What is wrong with Ampere's force law?

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In summary, the conversation discusses the difference between Ampere's force law and the Lorentz force equation in calculating the force between two wires. The equations produce the same result once it is noted that Ampere's force law is calculating the force per unit length for infinitely long wires. However, the calculations do not work the same for shorter wires and the equations have different units in Newtons. The conversation also discusses the significance of using infinitely long wires in Ampere's force law and the possibility of calculating force per unit length per second.
  • #1
Dunnis
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0
Code:
+                   -     +          -
| I2= 1A            |     | I2= 1A   |        I2= 1A
|===================|     |==========|     + ==================infinite >> -
          |                     |                    |
          |r=1m                 |r=1m                |r=1m
          |                     |                    |
|===================|     |==========|     + ==================infinite >> -
| I1= 1A            |     | I1= 1A   |        I1= 1A
|<------- 9m ------>|     |<-- 1m -->| 
+                   -     +          -


Ampere's Force Law: F= 2* mu0/(4Pi*r) * I1*I2; is this the force felt by each one single wire, or the total force between the two? There is apparently something wrong with this equation, it gives the same result regardless of different wire lengths, therefore it seem useless, no? Also it will produce different result than what would you "normally" get by using the Lorentz force equation: F= I*L x B. Yes?
 
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  • #2
It's the force experienced by each 1-meter section of the two infinitely long wires.

It will produce the same result as the force equation.
 
  • #3
Antiphon said:
It's the force experienced by each 1-meter section of the two infinitely long wires.

Why "infinitely long wires"? Are you saying this equation can not actually be used in the real world with non-infinitely long wires?


F= 2* mu0/(4Pi*r) * I1*I2; where do you see any wire lengths in this equation? Do you not see it will produce the same result whether wires are actually 3cm, 5m or infinitely long?


It will produce the same result as the force equation.

No, it will not. Why did you say that? For start the Lorentz force does have the wire length as relevant parameter, F= I*L x B.


2* mu0/(4Pi*r)*I1*I2 = I1*L x B ?

So, you are saying these two equations are the same, and they both calculate the same thing - the force acting on one single wire?
 
  • #4
The equation you put up applies to infinitely long wires. You wires are short so the equation doesn't apply to your wires.

Forces always act on a pair of entities. Otherwise you would not conserve momentum and you could lift youself up by your own bootstraps.

Both equations should produce the force PER UNIT LENGTH of infinitely long wires.
 
  • #5
Dunnis said:
No, it will not. Why did you say that? For start the Lorentz force does have the wire length as relevant parameter, F= I*L x B.


2* mu0/(4Pi*r)*I1*I2 = I1*L x B ?

So, you are saying these two equations are the same, and they both calculate the same thing - the force acting on one single wire?

Yep, the equations work out exactly the same once you note that the left-hand side is the force per unit length as Antiphon stated above. If it wasn't per unit length then logically the total force would have to be either infinite or zero since you have an infinite number of elements over which the force can act.
 
  • #6
Antiphon said:
The equation you put up applies to infinitely long wires. You wires are short so the equation doesn't apply to your wires.

What equation? You mean Ampere's force law works ONLY with infinite wires? But that's the whole problem, that's what I said and you are now saying it as if that makes some sense or can be used for something, while it actually makes it incorrect and useless.

What do you do with equation that only applies to infinite wires?


Antiphon said:
Both equations should produce the force PER UNIT LENGTH of infinitely long wires.

Should? Well, they don't, they both actually have units in Newtons.


Lorentz force, PER ONE METER:

F= I*L x B = 1A * 1m x 10^-7 N/(A*m) = 1*10^-7 Newtons



Ampere's force law, PER %#@$?:

F= 2*mu0/(4Pi*r)*I1*I2 = 2* 10^-7 N/A^2 *1A*1A = 2*10^-7 Newtons



Per unit length, yes - PER ONE METER, why do you need infinity for that? Do you think two wires that ACTUALLY are one meter in length would attract more, less, or with the same force as two infinite wires, per meter? And, how about per meter per second?
 
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  • #7
Born2bwire said:
Yep, the equations work out exactly the same once you note that the left-hand side is the force per unit length as Antiphon stated above. If it wasn't per unit length then logically the total force would have to be either infinite or zero since you have an infinite number of elements over which the force can act.

No they do not work the same as I demonstrated above. Why did you say that? Do you know math? If yes then show me your calculation and your logic, there is no need to pull false assertions out of thin air. -- What equation, what left hand side is the 'force per one meter' and how am I supposed to see that? What are you talking about?


Ampere's force law works ONLY with infinite wires and is therefore useless and can not be applied in the real world with normal wires - is this what you are trying to say?
 
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  • #8
Dunnis said:
No they do not work the same as I demonstrated above. Why did you say that? Do you know math? If yes then show me your calculation and your logic, there is no need pull false assertions out of thin air. -- What equation, what left hand side is the 'force per one meter' and how am I supposed to see that? What are you talking about?


Ampere's force law works ONLY with infinite wires and is therefore useless and can not be applied in the real world with normal wires - is this what you are trying to say?

