-]

so the limit and the flux density are both 12?

 

[dx]

If by 'flux density' you mean 'divergence', then yes.

 

[-EquinoX-]

yea flux density is divergence