| Topic: |
Science > Physics |
| User: |
"Peter Christensen" |
| Date: |
19 Oct 2006 05:59:00 AM |
| Object: |
I get a divergent potential - What's wrong here? |
I know that it's obvious, that I've somehow done something incorrect in
my reasoning, when I try to solve my problem. -But I've gone through
things many times now, and I've still not found out what is wrong.
I would like to calculate the electric field from a circular conductor.
-It looks quite simple, and it should be, but I continue to get a
result, which can't be correct from a physical point of view.
Some assumptions (which I find reasonable):
* The conductor is very long (as for example an electrical cable). -so
I can assume, that the gradient of the potential is zero in the length
direction of the conductor. This means that dU/dz = 0, where U is the
electric potential and z is the length direction of the conductor. I
just solve the problems as a 2D problem, and I use r and phi for the
coordinates, which are defined as usual polar coordinates.
* The potential has no time dependence, so dU/dt = 0. The problem is
much simpler because of these static conditions.
* I assume, that there is a radial symmetry, as the conductor is round,
and that there are no wave-effects involved when dU/dt = 0.
* The potential on the surface of the conductor: U(r=1) = 1 (f.x. 1
Volt).
* At a distance far from the conductor, the electrical field from it
must vanish. I define the potential, such that U(r->00) = 0. Of course,
there could be a constant added or subtracted, but I use the condition
above to define the potential in this problem. (It's just the usual way
to define it, as I remember.)
The electric field (E) is related to the potential U by E = -grad(U).
-And it can be measured by measuring the force (F) on a charged
particle as F = q*E, where q is the charge of the particle. So the
potential U is well defined everywhere in space from U(r=1) = 1
(surface of the circular conductor) to U(r->oo) = 0, and also around
the conductor, of course.
If I should try to give some real example of the physics in the
problem, then one can simply think of a single power line (round) of
metal with a DC current. Maybe these dimensions: Diameter: 2 cm,
length: several kilometers, potential: Some kV and 'far distance': some
tenths of meters.
In the free space, the electric potential U(x,y), or U(r,phi), can be
calculated by the fact, that the Laplace Operator should be zero. In
xy-coordinates, this means that d^2U/dx^2 + d^2U/dy^2 = U_xx + U_yy = 0
(partial derivatives). First, this must be translated to polar
coordinates:
Laplace Equation: U_rr + (1/r)*U_r + (1/r^2)*U_PhiPhi = 0
(I have remembered to check the result, so I don't think that the
problem can be here.)
First, I write the solution as U(r,phi) = R(r)*T(phi). Seperation of
the variables like this is the usual method to use with this equation.
Because of the radial symmetry dT/dPhi is zero, so that T(phi) must be
a constant independent of phi. I insert U(r,phi) = constant*R(r), and
get this equation for R(r):
r*R''(r) + R'(r) = 0 (Special case of the Cauchy-Euler equation)
Now finally to the problem. The general solution for the differential
equation above is: R(r) = c1 + c2*ln(r), where the c's are just
constants. I can't get the ln(r) function to go to 0 as r goes to
infinity, which means that c2 must be equal to zero. On the other hand
the electric potential can't be just a constant (c1), because there
would be no electric field from the conductor at all, as E=-grad(U).
Both a divergent and a constant potential are impossible as solutions
to the problem. And I know, that I have the correct full solution to
the equation for R(r) above, so I simply don't understand what's wrong.
What's wrong here? It is quite frustrating, that I don't know what the
error is, so if you spot it, then please let me know. :-) -Thanks in
advance.
PC
.
|
|
| User: "Sorcerer" |
|
| Title: Re: I get a divergent potential - What's wrong here? |
19 Oct 2006 06:28:39 AM |
|
|
"Peter Christensen" <PeCh@MailAPS.org> wrote in message
news:1161255540.038762.51330@b28g2000cwb.googlegroups.com...
|
| I know that it's obvious, that I've somehow done something incorrect in
| my reasoning, when I try to solve my problem. -But I've gone through
| things many times now, and I've still not found out what is wrong.
|
| I would like to calculate the electric field from a circular conductor.
| -It looks quite simple, and it should be, but I continue to get a
| result, which can't be correct from a physical point of view.
|
| Some assumptions (which I find reasonable):
|
| * The conductor is very long (as for example an electrical cable). -so
| I can assume, that the gradient of the potential is zero in the length
| direction of the conductor. This means that dU/dz = 0, where U is the
| electric potential and z is the length direction of the conductor. I
| just solve the problems as a 2D problem, and I use r and phi for the
| coordinates, which are defined as usual polar coordinates.
|
| * The potential has no time dependence, so dU/dt = 0. The problem is
| much simpler because of these static conditions.
|
| * I assume, that there is a radial symmetry, as the conductor is round,
| and that there are no wave-effects involved when dU/dt = 0.
|
| * The potential on the surface of the conductor: U(r=1) = 1 (f.x. 1
| Volt).
|
| * At a distance far from the conductor, the electrical field from it
| must vanish.
Place a parallel conductor adjacent to the first. This is then effectively
a capacitor. At a distance not very far from the conductor, the electrical
field
from it must vanish.
