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.