using Ampere's law to find magnetic field of straight current-carrying conductor

[PG1995]

Hi

Is this possible to to find magnetic field of a straight current-carrying conductor using Ampere's law instead of using Biot-Savart's law as in the attachment? Please let me know. Thank you.

Regards
PG

 

[BioniC187]

Do you want to find the field strength at a point x from the conductor using ampere's law?

 

[Sceadwian]

https://en.wikipedia.org/wiki/Biot-...cuital_law.2C_and_Gauss.27s_law_for_magnetism

Biot-Savart's law is an extension of Ampere's law, anything that satisfies Biot-Savart's law also satisfies Ampere's law, the extra parts of the equation have to be added to model the real world field effects involved in an ACTUAL device where Ampere's law is pure theory.

 

[PG1995]

It's point P along x-axis. Yes, I want to use Ampere's law. In the attachment they have used Bio-Savart's law. I'm afraid using Ampere's law would lead to quite a few complications. Thanks.

 

[Sceadwian]

You can't use Ampere's law alone, the Bio-Savart's law incorporates Ampere's law and Gauss' law together for practical reasons. There are proofs of this in the Biot-Savart Wikipedia entry under the topic index "The Biot–Savart law, Ampère's circuital law, and Gauss's law for magnetism"

 

[BioniC187]

https://en.wikipedia.org/wiki/Biot-...cuital_law.2C_and_Gauss.27s_law_for_magnetism

Biot-Savart's law is an extension of Ampere's law, anything that satisfies Biot-Savart's law also satisfies Ampere's law, the extra parts of the equation have to be added to model the real world field effects involved in an ACTUAL device where Ampere's law is pure theory.

Yeah, i was trying to get at that. Do you need to prove this for a university tutorial? You say lead to complications, if so then why not just use biot savart's law?

 

[PG1995]

Thank you, Sceadwian, BioniC.

I was just curious to know if that was possible. It is not part of any course study.

Best wishes
PG

 

[Sceadwian]

It is possible, what I'm trying to say is that the differences in the Biot-Savart equation and the Ampere's law equation encompasses Gauss's law, which means that in a practical situation you use the Biot-Savart law, unless for some other reason you need to work all the math from the ground up for a specific situation.

I can not see this situation occurring unless you get into very small form factors or complex three dimensional interaction which is way outside of current experimental practice or theory.

 

[PG1995]

Hi Sceadwian

I think I get your point now. You are essentially saying that Biot-Savart's law is a ready-made mathematical tool to deal with the situations as the one under discussion. Although it's possible to solve the problem using Ampere's law without using Biot-Savart's law, it would take a lot of work and one would need to start from scratch. Thank you.

Regards
PG

 

[MrAl]

Hello there,


Ampere's Law requires certain conditions that Biot-Savart doesnt.

If we take a wire 6 feet long and look at the B just 1 foot above the wire at the center of the wire, we see a certain B we'll call B1. That B1 gets there due to contributions from both sides of the wire (rem we are at the center of the wire) and the distance to the end of the wire is 3 feet in either direction. Thus we have contributions from 3 feet of wire twice, and the mid point distance to either side is 1.5 feet.
Now if we move to the end of the wire we see a different B we'll call B2. It's lower than B1 because we have for one the same contribution from 3 feet of wire that we had when we were in the center, but this time only once and the mid point distance is 1.5 feet. We have contributions from another part of the wire that is also 3 feet long, but that second part is now farther way with a mid point distance of 4.5 feet.
This of course means that the B near the ends of the wire is less than the B near the center of the wire and therefore Ampere's Law doesnt work at only one place at the wire because any one place is not representative of the entire wire's B field.
When the wire is infinitely long then the B is the same everywhere so if we look at any one point along the wire at a distance of 1 foot we'll always see the same B, and thus Ampere's Law applies. This also works to some degree when the distance away from the wire is not too large relative to the length of the wire and we ignore the ends.

You'll also note that Biot-Savart produces the same result as Ampere when we integrate over the entire length of an infinitely long wire, or we stay close to the wire relative to the length.

It's as if Biot-Savart includes an extra dimension that Ampere doesnt, although that's not entirely correct. So to get Ampere's to work you'd have to basically produce Biot-Savart from Ampere's.

 

[Sceadwian]

Ampere's Law requires certain conditions that Biot-Savart doesnt.

That's seems a bit contradictory to me. If a Biot-Savart equation satisfies Ampere's law then how can there be a condition that Biot-Savart's doesn't use that Ampere's law does seeing that it will always make sense? I mean it has to be dealt with in some other way otherwise you couldn't come up with a proof that would satisfy both? Probably a bit over my head, calculus is definitely not my thing =>

 

[MrAl]

Hello there Scead,


Well, when we draw an Amperian loop around the center section of a finite wire we get a certain B using Amperes due to the current through the wire. When we do the same thing near the end of the wire we get the same B using Amperes because the current is still the same in the wire there.

When we use Biot-Savart near the center of the wire we get a certain B, but when we use Biot-Savart near the end of the wire we get a different B, and this second B is less than the center located B.

So looking along the wire say we get 2 using Biot-Savart at the center and 1 near the ends. With Ampere we would get 2 near the center and 2 near the ends also. So there has to be a difference.

It is quite intuitive that the B should be less near the ends of the wire because the midpoints of the two sections of wire (dividing the wire at the center so we have two equal lengths) are the same when we are at the center but when we are at the end one section is much farther away. We have two influences in both cases, but if the second influence is farther away we know that the field decreases with distance so the second influence cant contribute as much if it is farther away.

If the wire is very long however then we get the same result from both methods. But that's a very long wire not a finite wire. This shows that Ampere's does in fact work when we have certain conditions.