The magnetic field from an infinite wire is
[tex] \mathbf{B} = \frac{I\mu}{2\pi r}\hat{\phi} [/tex]
Which we can easily see works out to be the same when we calculate the per unit length force. They should of course work out to be the same as the force law is nothing more than a combination of Ampere's Law and the Lorentz Force. We use Ampere's Law to get the magnetic field of the wire and then the Lorentz Force to find the resulting force. So Ampere's Force Law works for any general case. However, you are using the case that was worked out assuming the magnetostatic case of infinite wires.
 
  • #9
Previously: What equation, what left hand side is the 'force per one meter' and how am I supposed to see that? Both equations have units in Newtons, yes/no?


Born2bwire said:
The magnetic field from an infinite wire is
[tex] \mathbf{B} = \frac{I\mu}{2\pi r}\hat{\phi} [/tex]
Which we can easily see works out to be the same when we calculate the per unit length force.

Easily, eh? Then demonstrate your calculation and show us exactly where do you get this "unit length", commonly known as 'one meter', from? That equation too does not have any such variables, what in the world are you talking about? -- "Infinite wire", again!? There is no such thing as infinite wire, so please - where in THE REAL WORLD can I use that equation of yours and what purpose can it possibly have in real life and practical applications?



They should of course work out to be the same as the force law is nothing more than a combination of Ampere's Law and the Lorentz Force.

Wrong. Not Ampere's law, but Biot-Savart law. Those equations of yours work ONLY with infinite wires as we both keep repeating, it is only that you need to realize infinity is UNDEFINED, while we need EXACT equation if we are to define the unit of ampere.

When you take an instrument and measure how many amperes there are flowing in some conductor, what equations do you think those instruments are based on, the ones for INFINITE WIRES of undefined lengths, or the equations for NORMAL WIRES with certain and defined length?


Biot-Savart law and Lorentz force:

d2F = mu0/4pi*r^3 * i1*dl1 x (i2*dl2 x r)

http://www.bipm.org/en/si/si_brochure/chapter1/1-2.html



We use Ampere's Law to get the magnetic field of the wire and then the Lorentz Force to find the resulting force. So Ampere's Force Law works for any general case.

Wrong. You keep contradicting yourself. If it works ONLY for infinite wires, then it is not general and it does not work for any other cases, but ONLY for infinite wires, hence it does not work at all. I already told you all this couple of weeks ago, do you ever learn?


Code:
+                   -     +          -
| I2= 1A            |     | I2= 1A   |        I2= 1A
|===================|     |==========|     + ==================infinite >> -
          |                     |                    |
          |r=1m                 |r=1m                |r=1m
          |                     |                    |
|===================|     |==========|     + ==================infinite >> -
| I1= 1A            |     | I1= 1A   |        I1= 1A
|<------- 9m ------>|     |<-- 1m -->| 
+                   -     +          -


So, what result do you get for these situation, the same one?
 
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  • #10
Dunnis said:
Previously: What equation, what left hand side is the 'force per one meter' and how am I supposed to see that? Both equations have units in Newtons, yes/no?




Easily, eh? Then demonstrate your calculation and show us exactly where do you get this "unit length", commonly known as 'one meter', from? That equation too does not have any such variables, what in the world are you talking about? -- "Infinite wire", again!? There is no such thing as infinite wire, so please - where in THE REAL WORLD can I use that equation of yours and what purpose and use can it possibly have in real life and practical application.





Wrong. Not Ampere's law, but Biot-Savart law. Those equations of your work ONLY with infinite wires as we both keep repeating, it is only that you need to realize that infinity is UNDEFINED, while we need EXACT equation if we are to define the unit of ampere.


When you take an instrument and measure how many amperes there are in some conductor, what equations do you think those instruments are based on, the ones for INFINITE WIRES of undefined lengths, or the equations for NORMAL WIRES with certain and defined length?


Biot-Savart law and Lorentz force:

d2F = mu0/4pi*r^3 * i1*dl1 x (i2*dl2 x r)

http://www.bipm.org/en/si/si_brochure/chapter1/1-2.html





I already told you all this couple of weeks ago, do you ever learn?


Code:
+                   -     +          -
| I2= 1A            |     | I2= 1A   |        I2= 1A
|===================|     |==========|     + ==================infinite >> -
          |                     |                    |
          |r=1m                 |r=1m                |r=1m
          |                     |                    |
|===================|     |==========|     + ==================infinite >> -
| I1= 1A            |     | I1= 1A   |        I1= 1A
|<------- 9m ------>|     |<-- 1m -->| 
+                   -     +          -


So, what result do you get for these situation, the same one?

*Shrug* It's trivial. The force from the Lorentz Force for a given length L over infinite wires is
[tex] F = \left| \int^L_0 dz I_1\hat{z} \times \mathbf{B} \right| = \frac{I_1I_2\mu L}{2\pi r} [/tex]
So dividing both sides by the length $L$ and we get back what is predicted by the equation given previously. The equation for the force between two infinite wires is the force per unit length, so its units are N/m, which we regain when we divide the force derived for a length L (in N) by the length L (m).

Biot-Savart Law is a special case of Ampere's Law. Ampere's Law is the general law applicable to any situation while Biot-Savart assumes magnetostatics.

As for the examples, of course the results will be different. Since you already know about the Biot-Savart law then you can derive Ampere's Force Law for magnetostatics and use them to find the forces in the first two examples while the last one can be handled in the case I show above.

