Animating magnetic field lines on a circle

Question

So I want to create a changing vector as a function of theta and have it's origin stay on the path of a circle. The vector I want to plot is:

Manipulate[
 Show[Graphics[
   Arrow[{{Cos[t*3.6 °], 
       Sin[t*3.6 °]}, {-Sin[t*3.6 °], 
       Cos[t*3.6 °]}}*{Sin[t*3.6 °], 
      2 Cos[t*3.6 °]}]]], {t, 0, 100, .25}]

I apologize that I'm not very good at mathematica; I'm pretty new and trying to learn. I know how to make a circle, but I don't know how to make the vector go in a circle. Any tips or ideas?

Btw, I did this in terms of time and wanted one full circle in 100 minutes, hence the t*3.6 ° part.

Updated code below:

Manipulate[
Show[Graphics[{Red, Circle[{0, 0}, 8]}], 
Graphics[Arrow[{{Cos[θ*3.6 °], 
Sin[θ*3.6 °]}, {-Sin[θ*3.6 °], 
Cos[θ*3.6 °]}}*{Sin[θ*3.6 °], 
2 Cos[θ*3.6 °]}]], 
PlotRange -> {{-10, 10}, {-10, 10}}], {θ, 0, 100, .11, 
Appearance -> "Open"}]

I want the origin of that vector to sit on the edge of the circle and in the two periods, make one full period around the circle such that it is pointing up at the top and bottom poles and down and the two equators. I was thinking of using offset, but it keeps giving me an error. What I have so far is this:

Manipulate[
Show[Graphics[{Red, Circle[{0, 0}, 8]}], 
  Graphics[Arrow[{{Cos[t*3.6 °], 
    Sin[t*3.6 °]}, {-Sin[t*3.6 °], 
    Cos[t*3.6 °]}}*{Sin[t*3.6 °], 
    2 Cos[t*3.6 °]}, 
    Offset[{-8 Sin[t*3.6 °], 8 Cos[t*3.6 °]}, {0, 
    0}]], PlotRange -> {{-10, 10}, {-10, 10}}]], {t, 0, 100, .11, 
    Appearance -> "Open"}]

I guess if it makes it more clear, I want the origin of the vector to line up with the dot at all times t in the following plot:

Manipulate[
 Show[Graphics[{Red, Circle[{0, 0}, 8]}], 
   Graphics[Arrow[{{Cos[t*3.6 °], 
     Sin[t*3.6 °]}, {-Sin[t*3.6 °], 
     Cos[t*3.6 °]}}*{Sin[t*3.6 °], 
     2 Cos[t*3.6 °]}], PlotRange -> {{-10, 10}, {-10, 10}}], 
   Graphics[Point[{-8 Sin[t*3.6 °], 
   8 Cos[t*3.6 °]}]]], {t, 0, 100, .1, 
Appearance -> "Open"}]

Does that make it more clear? I really really appreciate all of your help.

Answer 1

I embellished a bit to show the circle as well, but that should be easy to remove if you like. Also, note that since your vector points go as products of Sin and Cos, a full circle is achieved in only 180 degrees.

Manipulate[
 Show[
  Graphics[{Red, Circle[{0, 0.5}, .5]}],
  Graphics[
   Arrow[
    {{Cos[t 1.8 °], 
       Sin[t 1.8 °]}, {-Sin[t 1.8 °], 
       Cos[t 1.8 °]}}*{Sin[t 1.8 °], 
      2 Cos[t  1.8 °]}
    ]], PlotRange -> {{-1.1, 1.1}, {-0.1, 2.1}}, AspectRatio -> 1]
 , {{t, 0, Style["t", 25]}, 0, 100, .25, Appearance -> "Open"}]