.
|
|
|
| User: "Peter Christensen" |
|
| Title: Re: I get a divergent potential - What's wrong here? |
20 Oct 2006 03:33:07 AM |
|
|
"Sorcerer" <Headmaster@hogwarts.physics_b> skrev i en meddelelse
news:HjJZg.181028$wg.35952@fe1.news.blueyonder.co.uk...
| * At a distance far from the conductor, the electrical field from it
| must vanish.
Place a parallel conductor adjacent to the first. This is then effectively
a capacitor. At a distance not very far from the conductor, the electrical
field
from it must vanish.
Always nice with a good idea from Hogwarts, but that wouldn't be the
problem, which I'm interested in...
PC
.
|
|
|
| User: "Sorcerer" |
|
| Title: Re: I get a divergent potential - What's wrong here? |
20 Oct 2006 08:52:14 AM |
|
|
"Peter Christensen" <PeCh@MailAPS.org> wrote in message
news:453889cd$0$49201$14726298@news.sunsite.dk...
| "Sorcerer" <Headmaster@hogwarts.physics_b> skrev i en meddelelse
| news:HjJZg.181028$wg.35952@fe1.news.blueyonder.co.uk...
|
| > | * At a distance far from the conductor, the electrical field from it
| > | must vanish.
| >
| > Place a parallel conductor adjacent to the first. This is then
effectively
| > a capacitor. At a distance not very far from the conductor, the
electrical
| > field
| > from it must vanish.
|
| Always nice with a good idea from Hogwarts, but that wouldn't be the
| problem, which I'm interested in...
Understand something. Force acts between TWO points.
"At a distance far from the conductor, the electrical field from it must
vanish."
is the sound of one hand clapping.
Androcles
.
|
|
|
|
|
|
| User: "Timo A. Nieminen" |
|
| Title: Re: I get a divergent potential - What's wrong here? |
19 Oct 2006 01:49:40 PM |
|
|
On Thu, 19 Oct 2006, Peter Christensen wrote:
I would like to calculate the electric field from a circular conductor.
-It looks quite simple, and it should be, but I continue to get a
result, which can't be correct from a physical point of view.
Some assumptions (which I find reasonable):
* The conductor is very long (as for example an electrical cable). -so
I can assume, that the gradient of the potential is zero in the length
direction of the conductor. This means that dU/dz = 0, where U is the
electric potential and z is the length direction of the conductor. I
just solve the problems as a 2D problem, and I use r and phi for the
coordinates, which are defined as usual polar coordinates.
[cut]
* At a distance far from the conductor, the electrical field from it
must vanish. I define the potential, such that U(r->00) = 0. Of course,
there could be a constant added or subtracted, but I use the condition
above to define the potential in this problem. (It's just the usual way
to define it, as I remember.)
[cut]
If I should try to give some real example of the physics in the
problem, then one can simply think of a single power line (round) of
metal with a DC current.
Are you trying to do this for a uniformly charged infinite cylinder, or
for an infinite uncharged conductor? For the latter, you get
the usual logarithmic potential, for which you can't have U(infinity)=0.
For the former, you can't have dU/dz = 0 (since you have a potential
gradient along the conductor to drive the current).
Now finally to the problem. The general solution for the differential
equation above is: R(r) = c1 + c2*ln(r), where the c's are just
constants. I can't get the ln(r) function to go to 0 as r goes to
infinity, which means that c2 must be equal to zero. On the other hand
the electric potential can't be just a constant (c1), because there
would be no electric field from the conductor at all, as E=-grad(U).
Yes. But you only need E(infinity)=0, not U(infinity)=0. Recall that this
solution is for the unphysical case of a uniformly charged infinitely long
conductor. Compare an infinite charged sheet - E is uniform on each side,
so U = c|z|, going to infinity as you move away from the sheet.
U(infinity)=infinity is not a problem. For a physically realistic problem,
yes, the charge distribution is bounded, and you can have U(infinity)=0,
but that's not what you specified.
But, again, if you want this to be for a current-carrying conductor rather
than a charged conductor, dU/dz is non-zero.
--
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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|
|
| User: "" |
|
| Title: Re: I get a divergent potential - What's wrong here? |
19 Oct 2006 08:12:43 AM |
|
|
The real problem with your analysis is that your trial solution
R(r) = c1 + c2*ln(r)
does not fit the boundary conditions of the problem. That potential
applies to the mythical "infinitely long" cylindrical conductor but
does not apply to one of finite length.
At positions close to the conductor your trial solution is
approximately correct, but even in that case it does not rigorously
hold true, and the deviations of the correct potential from your trial
solution increase as you approach the end of the cable or as you move
radially away from the cable..
If I remember correctly the problem you describe (a cable of finite
length) is best analyzed by using a Bessel function expansion to
describe the potential. However, I will not attempt to do that here,
partly because I do not remember the details.
Alan
Peter Christensen wrote:
I know that it's obvious, that I've somehow done something incorrect in
my reasoning, when I try to solve my problem. -But I've gone through
things many times now, and I've still not found out what is wrong.
I would like to calculate the electric field from a circular conductor.