EDIT:
I would also note that the Lorentz force you give in your original post is incorrect. The Lorentz force is defined for a point charge, thus the force due to a current is defined as the force from an infinitesimal current element. That is,
[tex]\mathbf{F} = q\left(\mathbf{E} + \mathbf{v}\times\mathbf{B}\right) = Id\hat{\ell}\times\mathbf{B}[/tex]
If there is no electric field. To get the total force one must integrate over the entire current source as I have done above.
 
  • #11
Dunnis said:
F= 2* mu0/(4Pi*r) * I1*I2; where do you see any wire lengths in this equation? Do you not see it will produce the same result whether wires are actually 3cm, 5m or infinitely long?
Hi Dunnis, look at the units of your equation. In SI units F is in N/m. That is where the length comes in. Go ahead and work it out for yourself in detail, you always need to keep track of the units.

Dunnis said:
Both equations have units in Newtons, yes/no?
No, in the equation F= 2* mu0/(4Pi*r) * I1*I2 you can determine that F has units of N/m.
 
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  • #12
Dunnis said:
Wrong. You keep contradicting yourself. If it works ONLY for infinite wires, then it is not general and it does not work for any other cases, but ONLY for infinite wires, hence it does not work at all. I already told you all this couple of weeks ago, do you ever learn?

Or maybe you misunderstood the response that you got! Have you ever considered that as a possibility? You will note that there is an almost unanimous response here trying to correct you.

Ampere's law works in general! However, it doesn't mean that you can easily get an analytical solution for all cases! It is easiest when the problem has high symmetry, such as a long infinite wire, a long infinite solenoid, etc, where edge effect and other issues do not come into play. For other cases, you may have to resort to either a numerical solution, use Biot-Savart, or use Green's function approach.

Zz.
 
  • #13
ZapperZ said:
Or maybe you misunderstood the response that you got! Have you ever considered that as a possibility? You will note that there is an almost unanimous response here trying to correct you.

Ampere's law works in general! However, it doesn't mean that you can easily get an analytical solution for all cases! It is easiest when the problem has high symmetry, such as a long infinite wire, a long infinite solenoid, etc, where edge effect and other issues do not come into play. For other cases, you may have to resort to either a numerical solution, use Biot-Savart, or use Green's function approach.

Zz.

Everyone said Ampere's force law (and later Ampere's law for B field) work ONLY with infinite wires, but is that supposed to mean "work in general" or "useless"?

Let's find out, can they be applied in real world or not? Like this:

Code:
CASE A:                        CASE B:

+                   -          +          -
| I2= 1A            |          | I2= 1A   |
|===================|          |==========|
          |                          |     
          |r=1m                      |r=1m 
          |                          |     
|===================|          |==========|
| I1= 1A            |          | I1= 1A   |
|<------- 9m ------>|          |<-- 1m -->| 
+                   -          +          -

CASE A:
[tex]
B_1 = \frac{\mu_0*I_1}{2\pi r} = 2* \frac{10^-7_{N/A^2} * 1_A}{1_m} = 2 * 10^-7 \ N/A*m
[/tex]


CASE B:
[tex]
B_1 = \frac{\mu_0*I_1}{2\pi r} = 2* \frac{10^-7_{N/A^2} * 1_A}{1_m} = 2 * 10^-7 \ N/A*m
[/tex]


Are these correct results? Is that per unit length, per infinity, per what?
 
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  • #14
DaleSpam said:
Hi Dunnis, look at the units of your equation. In SI units F is in N/m. That is where the length comes in. Go ahead and work it out for yourself in detail, you always need to keep track of the units.

No, in the equation F= 2* mu0/(4Pi*r) * I1*I2 you can determine that F has units of N/m.

I already did that, see above, and here is again:

Lorentz force, PER ONE METER:

F= I*L x B = 1A * 1m x 10^-7 N/(A*m) = 1*10^-7 Newtons



Ampere's force law, PER %#@$?:

F= 2*mu0/(4Pi*r)*I1*I2 = 2* 10^-7 N/A^2 *1A*1A = 2*10^-7 Newtons



Can I see you "calculation" and dimensional analysis now, how did you come up with your conclusion?
 
  • #15
Dunnis said:
Everyone said Ampere's force law (and later Ampere's law for B field) work ONLY with infinite wires, but is that supposed to mean "work in general" or "useless"?

Let's find out, can they be applied in real world or not? Like this:

Code:
CASE A:                        CASE B:

+                   -          +          -
| I2= 1A            |          | I2= 1A   |
|===================|          |==========|
          |                          |     
          |r=1m                      |r=1m 
          |                          |     
|===================|          |==========|
| I1= 1A            |          | I1= 1A   |
|<------- 9m ------>|          |<-- 1m -->| 
+                   -          +          -

CASE A:
[tex]
B_1 = \frac{\mu_0*I1}{2\pi r} = 2* \frac{10^-7_{N/A^2} * 1_A}{1_m} = 2 * 10^-7 \ N/A*m
[/tex]


CASE B:
[tex]
B_1 = \frac{\mu_0*I1}{2\pi r} = 2* \frac{10^-7_{N/A^2} * 1_A}{1_m} = 2 * 10^-7 \ N/A*m
[/tex]


Are these correct results? Is that per unit length, per infinity, per what?

