Calculating Flux Through Cylinder w/ Vector Field

Click For Summary

Homework Help Overview

The discussion revolves around calculating the flux of a vector field through the surface of a closed cylinder, specifically one with a radius and height both equal to c, positioned along the z-axis. The problem involves applying the divergence theorem and understanding the implications of the divergence of the vector field.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants explore the use of the divergence theorem and question whether the divergence of the vector field needs to be constant. There is discussion about integrating the divergence if it is not constant. Some participants also consider the implications of applying Green's theorem and the relationship between divergence and flux density.

Discussion Status

The conversation is actively exploring the application of the divergence theorem, with some participants confirming the need to find the divergence of the vector field. There is a recognition of the relationship between divergence and flux density, although not all aspects of the problem have been resolved.

Contextual Notes

Participants are navigating assumptions about the nature of the divergence and its implications for calculating flux. There is also a reference to an external image that raises additional questions about limits related to the problem.

-EquinoX-
Messages
561
Reaction score
1

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?
 
Physics news on Phys.org
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.
 
so if it's not a constant then I should integrate it right
 
Yes!
 
if I have:

[tex]\vec{F} = 6x\vec{i} + 6y\vec{j}[/tex]

then according to greens theorem this is 0 right? so therefore the flux is 0 as well?
 
bump!
 
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)?
 
it's just 12... and then multiply it by the volume right, which is [tex]\pi c^3[/tex] so it's [tex]12\pi c^3[/tex]
 
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.
 
  • #10
Last edited by a moderator:
  • #11
What limit?
 
  • #12
read my attachment above.. I don't understand the question it self
 
  • #13
Yes, the answer is 12.
 
  • #14
so the limit and the flux density are both 12?
 
  • #15
If by 'flux density' you mean 'divergence', then yes.
 
  • #16
yea flux density is divergence
 

Similar threads

  • · Replies 1 ·
Replies
1
Views
2K
Replies
6
Views
2K
  • · Replies 3 ·
Replies
3
Views
2K
Replies
26
Views
2K
Replies
5
Views
1K
Replies
7
Views
2K
Replies
8
Views
3K
Replies
2
Views
2K
  • · Replies 6 ·
Replies
6
Views
3K
  • · Replies 1 ·
Replies
1
Views
2K