Flux of a Vector Field on a Sphere

In summary, The conversation discusses how to compute the flux of a vector field out of a sphere using F.n and the normal vector to the sphere. The suggested method is to use the divergence of the equation of the sphere, which is equal to 2x i + 2y j + 2z k, and the field is found to be 2/r.
  • #1
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Homework Statement


Consider the vector field:
F = r/r3

where r = xi + yj + zk

Compute the flux of F out of a sphere of radius a centred at the origin.

Homework Equations





The Attempt at a Solution


Hi everyone,

I have: flux = [tex]\int[/tex]F.dA

I can't use Gauss' Law, because the field will not be defined at the origin.

Instead, I want to use F.n, where n is the normal vector to the sphere.

Is it correct that the normal vector is the div of the equation of the sphere?

ie. n = div (x^2 + y^2 + z^2 = a^2)

= 2x i + 2y j + 2z k

and then F.n = 2/r

Is this correct so far?
Thanks for any help
 
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  • #2
Looks good. the divergence is how fast the field falls off, so it has to be in the direction of the normal vector.
 

1. What is flux in a vector field?

Flux in a vector field is a measure of the flow of a vector quantity, such as velocity or force, through a surface. It represents the amount of the vector field passing through the surface per unit area.

2. How is flux in a vector field calculated?

Flux in a vector field is calculated by taking the dot product of the vector field and the unit normal vector to the surface, and then integrating this over the surface. The resulting value is a measure of the flow of the vector field through the surface.

3. What is the difference between positive and negative flux in a vector field?

Positive flux in a vector field indicates that the vector field is flowing outwards through the surface, while negative flux indicates that the vector field is flowing inwards through the surface. This is determined by the direction of the unit normal vector to the surface.

4. How is flux in a vector field affected by the shape and orientation of the surface?

The shape and orientation of the surface can greatly affect the flux in a vector field. A larger surface area or a surface that is more perpendicular to the vector field will result in a higher flux, while a smaller surface area or a surface that is more parallel to the vector field will result in a lower flux.

5. What are some real-world applications of flux in vector fields?

Flux in vector fields has many practical applications, such as in fluid dynamics, electromagnetism, and heat transfer. It is also used in various engineering and scientific fields, such as in the design of wind turbines, predicting weather patterns, and studying fluid flow in blood vessels or airways.

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