Ampere's Force Law works for the general case. It is a marriage of Ampere's Law and the Lorentz Force, all of which are generally valid (though Ampere's Law is not sufficient by itself to describe an electrodynamic system in its entirety).

You, however, are using a special case of Ampere's Force Law that has been derived for the specific situation of two magnetostatic infinite wires.

Dunnis said:
I already did that, see above, and here is again:

Lorentz force, PER ONE METER:

F= I*L x B = 1A * 1m x 10^-7 N/(A*m) = 1*10^-7 Newtons



Ampere's force law, PER %#@$?:

F= 2*mu0/(4Pi*r)*I1*I2 = 2* 10^-7 N/A^2 *1A*1A = 2*10^-7 Newtons



Can I see you "calculation" and dimensional analysis now, how did you come up with your conclusion?

You did it incorrectly. As Dalespam stated, work out the units yourself explicitly and you will see that they do not match up. More specifically, the first equation, your Lorentz Force, is not the force per meter (N/m), that is the force (N). The second one is the force per meter (N/m). Oh yeah, and as I mentioned before you used an incorrect equation for the magnetic field in the first equation too.
 
Last edited:
  • #16
Dunnis said:
Everyone said Ampere's force law (and later Ampere's law for B field) work ONLY with infinite wire.
Actually, if you look back over the thread, you'll see only you said that. What the others said was the formula you wrote down, not Ampere's law, applies to the case of infinitely long wires.

Dunnis said:
Ampere's force law, PER %#@$?:

F= 2*mu0/(4Pi*r)*I1*I2 = 2* 10^-7 N/A^2 *1A*1A = 2*10^-7 Newtons
Since I imagine you're just going to keep repeating yourself, saying you already did work out the units, I'll just ask, What happened to the 4πr?
 
  • #17
I have to warn you all that Dunnis is very stubborn, arrogant and rude. I've seen this in his previous posts. He clearly does not know basic EM theory, and probably does not have the full foundation to properly learn the subject well. He is often asking basic questions and then refusing to accept the answers.

There is nothing wrong with being ignorant and asking questions to educate oneself. Afer all, this is how we all learn. However, this must be done with humility and not with arrogance, rudeness and disbelief.

Dunnis doesn't seem to understand that many of us have studied EM theory and practice for decades, even after taking several undergrad and grad level courses on the subject. Some of us even use EM theory in everyday engineering and scientific work throughout our lives. Why does he come here asking questions as if we know the answer, and then turn around and tell us that everything we say is wrong? It's very strange and very annoying.

Until Dunnis learns the basic principles of self-education, he will make little progress. I for one will not waste any more of my time trying to talk to him, at least until I see evidence that he has changed his attitude. Also, if he doesn't change his attitude, he should be banned from these forums, in my opinion. He is not helping anyone here, and is not allowing anyone here to help him.
 
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  • #18
elect_eng said:
I have to warn you all that Dunnis is very stubborn, arrogant and rude. I've seen this in his previous posts. He clearly does not know basic EM theory, and probably does not have the full foundation to properly learn the subject well. He is often asking basic questions and then refusing to accept the answers.

There is nothing wrong with being ignorant and asking questions to educate oneself. Afer all, this is how we all learn. However, this must be done with humility and not with arrogance, rudeness and disbelief.

Dunnis doesn't seem to understand that many of us have studied EM theory and practice for decades, even after taking several undergrad and grad level courses on the subject. Some of us even use EM theory in everyday engineering and scientific work throughout our lives. Why does he come here asking questions as if we know the answer, and then turn around and tell us that everything we say is wrong? It's very strange and very annoying.

Until Dunnis learns the basic principles of self-education, he will make little progress. I for one will not waste any more of my time trying to talk to him, at least until I see evidence that he has changed his attitude. Also, if he doesn't change his attitude, he should be banned from these forums, in my opinion. He is not helping anyone here, and is not allowing anyone here to help him.

True, but it is always a bit irresponsible to leave incorrect or misleading statements alone. It seems that people often pull up old posts here on the forums when they are looking for answers as indicated by the occasional necro thread coming up every so often. In addition, I find that Physics Forums has enough web presence to come up in response to Google searches. So while the immediate goal may never be obtained, we can at least provide a cogent explanation to the uninitiated about why and how he is wrong in his posts. Besides, most of this stuff is general enough that I only need to spend as much time in my answers as it takes to type out the post so little of my own time is lost.
 
  • #19
Dunnis said:
Ampere's force law, PER %#@$?:

F= 2*mu0/(4Pi*r)*I1*I2 = 2* 10^-7 N/A^2 *1A*1A = 2*10^-7 Newtons



Can I see you "calculation" and dimensional analysis now, how did you come up with your conclusion?
In your analysis you neglected r. So mu0 has units of N/A², r has units of m, and I1 and I2 each have units of A. So you get

2*mu0/(4Pi*r)*I1*I2 -> N/A²/m*A*A= N/m
 
  • #20
Born2bwire said:
True, but it is always a bit irresponsible to leave incorrect or misleading statements alone.

I agree completely. This is a Catch-22 for sure. I was forced to respond to him when he interrupted another thread I was talking in. I normally don't like to call the moderators, but it was necessary to shut down that thread just to prevent his misinformation from propagating.