-It looks quite simple, and it should be, but I continue to get a
result, which can't be correct from a physical point of view.
Some assumptions (which I find reasonable):
* The conductor is very long (as for example an electrical cable). -so
I can assume, that the gradient of the potential is zero in the length
direction of the conductor. This means that dU/dz = 0, where U is the
electric potential and z is the length direction of the conductor. I
just solve the problems as a 2D problem, and I use r and phi for the
coordinates, which are defined as usual polar coordinates.
* The potential has no time dependence, so dU/dt = 0. The problem is
much simpler because of these static conditions.
* I assume, that there is a radial symmetry, as the conductor is round,
and that there are no wave-effects involved when dU/dt = 0.
* The potential on the surface of the conductor: U(r=1) = 1 (f.x. 1
Volt).
* At a distance far from the conductor, the electrical field from it
must vanish. I define the potential, such that U(r->00) = 0. Of course,
there could be a constant added or subtracted, but I use the condition
above to define the potential in this problem. (It's just the usual way
to define it, as I remember.)
The electric field (E) is related to the potential U by E = -grad(U).
-And it can be measured by measuring the force (F) on a charged
particle as F = q*E, where q is the charge of the particle. So the
potential U is well defined everywhere in space from U(r=1) = 1
(surface of the circular conductor) to U(r->oo) = 0, and also around
the conductor, of course.
If I should try to give some real example of the physics in the
problem, then one can simply think of a single power line (round) of
metal with a DC current. Maybe these dimensions: Diameter: 2 cm,
length: several kilometers, potential: Some kV and 'far distance': some
tenths of meters.
In the free space, the electric potential U(x,y), or U(r,phi), can be
calculated by the fact, that the Laplace Operator should be zero. In
xy-coordinates, this means that d^2U/dx^2 + d^2U/dy^2 = U_xx + U_yy = 0
(partial derivatives). First, this must be translated to polar
coordinates:
Laplace Equation: U_rr + (1/r)*U_r + (1/r^2)*U_PhiPhi = 0
(I have remembered to check the result, so I don't think that the
problem can be here.)
First, I write the solution as U(r,phi) = R(r)*T(phi). Seperation of
the variables like this is the usual method to use with this equation.
Because of the radial symmetry dT/dPhi is zero, so that T(phi) must be
a constant independent of phi. I insert U(r,phi) = constant*R(r), and
get this equation for R(r):
r*R''(r) + R'(r) = 0 (Special case of the Cauchy-Euler equation)
Now finally to the problem. The general solution for the differential
equation above is: R(r) = c1 + c2*ln(r), where the c's are just
constants. I can't get the ln(r) function to go to 0 as r goes to
infinity, which means that c2 must be equal to zero. On the other hand
the electric potential can't be just a constant (c1), because there
would be no electric field from the conductor at all, as E=-grad(U).
Both a divergent and a constant potential are impossible as solutions
to the problem. And I know, that I have the correct full solution to
the equation for R(r) above, so I simply don't understand what's wrong.
What's wrong here? It is quite frustrating, that I don't know what the
error is, so if you spot it, then please let me know. :-) -Thanks in
advance.
PC
.
|
|
|
| User: "Peter Christensen" |
|
| Title: Re: I get a divergent potential - What's wrong here? |
20 Oct 2006 03:54:24 AM |
|
|
<alanrockwood2000@yahoo.com> skrev i en meddelelse
news:1161263562.971106.275940@i3g2000cwc.googlegroups.com...
The real problem with your analysis is that your trial solution
R(r) = c1 + c2*ln(r)
does not fit the boundary conditions of the problem. That potential
applies to the mythical "infinitely long" cylindrical conductor but
does not apply to one of finite length.
Yes, now I know, that I can't just assume an 'infinite cylinder' and then
just use the result in 'real physics'. In practice, all cylinders must have
a finite length. I'm making an error, when I say that R(r->00) = 0, because
my solution for R(r) is not valid 'out there'. At a distance very far from
the conductor, the problem will be a 3D problem, and therefore my solution
for the potential is simply wrong when R -> infinite.
I remember, that I've once learnt about calculating the electrical fields
for both metallic wave-guides and optical fibers. I'm quite sure, that then
the approximation of 'infinetely long cylinders' was used, without any
problems. That's why I got this idea with just an infinitely long cylinder.
That's also the reason, that I think, that it's somehow strange that it
doesn't work here.
At positions close to the conductor your trial solution is
approximately correct, but even in that case it does not rigorously
hold true, and the deviations of the correct potential from your trial
solution increase as you approach the end of the cable or as you move
radially away from the cable..
If I remember correctly the problem you describe (a cable of finite
length) is best analyzed by using a Bessel function expansion to
describe the potential. However, I will not attempt to do that here,
partly because I do not remember the details.
I thought (first) that my problem with the static potential was much easier
to solve, than the problems with time-dependent electrical fields. But I
should probably try to solve my problems using the E-field instead of the
electric potential. (E=-grad(U)) -But my problem is still static, so I
probably can't use the wave-equations (from Maxwells Equations), which are
usually used on waveguides.
PC
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