Magnetic Field Around Finite Conductor: Questions & Answers

[cataldo]

Hi everybody...
I have a simple question for you..
Where can i found the expression of the magnetic field around at the conductor which it has finite lenght? because i always found the megnetic filed in a conductor infinite length ..
Sorry for the stupid question...
tanks and best regard

 

[saunderson]

For such an unsymmetric problem, where no closed current flow exists it would be the best to solve Biot-Savart's equation for discrete points (if possible) or numerically. I don't think, that a analytical solution exists!?

In addition i don't want to waste hours/days to solve an integral, that not even provides some extra physical insights... I don't think, that your problem has any physical significance (though, maybe it is a good exercise for a numeric beginner)

 

[Gerenuk]

It's actually tricky, because current always has to be closed. You cannot have open ends. Somehow I have the feeling that using Biot-Savart even rely on currents being closed and the equation might be wrong without?
But feel free to try integrating the Biot-Savart law.

 

[saunderson]

It's actually tricky, because current always has to be closed. You cannot have open ends. Somehow I have the feeling that using Biot-Savart even rely on currents being closed and the equation might be wrong without?
But feel free to try integrating the Biot-Savart law.

Biot-Savart's law can handle open ends, but there exists no vector potential for non closed currents (since it's not physical). For non closed currents the vector potential diverges.

 

[Meir Achuz]

Hi everybody...
I have a simple question for you..
Where can i found the expression of the magnetic field around at the conductor which it has finite lenght? because i always found the megnetic filed in a conductor infinite length ..
Sorry for the stupid question...
tanks and best regard

Finding B for a finite length can be useful since the finite wire can be part of a larger closed circuit, for instance to find the field of a square loop.
If you look at any textbook derivation of B for an infinite wire using Biot-Savart, just change the limits on the integral from +/- infinity to finite limits.

 

[bjacoby]

Biot-Savart's law can handle open ends, but there exists no vector potential for non closed currents (since it's not physical). For non closed currents the vector potential diverges.

I don't understand why people are having trouble with this. Biot-Savart law is based upon tiny current segments whose contributions are added together (integrated) to find the contribution. I presume the problem is people here have not advanced to calculus yet. The idea is still pretty simple. You just add up the contributions of the very short segments to make a long segment. The wire segment does not have to be a complete circuit. You can calculate the contributions from the various segments of a circuit and get the correct answer for a complete circuit which is required to get a current flowing.

Biot-Savart is written: dB = (muo i sin(theta) dl)/(4 Pi r²)

dl is the tiny length of current that gives the magnetic field contribution dB. Look up Biot-Savart to see how the angle theta and the distance r from the field point and current element are defined.

For example a straight piece of wire of length 2L carrying a current I has a magnetic field contribution in the plane through it's center at a distance R from the wire of the integration of the Biot-Savart law from -L to L rather than plus and minus infinity (for a very long wire) as someone already noted.

And note that the magnetic vector potential also exists and is given in the same plane by:

A = [(muo I)/(2 pi)] ln[{L + (L² + R²)^1/2}/R]

Sorry, I'm not sure how one writes nice looking equations here.

 

[Gerenuk]

One point to consider is that the Biot Savart law cannot give "correct" results for forces between segments only, i.e. before you haven't added up all contributions, the force isn't a real physical force.

For example take two little wire segments which interact through the magnetic field. Now find the force between them. The magnetic field from segment 1 is
\vec{B}\propto d\vec{l}_1\times \hat{r}
The Lorenz force from this field on segment 2 is therefore
\vec{F}_{21}\propto d\vec{l}_2\times(d\vec{l}_1\times \hat{r})

OK, but what if you calculate the force from segment 2 on segment 1? You get
\vec{F}_{12}\propto d\vec{l}_1\times(d\vec{l}_2\times \hat{r})\neq \vec{F}_{21}

Due to action/reaction this forces should be equal, but they are not! So neither of these results can be the "real" force.