However, this guy will keep on going like the energizer bunny, so I just thought I'd warn you. You may be in for a rough ride. :eek:
 
  • #21
DaleSpam said:
In your analysis you neglected r. So mu0 has units of N/A², r has units of m, and I1 and I2 each have units of A. So you get

2*mu0/(4Pi*r)*I1*I2 -> N/A²/m*A*A= N/m

Oops, true. Thank you.


Lorentz force, PER ONE METER:

F= I*L x B = 1A * 1m x 10^-7 N/(A*m) = 1*10^-7 N


Ampere's force law, PER %#@$?:

F= 2*mu0/(4Pi*r)*I1*I2 = 2* 10^-7 N/A^2 *1A*1A = 2*10^-7 N/m


Now they are even more different. Anything else?
 
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  • #22
vela said:
Actually, if you look back over the thread, you'll see only you said that. What the others said was the formula you wrote down, not Ampere's law, applies to the case of infinitely long wires.

"formula you wrote down"? Say it properly please so we do not end up arguing semantics.

Are you saying the Lorentz force: F= I*L x B, applies only to infinitely long wires?



Since I imagine you're just going to keep repeating yourself, saying you already did work out the units, I'll just ask, What happened to the 4πr?

4Pi got canceled with magnetic constant, and r I forgot about.
 
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  • #23
Dunnis said:
Oops, true. Thank you.


Lorentz force, PER ONE METER:

F= I*L x B = 1A * 1m x 10^-7 N/(A*m) = 1*10^-7 \ N


Ampere's force law, PER %#@$?:

F= 2*mu0/(4Pi*r)*I1*I2 = 2* 10^-7 N/A^2 *1A*1A = 2*10^-7 \ N/m


Now they are even more different. Anything else?

You're still using the wrong equation for the magnetic field but dimensionally that is correct now.
 
  • #24
Born2bwire said:
You're still using the wrong equation for the magnetic field but dimensionally that is correct now.

No, I'm using Biot-Savart law. Either of these two will do just fine:

9a1d819b700e7811aab6a7d57f661136.png


6bb1d60bd48bb83ace488aa5e7b87cdf.png


http://en.wikipedia.org/wiki/Biot-savart_law


..and this for Lorentz Force, either will do fine (where B= Biot-Savart from above):

6824df1073be46237e36da197f855c04.png


[URL]http://upload.wikimedia.org/math/c/0/e/c0e8bcfe4daee2939ba4b805b1acbba1.png[/URL]

[tex]
F = q*v \times \mathbf{B}[/tex]

http://en.wikipedia.org/wiki/Lorentz_force



...together they produce this:
[URL]http://upload.wikimedia.org/math/1/c/e/1ce896fcc3d7fdd1d0203706b7599c40.png[/URL]

http://en.wikipedia.org/wiki/Ampère's_force_law



...which is integral form of this equation given by 'Le Bureau international des poids et mesures' (BIPM), the most general and exact definition for magnetic force:


d2F = mu0/(4pi*r^3) * i1*dl1 x (i2*dl2 x r)

http://www.bipm.org/en/si/si_brochure/chapter1/1-2.html


These equation above is what defines the unit of ampere and along with it all the rest of electrics and electronics, and any other equation that has anything to do with any el. currents. These two are Biot-Savart and Lorentz force equations in their full integral form, where arbitrary segments and relative geometry can be integrated. That's what "general" means.



Born2bwire said:
Ampere's Force Law works for the general case. It is a marriage of Ampere's Law and the Lorentz Force, all of which are generally valid (though Ampere's Law is not sufficient by itself to describe an electrodynamic system in its entirety).

You, however, are using a special case of Ampere's Force Law that has been derived for the specific situation of two magnetostatic infinite wires.

This all IS magnetostatics anyway, we are talking about steady currents and the definition of ampere unit. There is no such thing as "special case" of Ampere's Force Law. -- This is Ampere's Force Law: F = 2*mu0/(4Pi*r)*I1*I2; there is no other, or special version, and that particular equation is not general. General is something you can apply to any scenario, with arbitrary geometry and arbitrary wire, or wire-segment, lengths.


Code:
CASE A:                        CASE B:

+                   -          +          -
| I2= 1A            |          | I2= 1A   |
|===================|          |==========|
          |                          |     
          |r=1m                      |r=1m 
          |                          |     
|===================|          |==========|
| I1= 1A            |          | I1= 1A   |
|<------- 9m ------>|          |<-- 1m -->| 
+                   -          +          -


CASE A: dl1 || dl2, i1= 1A, i2= 1A, r= 1m, L= 9m
======================================

Mathematica - WIRE 1:
D[10^-7Newtons/1^2 * 9*{l1,0,0} cross 9*{l2,0,0} cross {0,-1,0}, l1,l2] = [0, + 8.1*10^-6 N, 0]

Mathematica - WIRE 2:
D[10^-7Newtons/1^2 * 9*{l1,0,0} cross 9*{l2,0,0} cross {0,1,0}, l1,l2] = [0, - 8.1*10^-6 N, 0]

Execute on-line, here: www.wolframalpha.com/input




CASE B: dl1 || dl2, i1= 1A, i2= 1A, r= 1m, L= 1m
======================================

* L = int(dl)= unit length = 1m
* d2F(single wire) = mu0/(4pi*r^3) * i1*dl1 x (i2*dl2 x r)

