Is it true that the field lines of an electric field are identical to the trajectories of a charged particle with initial velocity zero? If so, how can one prove it?
The claim is from a german physics book from Nolting "Grundkurs theoretische Physik 3 – Elektrodynamik" page 51, let me quote:
Man führt Feldlinien ein und versteht darunter die Bahnen, auf denen sich ein kleiner, positiv geladener, anfangs ruhender Körper aufgrund der Coulomb-Kraft (2.11) bzw. (2.20) fortbewegen würde.
In english:
One introduces field lines and means trajectories along which a small, positively charged, initially resting body moves due to the Coulomb-foce (2.11) resp. (2.20).
2.11 is just the coulomb law, 2.20 is $F = q E$.
(If someone has a better translation, feel free to edit it).
I don't see why this should be true. So it would be great to see a proof or a counterexample with solved equations of motion.
For a magnetic field this claim is obviously wrong since the Lorentz Force depends linearly on the velocity.
Are there other physical fields where the claim is analogously true?
Edit: The answers show that the claim is not true in general but holds in the special case of a highly viscous medium. Is this also the case for moving charged cotton along the field lines in air, as shown in this animation: http://www.leifiphysik.de/web_ph09_g8/grundwissen/01e_feldlinien/01e_feldlinien.htm ?
Do you have any references or more details for this viscous media limit?
Do you have any computational counter example why it doesn't hold in general or a simulation which shows that?
Best Answer
No, the statement is false even in the electric case. At the very beginning, the acceleration is $\vec a \sim \vec E$ so they have the same direction at $t=0$: the tangents agree.
However, as soon as the particle reaches some nonzero velocity $\vec v \neq 0$, its acceleration is still $\vec a\sim \vec E$, in the direction of the field lines, however its velocity – and it's the velocity that determines the tangent direction of the trajectory – is not proportional to the acceleration.
Again, the field lines have direction corresponding to the acceleration at the given point but the trajectories have directions given by the velocity and $\vec v$ isn't a multiple of $\vec a$ in general.
Imagine a simple example above. If you start with a positive charge, and $\vec v=0$ and very close to the positive-charge source above, they will repel and the moving charge will quickly achieve a huge speed. This speed will act as inertia that will make the trajectories much more straight than the field line and the discrepancy will become more extreme if the initial position of the moving charge will be even closer to the positive source.
You would only get the relationship "field lines are equal to trajectories" if you were stopping the moving test charge at every moment and slowly allowed the field to accelerate from scratch after each infinitesimal amount of time.