Flux of vector field proportional to 1/r^3 through sphere

In summary, the conversation discusses the computation of the flux of a vector field \vec F out of a sphere of radius "a" centered at the origin. The Gauss Divergence Theorem is mentioned, but it cannot be applied due to the singularity of the vector field at the origin. The solution is to compute the flux directly from the definition.
  • #1
jameson2
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0

Homework Statement


Consider the vector field [tex] \vec F=\frac{\vec r}{r^3}[/tex] with [tex]\vec r=x\hat{i}+y\hat{j}+z\hat{k} [/tex]. Compute the flux of [tex] \vec F[/tex] out of a sphere of radius "a" centred
at the origin.

Homework Equations


The Gauss Divergence Theorem [tex]\int_D dV \nabla \bullet F=\int_S F\bullet dA [/tex]


The Attempt at a Solution


I think I'm either missing or not understanding something in this question. When I compute [tex] \nabla \bullet F[/tex], I get zero, which means the flux is zero, I think. But this doesn't seem right at all. What am I missing?
 
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  • #2
The vector field [tex]\vec{F}[/tex] has a singularity at the origin, so you can't use the divergence theorem (at least not in its most common form). Compute the flux directly from the definition instead.
 

1. What is a flux of vector field?

A flux of vector field is a measure of the flow of a vector field through a given surface. It represents the amount of the vector field passing through the surface per unit area.

2. How is the flux of a vector field proportional to 1/r^3?

The flux of a vector field is proportional to 1/r^3 because the surface area of a sphere is also proportional to 1/r^3. As the distance from the center of the sphere increases, the surface area of the sphere decreases, resulting in a decrease in the flux of the vector field.

3. What is the significance of the proportionality constant in this equation?

The proportionality constant in this equation represents the strength of the vector field. It determines how much of the field will pass through the surface at a given distance from the center of the sphere.

4. How is the flux of a vector field through a sphere calculated?

The flux of a vector field through a sphere can be calculated by multiplying the strength of the vector field by the surface area of the sphere at a given distance from the center. This can be represented by the equation: Flux = k * (1/r^3).

5. What are some real-life applications of this concept?

This concept is commonly used in physics and engineering to understand and analyze the behavior of electric and magnetic fields, as well as fluid flow. It is also used in astronomy to study the gravitational forces between celestial bodies.

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