=> mu0/(4pi*|1m|^2) * 1A*1m x 1A*1m = 1*10^-7 N


Mathematica - WIRE 1:
D[10^-7Newtons/1^2 * 1*{l1,0,0} cross 1*{l2,0,0} cross {0,-1,0}, l1,l2] = [0, + 1*10^-7 N, 0]

Mathematica - WIRE 2:
D[10^-7Newtons/1^2 * 1*{l1,0,0} cross 1*{l2,0,0} cross {0,1,0}, l1,l2] = [0, - 1*10^-7 N, 0]

Execute on-line, here: www.wolframalpha.com/input
 
Last edited by a moderator:
  • #25
Dunnis said:
Lorentz force, PER ONE METER:

F= I*L x B = 1A * 1m x 10^-7 N/(A*m) = 1*10^-7 NAmpere's force law, PER %#@$?:

F= 2*mu0/(4Pi*r)*I1*I2 = 2* 10^-7 N/A^2 *1A*1A = 2*10^-7 N/mNow they are even more different. Anything else?
I don't know how you got your number for the Lorentz force, but it is wrong. Your number for the Ampere's force is correct. In any case, you can show that they are equivalent algebraically.

F = I*L x B
F/L = I1*B = I1*mu0*I2/(2Pi*r) = 2*mu0/(4Pi*r)*I1*I2
 
  • #26
DaleSpam said:
I don't know how you got your number for the Lorentz force, but it is wrong.

I know how he got it. Look above and he says he uses Biot Savart Law. He shows the integration, but then show a 1/(4pi*r^2) dependence for the magnetic field, even after the integration. He just doesn't understand basic calculus.

This is exactly the argument I got in with him in the thread he interrupted. He can't seem to understand that an infinitely long wire has a 1/r dependence of the field. We showed him derivations and quoted books etc. He just refuses to believe anybody. This is exactly what I was trying to warn you about.

No matter what you say, he just will not believe you.

Apply his formulas for the case with r=10 meters, and you will really see the discrepancy.
 
Last edited:
  • #27
Dunnis said:
No, I'm using Biot-Savart law. Either of these two will do just fine:

9a1d819b700e7811aab6a7d57f661136.png


6bb1d60bd48bb83ace488aa5e7b87cdf.png


http://en.wikipedia.org/wiki/Biot-savart_law
No, the first one will do just fine, and if you carry out the integration properly, you will find the the magnetic field of an infinite straight wire is [itex]\textbf{B}=\frac{\mu_0 I}{2\pi r}\hat{\mathbf{\phi}}[/itex], where [itex]r[/itex] is the perpendicular distance from the wire, and [itex]\hat{\mathbf{\phi}}[/itex] is a unit vector pointing circumferentially around the wire.

The second one only gives you the approximate magnetic field for a point charge moving at constant, non-relativistic velocity [itex]\textbf{v}[/itex], not the field of a long straight wire.
 
  • #28
Dunnis said:
No, I'm using Biot-Savart law. Either of these two will do just fine:

9a1d819b700e7811aab6a7d57f661136.png


6bb1d60bd48bb83ace488aa5e7b87cdf.png


http://en.wikipedia.org/wiki/Biot-savart_law


..and this for Lorentz Force, either will do fine (where B= Biot-Savart from above):

6824df1073be46237e36da197f855c04.png


[URL]http://upload.wikimedia.org/math/c/0/e/c0e8bcfe4daee2939ba4b805b1acbba1.png[/URL]

[tex]
F = q*v \times \mathbf{B}[/tex]

http://en.wikipedia.org/wiki/Lorentz_force



...together they produce this:
[URL]http://upload.wikimedia.org/math/1/c/e/1ce896fcc3d7fdd1d0203706b7599c40.png[/URL]

http://en.wikipedia.org/wiki/Ampère's_force_law



...which is integral form of this equation given by 'Le Bureau international des poids et mesures' (BIPM), the most general and exact definition for magnetic force:


d2F = mu0/(4pi*r^3) * i1*dl1 x (i2*dl2 x r)

http://www.bipm.org/en/si/si_brochure/chapter1/1-2.html


These equation above is what defines the unit of ampere and along with it all the rest of electrics and electronics, and any other equation that has anything to do with any el. currents. These two are Biot-Savart and Lorentz force equations in their full integral form, where arbitrary segments and relative geometry can be integrated. That's what "general" means.





This all IS magnetostatics anyway, we are talking about steady currents and the definition of ampere unit. There is no such thing as "special case" of Ampere's Force Law. -- This is Ampere's Force Law: F = 2*mu0/(4Pi*r)*I1*I2; there is no other, or special version, and that particular equation is not general. General is something you can apply to any scenario, with arbitrary geometry and arbitrary wire, or wire-segment, lengths.


