What Is the Correct Approach to Calculate Flux Through a Sphere?

[renegade05]

Homework Statement

What is the flux of r through a spherical surface of radius a?

Homework Equations

I'm guessing I should use a surface integral? ∫v.da ?

The Attempt at a Solution

Plugging in: I would get ∫r.da ? but what is a small patch of a sphere?

I'm kind of confused. Not really sure what r is even. Is this just any vector going through the sphere?

Please help

thanks

 

[CAF123]

Plugging in: I would get ∫r.da ? but what is a small patch of a sphere?

The da there is a vector normal to an infinitesimal surface area element on the sphere. You can find it by parametrising the surface by some appropriate coordinates, thereby finding two vectors lying in the plane of the sphere which infinitesimally span a flat plane. This constitutes the area element you speak of. The derivation is done in most books.

I'm kind of confused. Not really sure what r is even. Is this just any vector going through the sphere?

Yes, it corresponds to the special case where the vector field coincides with position vector, so in Cartesian coordinates reads $\mathbf r = \langle x,y,z \rangle$. Your problem suggests working with the equivalent expression in a different set of coordinates.

 

[vanhees71]

Well, it might be simpler to use Gauss's integral theorem
$$\int_{V} \mathrm{d}^3 r \vec{\nabla} \cdot \vec{V} = \int_{\partial V} \mathrm{d} \vec{a} \cdot \vec{V}.$$

 

[HallsofIvy]

Homework Statement

What is the flux of r through a spherical surface of radius a?

Do you mean a sphere of radius a with center at the origin?

Homework Equations

I'm guessing I should use a surface integral? ∫v.da ?

Yes, you should. And to find what you call "da" (I would call it "$d\vec{S}$") write the surface in parametric equations- the standard equations for spherical coordinates with "$\rho$" set to "a":
$x= a cos(\theta) sin(\phi)$, $y= a sin(\theta) sin(\phi)$, and $z= a cos(\phi)$.
You can write that as a "position vector" for any point on the surface of the sphere depending on $\theta$ and $\phi$:
$$\vec{r}= a cos(\theta) sin(\phi)\vec{i}+ a sin(\theta)sin(\phi)\vec{j}+ a cos(\phi)\vec{k}$$.
The derivatives with respect to $\theta$ and $\phi$ are vectors in the tangent plane at each point on the surface of the sphere:
$$\vec{r}_\theta= -a sin(\theta) sin(\phi)\vec{i}+ a cos(\theta)sin(\phi)\vec{j}$$
$$\vec{r}_\phi= a cos(\theta)cos(\phi)\vec{i}+ a sin(\theta)cos(\phi)\vec{j}- a sin(\phi)\vec{k}$$.

Finally, the cross product of those two vectors
$$\left|\begin{array}{ccc}\vec{i} & \vec{j} & \vec{k} \\ a cos(\theta)cos(\phi) & a sin(\theta)cos(\phi) & -a sin(\phi) \\ -a sin(\theta)sin(\phi) & a cos(\theta)sin(\phi) & 0 \end{array}\right|= a^2cos(\theta)sin^2(\phi)\vec{i}+ a^2sin(\theta)sin^2(\phi)\vec{j}+ a^2sin(\phi)cos(\phi)\vec{k}$$
is perpendicular to the sphere and contains "area information"- the vector differential of surface area is $(a^2cos(\theta)sin^2(\phi)\vec{i}+ a^2sin(\theta)sin^2(\phi)\vec{j}+ a^2sin(\phi)cos(\phi)\vec{k})d\theta d\phi$

The Attempt at a Solution

Plugging in: I would get ∫r.da ? but what is a small patch of a sphere?

I'm kind of confused. Not really sure what r is even. Is this just any vector going through the sphere?

Please help

thanks

$\vec{r}$ is the "position vector" of a point: $x\vec{i}+ y\vec{j}+ z\vec{k}$ in Cartesian coordinates and $a cos(\theta)sin(\phi)\vec{i}+ a sin(\theta)sin(\phi)\vec{j}+ a cos(\phi)\vec{k}$ in these coordinates.

Take the dot product, $\vec{r}d\vec{S}$, of those two vectors and integrate with $0\le\theta\le 2\pi$ and $0\le\phi\le \pi$.

(of course, Vanhees71 may well be right- it might be easier to use Gauss' integral theorem.)

 

[renegade05]

so I get an answer of :

$$\frac{8\pi a^3}{3}$$

 

[CAF123]

so I get an answer of :

$$\frac{8\pi a^3}{3}$$

hey, try the question using the divergence theorem as vanhees71 suggested and compare your results :)

 

[renegade05]

hey, try the question using the divergence theorem as vanhees71 suggested and compare your results :)

Yup got the same answer :)

 

[vanhees71]

How did you come to this result?

 

[renegade05]

How did you come to this result?

By plugging in the volume of the sphere (4pi a^3 /3) into the divergence theorem. My answer is two times that. I think that makes sense since the flux would indicate the outward and inward flux?

 

[renegade05]

By plugging in the volume of the sphere (4pi a^3 /3) into the divergence theorem. My answer is two times that. I think that makes sense since the flux would indicate the outward and inward flux?

Is it not correct?

 

[CAF123]

Is it not correct?

What is $\nabla \cdot \mathbf r$, where $\mathbf r$ is the position vector? (A calculation most easily done in Cartesian coordinates)

 

[renegade05]

What is $\nabla \cdot \mathbf r$, where $\mathbf r$ is the position vector? (A calculation most easily done in Cartesian coordinates)

Whoops it's 4 pi a ^3

 

[vanhees71]

Now you've got it!:thumbs:

 

[vela]

Whoops it's 4 pi a ^3

So did you figure out why you got a different result from the surface integral?