Science > Physics > Charge conservation and surface currents (or line currents)
| Topic: |
Science > Physics |
| User: |
"" |
| Date: |
10 Sep 2006 02:54:23 PM |
| Object: |
Charge conservation and surface currents (or line currents) |
I would like some clarifications concerning the use of charge
conservation when someone is dealing with surface or line currents. The
integral form is:
/
|J dS = 0
/
S
Note that we are discussing time independent problems, thus there are
no derivatives of time.
Now, let's move to the point of discussion. Consider two line segments
that connect to an arbitrary surface. These line segments could be
either line currents or (if we consider that there is depth) they could
be surface currents. The surfaces are perpedicular to the screen:
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+---+
| |
| |--------
| |
+---+
By simple observation it can be stated that K_1=K_2 or i_1=i_2, where
indices denote the line segment. The same thing can be found by
applying the law of charge conservation. Now let's change the surface:
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+---+
/ /------
/ /
+---+
Again, K_1=K_2. But if we take the law of charge conservation we have:
/
|J dS=0 => K_1=K_2 Cos[phi]
/
S
Hmm, there is a problem here. According to my thoughts we are just
applying the law the wrong way. Even though we have to take the
perpedicular J to the surface, we shouldn't do this if we are dealing
with surface currents or line currents. Am I correct? How do we support
this? Can we say that K=J delta=>J=K/delta, where delta is the
surface's thickness so it follows that:
/ / dS /
|J dS=0 => |K -------=0 => | K dl=0
/ / delta /
S S curve
Note that we are diving dS which is a surface by delta which is a
length, so we get dl which is a length. That means that the surface
integral transforms into a curve integral, thus the need to take the
normal of J or K is eliminated.
And to restate my inquiry: Should we apply the law of charge
conservation differently when there are surface currents? Logic says
yes. But is that the truth?
Regards,
George Prekas
Undergraduate student at the National Technical University of Athens
.
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| User: "Sorcerer" |
|
| Title: Re: Charge conservation and surface currents (or line currents) |
10 Sep 2006 03:39:41 PM |
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<prekgeo@gmail.com> wrote in message
news:1157918062.654896.184450@i3g2000cwc.googlegroups.com...
|I would like some clarifications concerning the use of charge
| conservation when someone is dealing with surface or line currents. The
| integral form is:
| /
| |J dS = 0
| /
| S
| Note that we are discussing time independent problems, thus there are
| no derivatives of time.
Then charge is conserved BY DEFINITION. Obviously it requires
time to change any charge. Even "current" requires time.
| Now, let's move to the point of discussion.
Already discussed:
"clarification dealing with surface or line currents", "time independent".
Androcles
.
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| User: "" |
|
| Title: Re: Charge conservation and surface currents (or line currents) |
11 Sep 2006 02:05:08 AM |
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You are right, but in a simplified approach to the problem you can
consider that there is no charge increase or decrease. I think it's
called static approach or something like that. My question was how
should I calculate the flux of a surface or line current?
Sorcerer wrote:
<prekgeo@gmail.com> wrote in message
news:1157918062.654896.184450@i3g2000cwc.googlegroups.com...
|I would like some clarifications concerning the use of charge
| conservation when someone is dealing with surface or line currents. The
| integral form is:
| /
| |J dS = 0
| /
| S
| Note that we are discussing time independent problems, thus there are
| no derivatives of time.
Then charge is conserved BY DEFINITION. Obviously it requires
time to change any charge. Even "current" requires time.
| Now, let's move to the point of discussion.
Already discussed:
"clarification dealing with surface or line currents", "time independent".
Androcles
.
|
|
|
|
|
| User: "Timo A. Nieminen" |
|
| Title: Re: Charge conservation and surface currents (or line currents) |
11 Sep 2006 03:33:42 PM |
|
|
On Mon, 10 Sep 2006, wrote:
I would like some clarifications concerning the use of charge
conservation when someone is dealing with surface or line currents. The
integral form is:
/
|J dS = 0
/
S
Note that we are discussing time independent problems, thus there are
no derivatives of time.
Better to write the integrand as J.dS, since J is a vector quantity, and
dS is an infinitesimal vector normal to the surface.
[cut]
Hmm, there is a problem here. According to my thoughts we are just
applying the law the wrong way. Even though we have to take the
perpedicular J to the surface, we shouldn't do this if we are dealing
with surface currents or line currents. Am I correct? How do we support
this? Can we say that K=J delta=>J=K/delta, where delta is the
surface's thickness so it follows that:
/ / dS /
|J dS=0 => |K -------=0 => | K dl=0
/ / delta /
S S curve
Note that we are diving dS which is a surface by delta which is a
length, so we get dl which is a length. That means that the surface
integral transforms into a curve integral,
Yes ...
thus the need to take the
normal of J or K is eliminated.
.... and no.
In 3D, you have integral(J.dS)=0. In 2D, that is, for surface currents,
you have
integral(K.dl)=0,
where K is the surface current density, and dl is an
infinitesimal vector normal to the line. You still need to take normals.
However, what you did in your original attempt was to get rid of the wrong
dimension. You have to take the closed curve you integrate along in the
same surface that the surface current is in.
What about line currents? For a current along a single line, the 2D
integral for a surface current reduces to a 1D sum; I_in - I_out = 0.
For a circuit with junctions, this generalises to Kirchhoff's 2nd law.
Think about what a line current is. It isn't really an infinitely thin
line of current, it's a finite current density spread over some finite
area, typically the width of the wire the current is in. As an exercise,
start with the 3D conservation law for J, and obtain Kirchhoff's 2nd law
(ie sum(I_in) = sum(I_out)). Don't use identical wires, use different
radiii for some of them.
--
Timo Nieminen - Home page: http://www.physics.uq.edu.au/people/nieminen/
E-prints: http://eprint.uq.edu.au/view/person/Nieminen,_Timo_A..html
Shrine to Spirits: http://www.users.bigpond.com/timo_nieminen/spirits.html
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