Code:
CASE A:                        CASE B:

+                   -          +          -
| I2= 1A            |          | I2= 1A   |
|===================|          |==========|
          |                          |     
          |r=1m                      |r=1m 
          |                          |     
|===================|          |==========|
| I1= 1A            |          | I1= 1A   |
|<------- 9m ------>|          |<-- 1m -->| 
+                   -          +          -


CASE A: dl1 || dl2, i1= 1A, i2= 1A, r= 1m, L= 9m
======================================

Mathematica - WIRE 1:
D[10^-7Newtons/1^2 * 9*{l1,0,0} cross 9*{l2,0,0} cross {0,-1,0}, l1,l2] = [0, + 8.1*10^-6 N, 0]

Mathematica - WIRE 2:
D[10^-7Newtons/1^2 * 9*{l1,0,0} cross 9*{l2,0,0} cross {0,1,0}, l1,l2] = [0, - 8.1*10^-6 N, 0]

Execute on-line, here: www.wolframalpha.com/input




CASE B: dl1 || dl2, i1= 1A, i2= 1A, r= 1m, L= 1m
======================================

* L = int(dl)= unit length = 1m
* d2F(single wire) = mu0/(4pi*r^3) * i1*dl1 x (i2*dl2 x r)

=> mu0/(4pi*|1m|^2) * 1A*1m x 1A*1m = 1*10^-7 N


Mathematica - WIRE 1:
D[10^-7Newtons/1^2 * 1*{l1,0,0} cross 1*{l2,0,0} cross {0,-1,0}, l1,l2] = [0, + 1*10^-7 N, 0]

Mathematica - WIRE 2:
D[10^-7Newtons/1^2 * 1*{l1,0,0} cross 1*{l2,0,0} cross {0,1,0}, l1,l2] = [0, - 1*10^-7 N, 0]

Execute on-line, here: www.wolframalpha.com/input

Oh ok! I see now. The problem is you don't understand how to do calculus or read. That's ok. See you need to integrate that first equation over an infinite line current and then you will get the correct magnetic field. I know, but don't worry. The correct equation has been given to you by several people not only in this thread but in other threads too. I can see how you could have been confused by their explicit statements. I know I would be if I was told the same thing over and over again.

Now for your illiteracy. This is a tough one but I think we can handle it. You see, when you linked that weird equation from the wikipedia article, the article stated:

The general formulation of the magnetic force for arbitrary geometries is based on line integrals and combines Biot-Savart law and Lorentz force in one double differential, or alternatively a double integral equation as shown below.
[URL]http://upload.wikimedia.org/math/1/c/e/1ce896fcc3d7fdd1d0203706b7599c40.png[/URL]

You'll notice that this was from the Ampere's Force Law article, note that it says FORCE here. So that means that this is one version of the Ampere Force Law. Also notice that it is derived from the Biot-Savart Law which means that this applies to MAGNETOSTATICS. It's a big word but I know you can do it. See, this is the GENERAL MAGNETOSTATICS AMPERE FORCE LAW which you said does not exist. I mean that's what you seem to be saying here:

This all IS magnetostatics anyway, we are talking about steady currents and the definition of ampere unit. There is no such thing as "special case" of Ampere's Force Law. -- This is Ampere's Force Law: F = 2*mu0/(4Pi*r)*I1*I2; there is no other, or special version, and that particular equation is not general. General is something you can apply to any scenario, with arbitrary geometry and arbitrary wire, or wire-segment, lengths.

Well and good. You must have been confused by the extremely obscure statement in the wikipedia article, which I'm sure you read very carefully, that stated:

The best-known and simplest example of Ampère's force law, which underlies the definition of the ampere, the SI unit of current, states that the force per unit length between two straight parallel conductors is
[PLAIN]http://upload.wikimedia.org/math/5/2/0/5209bb9b332287128335b53e61f93bc1.png[/quote] [Broken]

This means that the equation which you have been claiming is the Ampere's Force Law is the SIMPLEST EXAMPLE. What they are trying to say with their big $10 words is that this is a special case for TWO STRAIGHT PARALLEL WIRES. Oh look, they also say that it is the FORCE PER UNIT LENGTH. Those guys and their hidden messages.

So, let's try to put it all together. I know that this is very hard to comprehend but if you read this slowly and repeatedly you might understand. Now that means you can apply this equation to any geometry or arrangement of wires that your heart desires as long as we assume MAGNETOSTATICS. Amazing isn't it? And wouldn't you know? If you integrate it out for THE CASE OF TWO INFINITE PARALLEL WIRES you get that darn nifty equation that you have been trying to use this whole time. So it would appear then that the equation that you have been using is a SPECIAL CASE OF AMPERE'S FORCE LAW FOR TWO INFINITE PARALLEL WIRES. Amazing!

And gosh darnit, look at that, they also gave the units for the special case units of (N/m). Just like if it was force per unit length. Ohh... those wacky wikipedia people.

EDIT: Looks like wikipedia left out the word "infinite" in the description of the two straight conductors. That's ok. I fixed it for them. I think it was a relatively new addition so sometimes these little slips get in there but it is important to be exact.
 
Last edited by a moderator:
  • #29
gabbagabbahey said:
No, the first one will do just fine, and if you carry out the integration properly, you will find the the magnetic field of an infinite straight wire is [itex]\textbf{B}=\frac{\mu_0 I}{2\pi r}\hat{\mathbf{\phi}}[/itex], where [itex]r[/itex] is the perpendicular distance from the wire, and [itex]\hat{\mathbf{\phi}}[/itex] is a unit vector pointing circumferentially around the wire.

The second one only gives you the approximate magnetic field for a point charge moving at constant, non-relativistic velocity [itex]\textbf{v}[/itex], not the field of a long straight wire.

