Textbook about Inductance

[max.wwwang]

I was reading this when I was confused by the introduction about inductance - How can we get to (2-52) from (2-51)?

This is a page from an old book -- Basic Electronics for Scientists. I found this book very good.

[Moriambar]

You basically multiply the first by the area occupied by the circuit (A).
Ampère's law states that B, the magnetic field is proportional to the current (first equation). Then the total magnetic field that is contained within the area of the circuit is the first equation multiplied by the area

[max.wwwang]

You basically multiply the first by the area occupied by the circuit (A).
Ampère's law states that B, the magnetic field is proportional to the current (first equation). Then the total magnetic field that is contained within the area of the circuit is the first equation multiplied by the area

In the face of it, it obviously is. The problem here is 2-51 is about a conductor (presumably straight), so how could it hook up with the area of a circular-shaped wire and get 2-52? Am I missing anything very obvious?

I also struggle to understand 2-51 -- how could the magnetic field strength at a point some distance away from the conductor dependent on the length of conductor(l)? Imagine a very long conductor the ends of which is very far away from the point of measurement. Will more length of the conductor add more magnetic field strength to the point (still proportional to the conductor length)?!

[Circlotron]

I also struggle to understand 2-51 -- how could the magnetic field strength at a point some distance away from the conductor dependent on the length of conductor, l? Imagine a very long conductor the ends of which is very far away from the point of measurement. Will more length of the conductor add more magnetic field strength to the point (still proportional to the conductor length)?!

If the inductor is a coil of wire then more turns with the same current = more field strength. Unroll it and the situation remains the same I expect.

[max.wwwang]

... Unroll it and the situation remains the same I expect.

At least this is extremely unintuitive - I know it would be silly, or at least premature, to say the formula in the book is incorrect. Admittedly the situation seems a bit different for a coil comparing with a straight conductor (still not very helping though). But imagine a straight conductor, would an increase of one inch of wire 1,000 miles away have the same effect as the one inch of wire added very close to the point of measurement?! Crazy!

I very much like the way this book introduces the idea of magnetic fields. It's very precise and responsible to say that -- albeit very much in line with the observations -- it's only an idea (or conceptual aid) of dealing with such things like electromagnetic induction. We cannot assume this really is how things work!

I cannot get away from the impression that this topic is treated without clarity unlike the rest of the same book (I dare to speculate it's a result of the author's lack of full understanding hence lack of confidence?). But as a reader I cannot be very sure. I'm also conscious and cautious about the fact that English is not my first language. Do you also find the reference to "circuit" (underlined, especially the first one) very abrupt? (I would be surprised but still happy if you all say NO, but even this would be very helpful to me.)

[westfw]

how could the magnetic field strength at a point some distance away from the conductor dependent on the length of conductor, l?

Well, of course it does.  At some fundamental level, the magnetic field is generated by moving electrons, and the longer the wire, the more moving electrons you have.  Presumably you could derive the equation by integrating the field caused by each electron over all the electrons involved.  (though I'm not exactly sure what "r" becomes in that situation - shortest distance?  distance to each point?)

As the length of the wire gets very long compared to the distance from the wire (as in almost any real electronic device), the r² term dominates.
There's a lot of this sort of thing that happens in physics.  Edge effects that are usually left unconsidered.  Another obvious example is the electric field between the plates of a capacitor: assumed to be a pretty constant value.  But it isn't, near the edges of the plates.

[max.wwwang]

Well, of course it does.  At some fundamental level, the magnetic field is generated by moving electrons, and the longer the wire, the more moving electrons you have.  Presumably you could derive the equation by integrating the field caused by each electron over all the electrons involved.

I see your point and agree that the length should have some effect (but my agreement is limited to this point). Its effect (as I understand), however, is not a stronger magnetic at the same given point, but rather more points (the same distance away from the conductor) you can have the same magnetic field (along more length of the wire).

Of course, I could be wrong.

(though I'm not exactly sure what "r" becomes in that situation - shortest distance?  distance to each point?)

I take it as the shortest distance from the point (of measurement) to the (straight) conductor with the line along the distance perpendicular to the conductor.

As the length of the wire gets very long compared to the distance from the wire (as in almost any real electronic device), the r² term dominates.

I'm unable to see how r^2 would dominate. This does not change the fact that the magnetic field effect is always linearly proportional to length.

There's a lot of this sort of thing that happens in physics.  Edge effects that are usually left unconsidered.  Another obvious example is the electric field between the plates of a capacitor: assumed to be a pretty constant value.  But it isn't, near the edges of the plates.