 

[Meir Achuz]

Due to action/reaction this forces should be equal, but they are not! So neither of these results can be the "real" force.

But if you include the charge buildup at the ends of the segments (required by conservation of charge), adding the electromagnetic momentum conserves overall momentum.

 

[Gerenuk]

But if you include the charge buildup at the ends of the segments (required by conservation of charge), adding the electromagnetic momentum conserves overall momentum.

Thanks, that might be a good way to think about it.

In any case, integrating just Biot-Savart over a segment doesn't give a real force, yet. However, as people have pointed out it might not be a problem if you plan to add other segments anyway.

 

[varga]

\vec{F}_{12}\propto d\vec{l}_1\times(d\vec{l}_2\times \hat{r})\neq \vec{F}_{21}

Due to action/reaction this forces should be equal, but they are not! So neither of these results can be the "real" force.

Are you sure those two don't equate?
What do you mean by "not real force"?
You mean there is a "leak" somewhere?



============================
Why would "ends" represent any specific problem? Biot-Savart law does describe magnetic field in full 3-D, as a shape. It is only the popular illustration of this field that leads people think there are some cuts or "ends" there.

So, this is it, magnetic field of a moving charge. This picture illustrates this field of a single charge in a single instant in time, so when you picture all the charges in that wire and their movement through time this will look like a cylinder. However, these cuts at the end of this cylinder are not 'flat ends' and should actually be very well defined, just like the rest.

The problem is this popular illustration would have you believe this magnetic field of a single charge in a single instant exist only in 2-D plane perpendicular to velocity vector, but if you look at Biot-Savart law you may notice that this potential is actually defined in full 3-D, behind and in front of that plane, there are no "undefined ends" actually.


So, how it looks like, how are those ends actually defined or shaped in 3-D? I'm not sure how to draw that precisely, but I think it looks something like 'biconcave lens' rather than "flat plane", makes sense?

 

[Gerenuk]

Are you sure those two don't equate?
What do you mean by "not real force"?
You mean there is a "leak" somewhere?

Yes, these expressions are different in general.

Clem gave a nice answer. Biot-Savart alone gives wrong answers for finite segments. You would need to include the pile up of charges at the end of the segments to get something that is a measureable force. Otherwise you just get a theoretical number which doesn't correspond to anything you would measure.

Fortunately for closed loops these pile up charges cancel, so that you do not need the contribution other than Biot-Savart anymore.

 

[Gerenuk]

The problem is this popular illustration would have you believe this magnetic field of a single charge in a single instant exist only in 2-D plane perpendicular to velocity vector, but if you look at Biot-Savart law you may notice that this potential is actually defined in full 3-D, behind and in front of that plane, there are no "undefined ends" actually.

Maybe someone else can clarify this. I think Biot-Savart does not exactly coincide with the relativistic field of a moving charge?!

 

[varga]

Maybe someone else can clarify this. I think Biot-Savart does not exactly coincide with the relativistic field of a moving charge?!

I do not think any relativistic effects come into this as average electron velocity (~amperes) in a wire conductor does not reach relativistic speeds. Biot-Savart law is obviously good enough for most of electric and electronics circuits and practical application, like Coulomb's law, but even if there was any error correction needed, that would not change the general shape of the field, it would only distort it.

Yes, these expressions are different in general.

Ok, can you say something about force vectors, are they opposite in direction, where do they point? Does F(1-2) point from 1 to 2, from somewhere else to 2, or from 1 to somewhere else?

Clem gave a nice answer. Biot-Savart alone gives wrong answers for finite segments.

According to what experiment? I do not see anyone asserted Biot-Savart alone gives wrong answers for finite segments. I do not see why would Biot-Savart be any more inaccurate than Coulomb's law visa vi SR and why would not fields of individual particles add up to give the correct result as a 'compound field' just because wire makes a turn or "ends".