1C * 1m/s = 1C/s * 1m = 1A * 1m

One ampere IS one coulomb per meter per second, those two equations are identical. If a wire is carrying a current of one ampere there can not be more than one coulomb per meter of wire per second anyway, and so there is nothing to integrate, but if you integrated it over infinity you would only get infinity in any case. -- Two wires that ACTUALLY are one meter in length would attract more, less, or with the same force as two infinite wires, per meter?
 
Last edited:
  • #30
Born2bwire said:
5209bb9b332287128335b53e61f93bc1.png


1.) ...this is a SPECIAL case for TWO STRAIGHT PARALLEL WIRES.

2.) Now that means you can apply this equation to any geometry or arrangement of wires that your heart desires as long as we assume MAGNETOSTATICS. Amazing isn't it?


It's fascinating. SPECIAL is opposite to GENERAL and you are saying that equation is both, special and general?!

Code:
CASE A:                        CASE B:

+                   -          +          -
| I2= 1A            |          | I2= 1A   |
|===================|          |==========|
          |                          |     
          |r=1m                      |r=1m 
          |                          |     
|===================|          |==========|
| I1= 1A            |          | I1= 1A   |
|<------- 9m ------>|          |<-- 1m -->| 
+                   -          +          -


CASE A:
[tex]B_1 = \frac{\mu_0*I_1}{2\pi r} = 2* \frac{10^-7_{N/A^2} * 1_A}{1_m} = 2 * 10^-7 \ N/A*m[/tex]


CASE B:
[tex]B_1 = \frac{\mu_0*I_1}{2\pi r} = 2* \frac{10^-7_{N/A^2} * 1_A}{1_m} = 2 * 10^-7 \ N/A*m[/tex]



Is that correct real-world result? Wire length does not matter, is that the point?

Is that "apply this equation to any geometry or arrangement of wires that your heart desires"?





Born2bwire said:
5209bb9b332287128335b53e61f93bc1.png


1.) If you integrate it out for THE CASE OF TWO INFINITE PARALLEL WIRES you get that darn nifty equation that you have been trying to use this whole time.

2.) So it would appear then that the equation that you have been using is a SPECIAL CASE OF AMPERE'S FORCE LAW FOR TWO INFINITE PARALLEL WIRES. Amazing!


1.) Hahaa. You can not integrate that equation, it is not INTEGRAL, and you previously said it is already "integrated" (for cases with infinite wires). -- It is the other way around, that equation is DERIVATION of Biot-Savart law full integral form. You are confusing symbolic derivation and wonderland approximation with NUMERICAL INTEGRATION.


2.) Amazing indeed, quite funny too.


http://en.wikipedia.org/wiki/Ampère's_force_law
- "The general formulation of the magnetic force for arbitrary geometries is based on line integrals and combines Biot-Savart law and Lorentz force in one double differential, or alternatively a double integral equation as shown below.

1ce896fcc3d7fdd1d0203706b7599c40.png



This is Biot-Savart and Lorentz force in their FULL INTEGRAL FORM, that is what is general and why it is this particular equation that is given by the BIPM, and not any of the Ampere's laws. This equation needs not only to be GENERAL, but it also must be EXACT as it is what defines ampere unit and with it all the rest of electrics, electronics and any other equation that has anything to do with any el. currents.


Do you not understand Mathematica?
 
  • #31
Dunnis said:
One ampere IS one coulomb per meter per second

No, it is not. It is 1 C/s. As the equations you posted said.

Anyway, your question has been answered, over and over again. If you choose not to accept the answer, there's not much more to be done. We might as well close this thread.
 

1. What is Ampere's force law?

Ampere's force law, also known as Ampere's law, is a fundamental law of electromagnetism that describes the magnetic field generated by a current-carrying wire. It states that the magnetic field at a point is directly proportional to the current passing through a small loop around the point, and inversely proportional to the distance from the point to the wire.

2. What is wrong with Ampere's force law?

There are two main issues with Ampere's force law. Firstly, it only applies to steady currents, and does not account for time-varying currents. Secondly, it does not take into account the effects of displacement current, which is a type of current that arises from changing electric fields. This led to the development of Maxwell's equations, which provide a more complete description of electromagnetism.

3. How does Ampere's force law differ from Maxwell's equations?

Ampere's force law is a simplified version of Maxwell's equations, which take into account the effects of time-varying currents and displacement current. Maxwell's equations also include the concept of electric fields, which interact with magnetic fields to create electromagnetic waves. Ampere's law is a special case of one of Maxwell's equations, known as the Ampere-Maxwell law.

4. Why was Ampere's force law initially accepted if it is incomplete?

At the time of its development, Ampere's force law was a groundbreaking discovery and provided a significant understanding of the relationship between electricity and magnetism. It was only later, with the development of more advanced experiments and technologies, that the limitations of Ampere's law became apparent. Additionally, Maxwell's equations were not fully developed until several decades after Ampere's law was first proposed.

5. How does the failure of Ampere's force law impact our understanding of electromagnetism?

The failure of Ampere's force law highlights the importance of continually refining and improving scientific theories and models. It also demonstrates the interconnectedness of different branches of science, as the development of Maxwell's equations was crucial in understanding the relationship between electricity and magnetism. The failure of Ampere's law also reminds us that scientific theories are not absolute truths, but rather our best current understanding of the natural world.

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