I don't have any problem with the edge effect. There is Saint Venant's Principle in mechanics and many other similar things in other disciplines. That's exactly why I use an extremely long wire (more than 1,000 miles long) as an example.

[max.wwwang]

At one point I asked myself - is it fundamentally incorrect to think of a straight conductor without a loop because current only exists in a loop? This isn't true - imagine a very long straight wire (one end in Beijing the other in Washington. And to make this precisely accurate it goes through the earth, not over its surface which would make it curve-shaped) and have a metal plate at the end of Washington. Suppose one man at this side approaches the plate with a ball with a great amount of negative charge. There will be a current flowing from Beijing to Washington along this wire, won't there?

[Mechatrommer]

At least this is extremely unintuitive - I know it would be silly, or at least premature, to say the formula in the book is incorrect. Admittedly the situation seems a bit different for a coil comparing with a straight conductor (still not very helping though). But imagine a straight conductor, would an increase of one inch of wire 1,000 miles away have the same effect as the one inch of wire added very close to the point of measurement?! Crazy!

that formula is a good recap for me to remind/correct the fundamental idea of an "inductance" that it is actually proportional to the length and the idea of "magnetic flux". maybe we are confusing things with "flux density". flux density should be different when one wire is straight than when its coiled into compact geometry. but both will have the same amount of "flux". no? so for your wire washington to beijing, there will be a great deal amount of fluxes involved and a great "inductance" ie opposing to current change (di/dt) so when one applied a statics on the other end, you should feel it few seconds later on the other end, not instantaneously, due to inductance effect.

[max.wwwang]

that formula is a good recap for me to remind/correct the fundamental idea of an "inductance" that it is actually proportional to the length and the idea of "magnetic flux". maybe we are confusing things with "flux density". flux density should be different when one wire is straight than when its coiled into compact geometry. but both will have the same amount of "flux". no? so for your wire washington to beijing, there will be a great deal amount of fluxes involved and a great "inductance" ie opposing to current change (di/dt) so when one applied a statics on the other end, you should feel it few seconds later on the other end, not instantaneously, due to inductance effect.

No, I'm not mixing up flux density (i.e. magnetic field strength) with flux. I understand the inductance should be proportional to length  - particularly easy to understand for a coil because similar more rounds of wires are working together to multiply the flux (and flux density). But the flux density for a straight conductor is different to me - the extra length of wire only creates another similar field (of the same density in its own surrounding space) but not adding to the field of its adjacent segment of wire.

[max.wwwang]

Not to sidetrack the discussion, but I watched this video which is also about electromagnetic induction and found he experiment from 48:20 or so is very eye-opening and thought-provoking.

Not only other professors referred to by professor Lewin, I deeply double the experiment (though, of course, I don't think he is cheating; and my doubt or difficulty in understanding is not a surprise). But I think he may have been misled by some physical setup of the experiment. How arrogant I sound!

[Mechatrommer]

At least this is extremely unintuitive - I know it would be silly, or at least premature, to say the formula in the book is incorrect. Admittedly the situation seems a bit different for a coil comparing with a straight conductor (still not very helping though). But imagine a straight conductor, would an increase of one inch of wire 1,000 miles away have the same effect as the one inch of wire added very close to the point of measurement?! Crazy!

r is the distance of wire from the field being measured. if you add 1 inch wire in beijing, divided by great distance of r squared, it doesnt matter anyway, µ is already a very small constant.

[max.wwwang]

At least this is extremely unintuitive - I know it would be silly, or at least premature, to say the formula in the book is incorrect. Admittedly the situation seems a bit different for a coil comparing with a straight conductor (still not very helping though). But imagine a straight conductor, would an increase of one inch of wire 1,000 miles away have the same effect as the one inch of wire added very close to the point of measurement?! Crazy!

r is the distance of wire from the field being measured. if you add 1 inch wire in beijing, divided by great distance of r squared, it doesnt matter anyway, µ is already a very small constant.

That's true. But the effect of this one more inch of wire has exactly the same effect of the inch of wire right crossing the plane containing the point and perpendicular to the conductor (i.e. closest to the point of measure), no matter how great 'r' might be and how small effect it is for any one inch of length. Let's assume the point of measurement has the same distance to both ends of the conductor (i.e. in a plane that is perpendicular to the wire and cuts the wire into two segments of equal length).

Let me clarify, 'r' does not change for a given point if you add more wire along either or both ends of the wire (fixed relative to the point). In other words, adding another 1,000 miles of the wire at one end of it only doubles the flux density at the same point, according to 2-51 (which does not seem right to me).

[Edit - this is rightly presumed based on the wording of the book but it's incorrect. Refer to #18 for the correct meaning of "r".]

