| Re: Are the field lines the trajectory of a particle with mass M? [message #38565 is a reply to message #38564] |
Sun, 14 March 2004 08:01  |
James Kuyper
Messages: 425
Registered: March 2000
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Senior Member |
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pakachunka wrote:
>
> I believe the field lines are not the trajectories... but a friend of
> mine is driving me crazy, because he says they are.
>
> How can I demonstrate that field lines are not the trajectories?
>
> I mean: what are field lines, to start?
Your question is too broad. There are many kinds of fields. Field lines
are associated with vector fields. A typical vector field may be
described by a vector-valued function v(x,y,z,t), which means it has a
single size and direction for every meaningful combination of x,y,z, and
t. Field lines are lines associated with a vector field that are
arranged so that at every point, the tangent to the line at that point
is in the same direction as the vector field at that point. In other
words, the vector field tells the lines where to go.
Now, if the two of your are talking about the velocity field of a fluid,
then the field lines are indeed exactly the trajectories of the
individual particles that make up the fluid.
However, fluid dynamics is fairly complicated, and your question gives
the impression that you're at a fairly elementary level in physics. In
that case, the fields you're most likely to run into aren't velocity
fields, but electrical or gravitational fields. For example, the
electrical field in the vicinity of a particle with charge Q, at a
position <x0,y0,z0> has an associated static electrical field at a point
<x,y,z> which is given by
E(x,y,z) = kQ<x-x0,y-y0,z-z0>/r^2
where r^2=(x-x0)^2+(y-y0)^2+(z-z0)^2.
The important thing is that the electrical field is NOT a velocity
field, and therefore the field lines are not in general the same as the
particle trajectories. If the electrical field provides the only force
that is acting on a particle with a charge of q and a mass of m, then it
will feel an acceleration of Eq/m. Acceleration is not velocity, it's
the first derivative of the velocity. There's a connection between the
electrical field lines and the trajectories of the particals, but it's
not a simple one.
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