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13.6 - Flux of a Vector Field
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[13.6] - The flux of a vector field
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Calculate $\unicode{x222F}_{S} \myv F(x,y,z)\cdot d\myv S$ for 
$\myv F(x,y,z)=\frac{1}{(x^2+y^2+z^2)^{3/2}}(x\uv i+y\uv j+z\uv k)$
 integrated over the surface, $S$ of a sphere of radius $R$.

<ol start="0">
<li>The position vector, $\myv r$ has components $\myv r=\langle x,y,z \rangle$.
 Its magnitude is $|\myv r|=\sqrt{x^2+y^2+z^2}=(x^2+y^2+z^2)^{1/2}$. Express $\myv F$ in terms of $\uv r$ and $r$ alone.
<p class="answer">Using the definitions, $F$ can be written as:
$$\begineq \myv F(x,y,z)&=&\frac{1}{(x^2+y^2+z^2)^{3/2}}(x\uv i+y\uv j+z\uv k)\\
&=&\frac{1}{[(x^2+y^2+z^2)^{1/2}]^3}\myv r\\
&=&\frac{\myv r}{[r]^3}=\frac{1}{r^2}\frac{\myv r}{r}=\frac{1}{r^2}\uv r.\\
\endeq$$

<li>We'll need a surface normal (unit) vector which is perpendicular to the surface of the sphere.  

<p>Consider the sphere $x^2+y^2+z^2=R^2$ to be a level surface of the function $G(x,y,z)=x^2+y^2+z^2$. The gradient of a function is always perpendicular to a level curve or level surface, so


compute the gradient $\myv \grad G(x,y,z)$ to this surface. 
<p class="answer">$$\myv \grad G = \myc{G_x,G_y,G_z}=\myc{2x,2y,2z}$$
<li style="margin-top: 12ex">
Does $\myv \grad G$ point inward or outward from the surface of the sphere?
<p class="answer">outward.
<li style="margin-top: 4ex">
Compute an outward *unit* normal vector, $\uv n$ to the sphere.

<p class="answer">That gradient vector above is a normal vector, $\myv n=\myv\grad G$, perpendicular to the sphere.  So, a unit normal vector would be

$$\begineq \uv n&=&\frac{\myv n}{|\myv n|}=
\frac{\myc{2x,2y,2z}}{\sqrt{4x^2+4y^2+4z^2}}\\
&=&\frac{2\myc{x,y,z}}{2\sqrt{x^2+y^2+z^2}}\\
&=&\frac{\myv r}{r}=\uv r.\endeq$$



<li style="margin-top: 12ex">
Now consider the vector field $\myv F(x,y,z)=\frac{1}{(x^2+y^2+z^2)^{3/2}}(x\uv i+y\uv j+z\uv k)$.
  Where is $\myv F$ defined?
<p class="answer">Everywhere except at the origin (where the denominator is 0).

<li style="margin-top: 4ex">
Show that the flux $\unicode{x222F}_S \myv F\cdot d\myv S = 
\unicode{x222F}_S\myv F\cdot \uv n\,dS=4\pi$,
 where $S$ is the spherical surface
$x^2+y^2+z^2=R^2$.  It may be useful to 
know that the surface area of a sphere of radius $R$ is $\unicode{x222F}_S dS=\int_0^\pi
\int_0^{2\pi}(R\,d\theta)(d\phi)=
4\pi R^2$. 
<p class="answer">
We can write $\myv F=\frac 1{r^2}\uv r$ and we figured out that $\uv n=\uv r$. 
So, $$\myv F\cdot \uv n = \frac{1}{r^2}\uv n\cdot\uv n=
\frac{1}{r^2}.$$  
On the surface of the sphere of radius $R$, $r=R$ is constant.
So, The flux integral is:
$$\iint_S\myv F\cdot \uv n\,dS
=\iint_S \frac{1}{R^2}\,dS
=\frac{1}{R^2}\iint_S\,dS
=\frac{1}{R^2} 4\pi R^2 = 4\pi.
$$
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