Ampere's and Biot Savart Law for finite straight conductors

[aim1732]

Are magnetic field lines around a finite current carrying straight conductor concentric circles in plane perpendicular to length of wire? I have seen texts derive an expression for it :
B = μ₀.i/4πd [cos Φ₁-cosΦ₂]
where d is perpendicular distance of separation of the point from the wire and Φ₁ and Φ₂ are angles the end points of the wire subtend at the point.
Clearly all points at distance d from the wire have equal field magnitude, in direction perpendicular to the plane containing the wire and the point.
This is from Biot Savart Law.
The reason I doubt this is that if we imagine an Amperian loop along one of these circular field lines and calculate the line integral of B.dl it does not give me μ0.i. This procedure works well for infinitely long conductors---- I know that is a symmetrical situation suited for Ampere's Law but I see nothing different for finite conductors. According to the equation I wrote above field lines for finite conductors are identical to those for infinite conductors.

I am guessing I am missing something obvious. I would rather have someone tell me how symmetry manifests in Ampere's Law as against Gauss's Law-it was much simpler for Gauss's Law I guess.


All comments are welcome.

 

[jtbell]

In order for Ampere's Law to work, the net current has to be the same through all surfaces that are bounded by the Amperian loop. If the conductor has finite length this is not true. Suppose the Amperian loop is a circle going around the conductor at its midpoint. If the surface is a disk pierced by the conductor, then the current through that surface is obviously not zero. If the surface is a "pouch" that extends beyond one end of the conductor, so that the conductor does not pierce the surface, then the current through that surface is zero.

 

[yungman]

My understanding is ampere's law only concern with current I that pass through the surface, the length don't matter. Remember you have to think of current cannot begin or end at a point, that is current is in a loop too! No mater how you consider the serface, you only look at whether the current pierce through the surface and Ampere's law apply.

 

[Meir Achuz]

Ampere's law is derived from Curl B ~ j, which requires that div j=0. This is not true for a finite length straight wire, so Ampere's law does not hold for this case. The law of Biot-Savart also holds only for a closed circuit.
The formula for B from a finite length straight wire can only be used when the straight segment is part of a closed circuit.

 

[yungman]

Ampere's law is derived from Curl B ~ j, which requires that div j=0. This is not true for a finite length straight wire, so Ampere's law does not hold for this case. The law of Biot-Savart also holds only for a closed circuit.
The formula for B from a finite length straight wire can only be used when the straight segment is part of a closed circuit.

What is "Curl B ~ j"? I just don't understand the symbol.

Finite length of wire with current cannot exist and I think the original question implies that it is a closed loop with current flow.

 

[yungman]

Are magnetic field lines around a finite current carrying straight conductor concentric circles in plane perpendicular to length of wire? I have seen texts derive an expression for it :
B = μ₀.i/4πd [cos Φ₁-cosΦ₂]
where d is perpendicular distance of separation of the point from the wire and Φ₁ and Φ₂ are angles the end points of the wire subtend at the point.
Clearly all points at distance d from the wire have equal field magnitude, in direction perpendicular to the plane containing the wire and the point.
This is from Biot Savart Law.
I don't think that is true. Ampere's law only said the integration of a closed path around a current carrying element is equal to the current, but nothing about the magnitude of the field at any point of the closed path is equal even if it is equal distance from the center of the wire. For one you cannot have a simple section of straight wire with current I, it has to be a closed circuit. YOu can approximate an infinite long straight current carrying wire by making the straight section very long, but not a finite length of straight wire.


The reason I doubt this is that if we imagine an Amperian loop along one of these circular field lines and calculate the line integral of B.dl it does not give me μ0.i. This procedure works well for infinitely long conductors---- I know that is a symmetrical situation suited for Ampere's Law but I see nothing different for finite conductors. According to the equation I wrote above field lines for finite conductors are identical to those for infinite conductors.
According to Ampere's law, the integration over the closed path surrounding the wire is I, that don't mean the magnitude of the flux is equal at every point on the closed path.


I am guessing I am missing something obvious. I would rather have someone tell me how symmetry manifests in Ampere's Law as against Gauss's Law-it was much simpler for Gauss's Law I guess.


All comments are welcome.

I am just jointing in, I am no expert but I am studying this also and I find this very interesting think about a real life question and hitting the book to try to understand more. Join me in my post and put in your comments also.

 

[jtbell]

What is "Curl B ~ j"?.

$$\vec \nabla \times \vec B = \vec j$$

Have you studied vector calculus yet?

 

[yungman]

$$\vec \nabla \times \vec B = \vec j$$

Have you studied vector calculus yet?

Not with this symbol "~" in "Curl B ~ j"! I studied vector calculus, PDE, ODE and never seen that symbol used as "="! I thought it's a new operator I don't know. Besides

$$\nabla X \vec{B} = \mu_0 \vec{J}$$.

 

[aim1732]

Sorry the things you all are talking make no sense to me----all this is way above my level at the moment. It was just a question that my teacher asked us to consider, just to stimulate our thinking I guess---beyond the scope of my present study and any math I currently know.
So without interrupting your informed debate could you please explain in simpler terms what is wrong?

If the surface is a "pouch" that extends beyond one end of the conductor, so that the conductor does not pierce the surface, then the current through that surface is zero

.


Sorry I do not get what you mean by "pouch"?
Regrets for being the layman...

