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Unit normal of sphere

  1. Oct 4, 2012 #1
    I was looking at this example:

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

    and was confused between the difference between [itex]\hat{}n[/itex] and [itex]\vec{}r[/itex]

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

    Thanks :)
     
  2. jcsd
  3. Oct 4, 2012 #2
    Sorry thats supposed to be n hat and the vector r, Im a latex noob.
     
  4. Oct 4, 2012 #3
    I also dont quite get why the is unit vector is not just r hat
     
  5. Oct 4, 2012 #4

    HallsofIvy

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    "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.
     
  6. Oct 4, 2012 #5
    Thank you Ivy. Very helpful.
     
  7. Oct 4, 2012 #6
    I got [itex]\vec{f}[/itex].[itex]\hat{n}[/itex] as 1/[itex]r^{4}[/itex] not 1/[itex]r^{2}[/itex] as they got. What have I done wrong?
     
  8. Oct 4, 2012 #7

    HallsofIvy

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    There is no "f" so I assume you mean "F" at the site linked to. That is defined by
    [tex]\vec{F}= \frac{\vec{r}}{r^3}[/tex]
    [itex]\frac{\vec{r}}{r}[/itex] is the unit vector [itex]\vec{n}[/itex] normal to the sphere so the length of [itex]\vec{F}[/itex] is [itex]1/r^2[/itex]. I don't know how you would have gotten [itex]1/r^4[/itex].
     
  9. Oct 5, 2012 #8
    I got it by doing:

    [itex]\vec{F}[/itex] . [itex]\hat{n}[/itex] = [itex]\frac{\vec{r}}{r^{3}}[/itex] .[itex]\frac{\vec{r}}{r}[/itex] = [itex]\frac{1}{r^{4}}[/itex]

    I dont see what is wrong with that.
     
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