# Magnetic field lines

1. Sep 6, 2010

### jostpuur

Suppose that a function $B:\mathbb{R}^n\to\mathbb{R}^n$ and $c:\mathbb{R}\to\mathbb{R}^n$ are defined such that $c$ is differentiable, and

$$\dot{c}(t) = B(c(t))$$

for all $t$. The question is that what must be assumed of $B$, so that it would become possible to prove that

$$c(T)=c(0)$$

with some $T\neq 0$?

2. Sep 7, 2010

### n1person

Do you mean prove that c(dot)(T) = c(dot)(0)?

3. Sep 7, 2010

### jostpuur

No. I mean that the curve comes back to where it started from. (Not that it would point in the same direction at least twice.)

4. Sep 7, 2010

### klondike

$$\oint _{\partial S}B \bullet ndS = 0$$ or B must be divergence free.

5. Sep 7, 2010

### jostpuur

That answer is incorrect.

$n=2$, $B(x)=(x_1,-x_2)$, $c(t)=(e^t,e^{-t})$ give a counter example.

Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Threads - Magnetic field lines Date
I Simulation of magnetic field Oct 29, 2016
I Magnetic vector potential of infinite straight wire Aug 30, 2016
Simple divergence theorem questions Oct 6, 2014
Cross Product and Magnetism Jul 26, 2009
Magnet faling near an iron wll Mar 4, 2009