 

[yungman]

Sorry the things you all are talking make no sense to me----all this is way above my level at the moment. It was just a question that my teacher asked us to consider, just to stimulate our thinking I guess---beyond the scope of my present study and any math I currently know.
So without interrupting your informed debate could you please explain in simpler terms what is wrong?

.


Sorry I do not get what you mean by "pouch"?
Regrets for being the layman...

Think of a closed loop and the surface bounded inside the closed loop. Say the loop is a simple circle ( not limit to this ) and the surface can be just the circle inside or it can be a pouch ( like a bag ) where the opening is bounded by the same closed loop. Both are surfaces bounded by the closed curve.

In the ampere's law, steady current itself has to be a closed loop no matter what. If the current loop "string" through the boundary of the surface ( pierced through the surface ), it will pierce the surface no matter what is the shape of the surface.

My guess of his explanation is that if the shape of the surface is a pouch, then the current go inside the pouch and curve around and come back out of the pouch without "stringing" the closed boundary. In this case, the current according to ampere's law is still zero. The current loop has to "string" through the closed curve of the boundary before it counts.

Another way to think of it is consider a paper bag with the opening as the closed boundary. If you take a piece of rope and hold both ends and let the middle hang down to form a loop. You lower the loop of the rope through the opening of the bag into the bag. The rope actually pass the boundary of the bag, but if you follow one end of the rope and trace the rope, it will come out from the other end of the rope. the rope never penatrade the surface of the bag. It just go in and come out of the bag through the opening. On the other hand if you make a hole in the bag and string the rope through the hole so the rope go into the mouth of the bag, through the hole and come out on the side of the bag, then you string through the boundary of the bag.

 

[jtbell]

Not with this symbol "~" in "Curl B ~ j"! I studied vector calculus, PDE, ODE and never seen that symbol used as "="! I thought it's a new operator I don't know. Besides

$$\nabla X \vec{B} = \mu_0 \vec{J}$$.

Yeah, I left out the constant because I was in a hurry. Now that you've reminded me, I think the '~' in this content simply means something like "proportional to".

Sorry I do not get what you mean by "pouch"?

See the attached picture.

 

[yungman]

Yeah, I left out the constant because I was in a hurry. Now that you've reminded me, I think the '~' in this content simply means something like "proportional to".



See the attached picture.

Thanks

 

[jtbell]

Now that I think of it, your question actually ties in with an important discovery that Maxwell made when he formulated his equations for the electric and magnetic fields.

In a finite wire segment, you cannot have just a current, because you can't magically create charge at one end of the wire and make it disappear at the other end. However, you can have a net negative charge at one end and a net positive charge at the other end, both increasing or decreasing in magnitude, as the current carries charge from one end to the other. The charges at the ends of the wire produce electric fields which increase or decrease as time passes.

It turns out that Ampere's law is actually incomplete. Maxwell was the first to recognize this. He "completed" the law by adding what he called the "displacement current" to the "conduction current" that the wire carries. The displacement current exists everywhere in space (not just in the wire) and is related to the rate at which the electric field changes.

If we think in terms of displacement current, we can visualize a finite current-carrying wire segment with growing charges on the ends as follows: A displacement current comes in through space from infinity in all directions towards the negative charge. There it becomes a conduction current and travels along the wire to the positive charge. There, it becomes a displacement current again and spreads out through space towards infinity.

Ampere's law now works for this situation, provided that you use the total (conduction plus displacement) current through a surface bounded by the Amperian loop. This total current is the same for any surface bounded by the loop, including my "pouched" one.

Including the displacement current in Ampere's law and writing it in terms of the time derivative of the electric field, gives you the final form of what is sometimes called the Ampere-Maxwell law.

 

[yungman]

To aim1732

If you are interested in EM, I really suggest you to enroll in more calculus classes beyond the third semister calculus ( multi variables). Even in that class, it barely touch the important topics like vector fields, line integrals, green's theorem, divergence and stoke's theorem, coordinate systems. These by itself worth a separate class...I guess called "Vector Calculus". I studied these over like three times to get the "feel" of them. It is not enough to just knowing how to solve the problems, I am sure I can get "A" after studying the first time. I just finish studying this the third time after studying ODE and PDE and I am amaze the insight I got out of it. It is the insight of this that can really help you understand EM. Like the pouch thingy we are talking about is covered in the "Oriented surface" and in all the theorems I talked above. And they explain about why the shape of the surface is not important and you get the same result with all surface bounded by the same closed boundary.

And to really understand EM, I found ODE and particular PDE are very important also together with D'Alembert and Green's function. I am plenning to study the Complex Analysis in the future and then go back to study the Advance EM.

EM is a very difficult subject. I studied the first two semisters twice before and I am repeating it again. I know most of the math and derivations by heart and this time I am reading it as English and try to explain back to myself in English. I come up with so much question even at the beginners level. I have a thread right now here about the static magnetic boundary and I am still waiting for a reasonable answer. They all sound so easy until you get deeper into it.

I am retired and doing this as a hobby. I have the time! Math, math and more math! Take it from someone that had a full and successful career. I got very far without all the math but I always feel there is a hole somewhere. I am making it all up now. Math is the language of science ( not just EM) , all the advanced scientific books speak in terms of math formulas. Without that, you cannot fully appreciate the books.

 

[Meir Achuz]

I just used ~ as proportional, because I was too lazy to put in all the constants.