The usual way this is explained is to use Ampere's there has to be a high degree of symmetry. This simplifies some problems, but then again i dont have a problem with using Biot-Savart myself as it seems easier to apply when we dont have very simple situations. Note that whenever we see Ampere's being used directly there's always a "very long wire" or "infinite wire".

We can also consider this:
What if we want to know the B at a point *beyond* the end of the finite wire. There's no current here so how do we use Amperes? If we draw a 3d surface that still allows the current to pass though it even though the loop is beyond the end of the wire, then we get that same value "2" as before which cant be right because all of the influences can now be very far away. Biot-Savart would tell us we have a very low B there, maybe as low as 0.01 (as compared to 2 at the center of the wire).

Some of these issues are just nature and how we try to describe it. Biot-Savart has it's quirks too when we use it for a finite wire. For example, how exactly do we get a finite straight wire to pass a constant current without introducing other influences that will also influence the total outcome of the field? There's no such thing as a finite straight wire that can pass current as there has to be a loop of current. So when we use Biot-Savart we are still implying that there are some practical constraints or rather some assumptions when we look at a straight wire. But the idea helps us solve practical problems anyway.

So it's not that Ampere's is not correct, it is just too hard to apply unless there are certain circumstances.

 

[Sceadwian]

Thanks for the laymens explanation MrAl I see your point now.

 

[PG1995]

Now if we move to the end of the wire we see a different B we'll call B2. It's lower than B1 because we have for one the same contribution from 3 feet of wire that we had when we were in the center, but this time only once and the mid point distance is 1.5 feet. We have contributions from another part of the wire that is also 3 feet long, but that second part is now farther way with a mid point distance of 4.5 feet.

MrAl, could you please tell me how the midpoint distance is 4.5? I don't seem to get it. Thank you.

It's as if Biot-Savart includes an extra dimension that Ampere doesnt, although that's not entirely correct. So to get Ampere's to work you'd have to basically produce Biot-Savart from Ampere's.

So, my conclusion in this post was correct.

In my view, the essence of Ampere's law is that it considers that the field is uniform over a certain distance everywhere. For instance, it assumes that the field is uniform over the radius of, say 2 units, along a length of wire. And this assumption can only be true when we have an infinite length of wire. If the geometry is such that this assumption cannot be really correct, then Ampere's law will give incorrect result as MrAl has stated. While on the other hand, Biot-Savart's law will give correct result because it summates contribution of all the infinitesimal segments of a wire which could be of finite of infinite length.

Thank you.

Regards
PG

 

[MrAl]

Hello again,


That's pretty much it yes

There's one tiny detail about Ampere's we can take a quick look at. Ampere's works in three dimensions so it describes what the field should be for any circumstance. However, the way we "usually" apply Ampere's is where we assume that B is not really a vector, and this limits the geometry scenarios where we can apply Ampere's. Well, we assume it is a vector but has only one component and that the curl of B is also a vector with only one non zero component. What this means for a straight wire is that we have to make sure that the curl of B only has one constant component in the direction related to B itself, and the only way we can get that is with certain circumstances like with an infinite wire. An infinite wire works because everywhere the curl of B has just one non zero component. With a finite length wire however, the curl has at least two components (really three but two would be the same) and that means we'd have to find a function that describes the behavior of B as we move out from the center of the wire. Normally all we have to do is find what B is wrapping around the wire (infinite wire), which is only one dimension. Funny, we could probably do this using Biot-Savart but that might be cheating Perhaps there is a more elementary theory we can use to do this. What i think we would end up with is a rule like Biot-Savart where we relate the B at every point out from the wire to the length and radius. That would make sense because that's what BS does already
Maybe we can find some history on Biot-Savart to find out how they did it (experimentally or functionally starting with Ampere's).

I think i've explained this correctly but keep in mind it has been some years since i actually did a lot of calculations like this. Biot-Savart seems simple enough so that's my favorite

Maybe if we started from BS and worked backwards using vectors we might find Ampere's at the heart

Here's a view of the magnetic field 'equipotential' lines out from a wire looking in one plane. Note that they do curve. With an infinite wire we'd see just straight horizontal lines.
These points were calculated using Biot-Savart. The points to the far left and right are just beyond the ends of the wire, the wire is red.

 

[PG1995]

Thank you, MrAl.

I understand it now. Your analysis was really helpful. There are still some points I don't exactly understand but I'm sure I will get over them as I learn more. I'm afraid you have missed one question from my previous posting. Please see below. By the way, did you use MATLAB to create **broken link removed**. Thank you.

Now if we move to the end of the wire we see a different B we'll call B2. It's lower than B1 because we have for one the same contribution from 3 feet of wire that we had when we were in the center, but this time only once and the mid point distance is 1.5 feet. We have contributions from another part of the wire that is also 3 feet long, but that second part is now farther way with a mid point distance of 4.5 feet.

MrAl, could you please tell me how the midpoint distance is 4.5? I don't seem to get it. Thank you.

Regards
PG

 

[MrAl]

Hi,

The mid point distance is the distance along the wire from one end to the opposite section of wire at it's midpoint. See diagram.
Ignore spelling errors

No i dont have MATLAB. I use other drawing programs mostly ones that i have written myself for Windows (well actually in a programming language that runs on Windows). The function plotted was found from Biot-Savart, converted into functional form that takes x and R as arguments, the full space calculated (with small dR and dx), then the points of equal potential picked out which is really done with a function like:
y=B, {B: |B| mod(0.5)<=0.05}
I solved for the lines of constant R but found it to be a 10th degree equation in R, which reduced to a 5th degree equation in R^2, but it was still a little too cumbersome to use directly so i went back to the full space method (it can be done but i didnt feel like spending the time).