A constant motion
This is a mathematical physics question, which is a little different from our usual fare. We suspect some of you will need to do a little research on this and so we hope you learn a bit.
You live in two spatial dimensions, and your world is a smooth, closed two-dimensional surface (instead of our usual world with 3 spatial dimensions). A closed two-dimensional surface could be a sphere, torus, torus with more than one hole, Klein bottle, etc. In your world, you possess something unique - you are the only person to possess an electric charge. Unfortunately, your charge is very, very small in magnitude: it is a test charge with magnitude . You'd like to put your test charge down and take a nap, but you notice that no matter where you place your test charge (anywhere on the surface), it will accelerate (albeit with a small acceleration) away from you in some direction.
What is the Euler characteristic of the surface you live on?
The answer is 0.
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
Since the test charge accelerates at every point of the surface, there must be a non-vanishing electric field over the surface. By the Poincare-Hopf index theorem, the only Euler characteristic for which this is possible is 0.