| Topic: |
Science > Physics |
| User: |
"Jeremy Price" |
| Date: |
08 Dec 2003 11:40:33 PM |
| Object: |
Flux? |
I was thinking about flux the other day, and I was wondering if you could
define flux for other fields. For example, could you talk about
gravitational flux? Can you talk about flux with any vector field? I would
think so, I would think gravitational flux would look a lot like electric
and magnetic...
phi = integral F*dA
where F and A are vectors, if we take a sphere, and a constant F,
phi = F integral dA = F 4pi r^2 = (Gm/r^2) (4pi r^2) = 4piGm.
So we'd have something like Gauss' law, but for gravity:
phi_G = integral F*dA = 4piGm
Which seems to make sense, a sphere with no mass inside it should have no
net flux through it. And also, it shows that a hollow sphere would have no
net gravitational field inside it, just like Gauss' law says about the
electric field inside a charged sphere! That's a much easier way to prove it
than with the complicated integrals that you might otherwise do!
What would d phi_G/dt represent? Obviously, the change in flux... but I
mean, is it related to anything else, like how Faraday's law says -d phi_B /
dt = emf?
Could anything else be done with this? Is there a reason we don't seem to
talk about gravitational flux?
Thanks,
Jeremy
.
|
|
| User: "Bilge" |
|
| Title: Re: Flux? |
09 Dec 2003 12:37:30 PM |
|
|
Jeremy Price:
I was thinking about flux the other day, and I was wondering if you could
define flux for other fields. For example, could you talk about
gravitational flux? Can you talk about flux with any vector field? I would
think so, I would think gravitational flux would look a lot like electric
and magnetic...
You can talk about flux with any _vector_ field, since the flux is
just the integral of the divergence of the vector field. Gravity is
not a vector field. Right off-hand, I'm not certain how one would
define the flux of the gravitational field, except perhaps by the
flux vector for dust or the stress-energy tensor.
phi = integral F*dA
where F and A are vectors, if we take a sphere, and a constant F,
phi = F integral dA = F 4pi r^2 = (Gm/r^2) (4pi r^2) = 4piGm.
So we'd have something like Gauss' law, but for gravity:
phi_G = integral F*dA = 4piGm
Yes, you can do this, but you are considering newtonian gravity which
is derivable from a scalar potential.
.
|
|
|
|
| User: "Gregory L. Hansen" |
|
| Title: Re: Flux? |
09 Dec 2003 09:40:21 AM |
|
|
In article <br3n91$ce2$1@nntp6.u.washington.edu>,
Jeremy Price <cfgauss@u.PANTS.washington.edu> wrote:
I was thinking about flux the other day, and I was wondering if you could
define flux for other fields. For example, could you talk about
gravitational flux? Can you talk about flux with any vector field?
Yes.
I would
think so, I would think gravitational flux would look a lot like electric
and magnetic...
Coulomb's law: div E = 4 pi rho
Newton's law: div F = -4 pi G rho_m
with rho_m the mass distribution. And now pretty much everything you know
about electromagnetism can be translated to gravity. Including a
gravitomagnetic force if you define a speed of propagation c_g.
--
"The preferred method of entering a building is to use a tank main gun
round, direct fire artillery round, or TOW, Dragon, or Hellfire missile to
clear the first room." -- THE RANGER HANDBOOK U.S. Army, 1992
.
|
|
|
|
| User: "Bjoern Feuerbacher" |
|
| Title: Re: Flux? |
09 Dec 2003 03:08:03 AM |
|
|
Jeremy Price wrote:
I was thinking about flux the other day, and I was wondering if you could
define flux for other fields. For example, could you talk about
gravitational flux? Can you talk about flux with any vector field?
Yes.
I would
think so, I would think gravitational flux would look a lot like electric
and magnetic...
phi = integral F*dA
where F and A are vectors, if we take a sphere, and a constant F,
phi = F integral dA = F 4pi r^2 = (Gm/r^2) (4pi r^2) = 4piGm.
Careful here: F is an attracting force, so you have to include
a minus sign!
So we'd have something like Gauss' law, but for gravity:
phi_G = integral F*dA = 4piGm
Correct if you include a minus sign.
Which seems to make sense, a sphere with no mass inside it should have no
net flux through it. And also, it shows that a hollow sphere would have no
net gravitational field inside it, just like Gauss' law says about the
electric field inside a charged sphere!
Err, not necessarily. From
integral_(closed surface) F*dA = 0,
it does *not* follow directly that
F = 0
inside the volume enclosed by the surface.
However, what you could do is: assume that F is spherically symmetric
and perpendicular to the surface; then you get for any sphere
inside the hollow sphere:
integral_(closed surface) F*dA = F(r)*A = 0,
and because A is not zero, this then implies that F(r) = 0. Because this
work for *every* r inside the hollow sphere (well, with the exception
of r = 0, because there A = 0), you have then proven that F = 0
everywhere
inside the hollow sphere. But remember that this conclusion is based
on the assumptions above! (which are admittedly quite sensible)
That's a much easier way to prove it
than with the complicated integrals that you might otherwise do!
Agreed.
What would d phi_G/dt represent? Obviously, the change in flux... but I
mean, is it related to anything else, like how Faraday's law says -d phi_B /
dt = emf?
Not in Newtonian physics. If you try to do analogous things with
Einstein's
equations, I think it would turn out that there indeed is something like
"gravitational induction" (sorry, I don't know enough about GRT to give
you
a clear answer here).
Could anything else be done with this?
Probably, yes.
Is there a reason we don't seem to
talk about gravitational flux?
I think that people who work often with gravitational fields (people
in geophysics who want to measure the shape of the earth, for example)
probably *do* use this concept.
Bye,
Bjoern
.
|
|
|
|
|
Related Articles |
|
|
