Why is the unit normal of a sphere important in vector calculations?

[Hypersquare]

I was looking at this example:

http://keep2.sjfc.edu/faculty/kgreen/vector/block3/flux/node10.html

and was confused between the difference between $\hat{}n$ and $\vec{}r$

Why is the original vector field not given in terms of a unit vector? And what difference does this make?

Thanks :)

 

[Hypersquare]

Sorry that's supposed to be n hat and the vector r, I am a latex noob.

 

[Hypersquare]

I also don't quite get why the is unit vector is not just r hat

 

[HallsofIvy]

"r", with the arrow over it is the "position vector" at a given point on the sphere. n with a hat is the unit vector in that direction. I presume they are using "n" to represent the unit vector because it is "normal" to the spherical surface and "n" is the standard notation for a normal vector.

For a sphere with center at the origin, the normal vector at any point is in the direction of the position vector. For any other surface that would not be true.

 

[Hypersquare]

Thank you Ivy. Very helpful.

 

[Hypersquare]

I got $\vec{f}$.$\hat{n}$ as 1/$r^{4}$ not 1/$r^{2}$ as they got. What have I done wrong?

 

[HallsofIvy]

There is no "f" so I assume you mean "F" at the site linked to. That is defined by
$$\vec{F}= \frac{\vec{r}}{r^3}$$
$\frac{\vec{r}}{r}$ is the unit vector $\vec{n}$ normal to the sphere so the length of $\vec{F}$ is $1/r^2$. I don't know how you would have gotten $1/r^4$.

 

[Hypersquare]

I got it by doing:

$\vec{F}$ . $\hat{n}$ = $\frac{\vec{r}}{r^{3}}$ .$\frac{\vec{r}}{r}$ = $\frac{1}{r^{4}}$

I don't see what is wrong with that.