By the way I found a better description for this shape of magnetic field around a single moving charge. If we take a ball to represent the shape of electric field, then the shape of magnetic field would look like the ball we squeeze from two points, from behind and from the front along its velocity vector until the points touch in the middle. It is sort of doughnut shape and there is a very small singularity hole along the velocity vector, i.e. there is no magnetic filed immediately behind and in front of the moving charge.

 

[Gerenuk]

Ok, can you say something about force vectors, are they opposite in direction, where do they point? Does F(1-2) point from 1 to 2, from somewhere else to 2, or from 1 to somewhere else?

Not sure what you mean. But you can make your own conventions and repeat the calculation. In the end you will get something like
\vec{F}_{12}\propto d\vec{l}_1\times(d\vec{l}_2\times\hat{r})
\vec{F}_{21}\propto d\vec{l}_2\times(d\vec{l}_1\times\hat{r})
which is just mathematically not the same. A real physical law would require \vec{F}_{12}=-\vec{F}_{21}
You agree?
One cannot decide which force to take. In fact both a wrong.

According to what experiment? I do not see anyone asserted Biot-Savart alone gives wrong answers for finite segments.

No textbook claims that finite segment Biot-Savart makes sense.

By the way I found a better description for this shape of magnetic field around a single moving charge. If we take a ball to represent the shape of electric field, then the shape of magnetic field would look like the ball we squeeze from two points, from behind and from the front along its velocity vector until the points touch in the middle. It is sort of doughnut shape and there is a very small singularity hole along the velocity vector, i.e. there is no magnetic filed immediately behind and in front of the moving charge.

It's still a doughnut and I suppose due to the above argumentation Biot-Savart should be in conflict with the doughnut field you found.

 

[Born2bwire]

I do not think any relativistic effects come into this as average electron velocity (~amperes) in a wire conductor does not reach relativistic speeds. Biot-Savart law is obviously good enough for most of electric and electronics circuits and practical application, like Coulomb's law, but even if there was any error correction needed, that would not change the general shape of the field, it would only distort it.




Ok, can you say something about force vectors, are they opposite in direction, where do they point? Does F(1-2) point from 1 to 2, from somewhere else to 2, or from 1 to somewhere else?




According to what experiment? I do not see anyone asserted Biot-Savart alone gives wrong answers for finite segments. I do not see why would Biot-Savart be any more inaccurate than Coulomb's law visa vi SR and why would not fields of individual particles add up to give the correct result as a 'compound field' just because wire makes a turn or "ends".


By the way I found a better description for this shape of magnetic field around a single moving charge. If we take a ball to represent the shape of electric field, then the shape of magnetic field would look like the ball we squeeze from two points, from behind and from the front along its velocity vector until the points touch in the middle. It is sort of doughnut shape and there is a very small singularity hole along the velocity vector, i.e. there is no magnetic filed immediately behind and in front of the moving charge.

It is always dangerous to assert that special relativity is not applicable in a electrodynamic problem. Electromagnetics ALWAYS satisfies special relativity in their full forms. This means that even if the sources are moving at non-relativistic speeds that ignoring the appropriate Lorentz transforms can result in very inaccurate answers because the fields will always transform via Lorentz transformations. One simple example is the calculation of the force between two current carrying wires. In the lab frame, the force is due to the magnetic fields acting on the currents in the wires. However, if we move to a moving frame where the electrons are at rest, then the appropriate Lorentz transformations result in different length contractions between the electrons and positive ion streams. This results in a net charge density which creates a Coulombic force. This effect arises regardless of how slow the charges are moving because you always have to use Lorentz transformations on fields.

Clem is correct about charge build-up being a valid occurrence. It is common to seen "endcaps" on your typical whip antennas in portable radios. This metal cap not only functions as a handy grip for extending the antenna, but it acts as a charge resevoir. This allows the electrical length of the antenna to be longer than its physical length because the currents do not need to be zero at the end of the antenna. Instead, the currents are non-zero because there is a chunk of metal which the currents can flow around and deposit the charges.