Calculating Flux Across a Sphere Surface | Vector Field F=r+grad(1/magnitude(r))

In summary, the task is to compute the flux of a given vector field across the surface of a sphere. The flux is equal to the double integral of the vector field multiplied by the unit normal of the surface. This can be simplified into two parts, with the first part resulting in a flux of 4*pi*a^3. The second part involves using polar coordinates and integrating to get a final answer of (4*pi*a^3)-(4*pi*a^(-1/2)).
  • #1
JaysFan31

Homework Statement


Consider the vector field
F= r + grad(1/magnitude(r)).
Compute the flux (double integral F ndS) of F across the surface of the sphere x^2+y^2+z^2=a^2 where a>0. ndS is the vector element of surface with n the unit normal which here is assumed to point away from the enclosed volume.

Homework Equations


Flux equals the double integral of F*n dS where * is the dot product between the vector field F and the unit normal n.

The Attempt at a Solution


If I break it into two parts, I get F=r and F= grad(1/magnitude(r))

F=r just has flux 4*pi*a^3 since F=xi+yj+zk and F*N=a where * is the dot product. Note that N=(x/a)i+(y/a)j+(z/a)k. I know this just by looking at it, but how would I set up this double integral to get 4*pi*a^3?

F=grad(1/magnitude(r))=(-x)/(x^2+y^2+z^2)^{3/2}i-y/(x^2+y^2+z^2)^{3/2}j-z/(x^2+y^2+z^2)^{3/2}k

I can simplify it by using a^2=x^2+y^2+z^2 and use polar coordinates, but how do I do this?

I get that grad(1/magnitude(r))=
[(-x)/a^(3/2)]i + [(-y)/a^(3/2)]j + [(-z)/a^(3/2)]k

I know that the unit normal N=(x/a)i + (y/a)j + (z/a)k.

Thus the dot product of grad(1/magnitude(r)) and the unit normal=
(-x^2)/a^(5/2) + (-y^2)/a^(5/2) + (-z^2)/a^(5/2) =
(-a^2)/(a^5/2)= -1/a^(3/2).

Now how do I integrate this to get the flux of the surface?

Is it,
integral from 0 to 4pi, integral from 0 to a^2 of (-a^(-3/2))dr dtheta?
 
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  • #2
Is spherical coordinates easier? How do I this?

OK, does an answer of (4*pi*a^3)-(4*pi*a^(-1/2))? make sense for the final answer?
 
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1. What is flux?

Flux is a measure of the flow of a vector field through a given surface. It is represented by the symbol Φ and can be calculated by taking the dot product of the vector field and the surface's normal vector.

2. How is flux calculated across a sphere surface?

To calculate flux across a sphere surface, we use the formula Φ = ∫∫ F ⋅ dS, where F is the vector field and dS is the differential surface element of the sphere. This integral can be simplified using the divergence theorem to Φ = ∫∫∫ (div F) dV, where div F is the divergence of the vector field.

3. What is the meaning of the vector field F=r+grad(1/magnitude(r))?

This vector field represents the force of attraction towards the origin with a decreasing magnitude as the distance from the origin increases. The first term, r, represents the direction towards the origin and the second term, grad(1/magnitude(r)), represents the magnitude of the force.

4. What is the significance of using a sphere surface for calculating flux?

A sphere is a symmetrical shape, which makes it easier to calculate flux compared to other irregular shapes. Additionally, using a sphere allows us to apply the divergence theorem, which simplifies the calculation of flux.

5. Can flux be negative?

Yes, flux can be either positive or negative depending on the direction of the vector field and the surface's orientation. A positive flux indicates a flow out of the surface, while a negative flux represents a flow into the surface.

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