[max.wwwang]

I guess perhaps formula 2-51 holds only for a coil (not applicable to a straight conductor) but this has been made clear in the book (I mean this page; this is the very beginning of the discussion about inductance in this book)?

[max.wwwang]

I found this formula here, seems more convincing than 2-51 --

It doesn't include the length of conductor in it, only I and r.

[T3sl4co1l]

At one point I asked myself - is it fundamentally incorrect to think of a straight conductor without a loop because current only exists in a loop?

When this is done mathematically, the return path happens at infinity.  There must always be a loop, yes.  Magnetism is meaningless without loops (at least until monopoles are discovered).

If you're looking for the inductance per length of wire in free space, it depends on geometry.  A multi-turn coil has much higher inductivity than a loop, and a large loop has more than a small loop (um, hm, relative to wire diameter I think?).  There is no single answer, but they are all proportional to the inductivity of free space, mu_0.  For conductor pairs of prismatic cross-section, this is a transmission line property; together with e_0 and the geometry factors, you get Zo and c_0 (impedance and velocity).

Tim

[Mechatrommer]

and then there's permeability that can affect inductance... formula is for air... here if you want the more truth, it will involve derivation... have fun...
http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/amplaw.html

and it seems somehow l.sin(Ø) = 4πr (assuming infinitely long wire?), and then it can be deduced to your later formula. go figure... i want to make circuit...

[pwlps]

I cannot get away from the impression that this topic is treated without clarity unlike the rest of the same book (I speculate it's a result of lack of full understanding hence lack of confidence?). But as a reader I cannot be very sure. I'm also conscious and cautious about the fact that English is not my first language. Do you also find the reference to "circuit" (underlined, especially the first one) very abrupt? (I would be surprised but still happy if you all say NO, but even this would be very helpful to me.)

You are perfectly right, the explanation is misleading.  But the problem is rather subtle and I have seen many bad explanations.

Ampere's law is an integral version of a local law (Maxwell equation) : integral of B on a contour around a conductor is proportional to the current in the conductor. From this you can calculate the magnetic field only in a situation of infinite straight conductor where the cylindrical symmetry simplifies the problem. There shouldn't be any "l" in 2.52 if it is said to be a consequence of Ampere's law, period.

I also struggle to understand 2-51 -- how could the magnetic field strength at a point some distance away from the conductor dependent on the length of conductor(l)? Imagine a very long conductor the ends of which is very far away from the point of measurement. Will more length of the conductor add more magnetic field strength to the point (still proportional to the conductor length)?!

Actually 2.51 is the Biot-Savart law. This is also an integral law telling how to sum contributions from different portions of a circuit so that it is usually written as dB=const*I dl sin(theta)/r^2.
In the limit of very long conductor the contributions from the ends will progressively vanish because of sin(theta) which goes to zero, therefore B is NOT proportional to the conductor length.

The Biot-Savart law is ONLY correct (equivalent to Ampere law) if integrated over either an infinite or a closed loop conductor. Otherwise it would obviously violate the charge conservation.

But the funny thing is that an incomplete (finite conductor length) integration has a well defined physical meaning, this comes from the fact that combining Biot-Savart with the continuity equation (charge conservation) yields Ampere's law:

https://physics.stackexchange.com/questions/495625/derivation-of-amperes-law-from-biot-savart

As they point out, Biot-Savart "is inconsistent with the continuity equation. We don't know how to generalize Biot-Savart, but patching up the problem with the continuity equation in the simplest possible way yields the correct Ampere's law".

What does this means physically ? A simple interpretation is that integration of Biot-Savart over a finite length conductor will yield a magnetic field that would be measured if we put e.g. two charged spheres (acting as a charge source/sink) at both ends of the conductor.
The Ampere's law is recovered because we now have varying electric field from the spheres, this varying electric field gives a contribution to the total magnetic field which happens to compensate exactly for the missing part of the conductor, from the actual ends to infinity.  Finally electromagnetism laws are very coherent and robust if we know how to interpret them correctly

[max.wwwang]

You are perfectly right, the explanation is misleading.  But the problem is rather subtle and I have seen many bad explanations.

Ampere's law is an integral version of a local law (Maxwell equation) : integral of B on a contour around a conductor is proportional to the current in the conductor. From this you can calculate the magnetic field only in a situation of infinite straight conductor where the cylindrical symmetry simplifies the problem. There shouldn't be any "l" in 2.52 if it is said to be a consequence of Ampere's law, period.

Thanks for seconding my understanding! That's very helpful and encouraging. I also found "l" in 2-52 very difficult to understand. With below in mind, I daresay both 2-51 and 2-52 are not only misleading; they are incorrect.

Actually 2.51 is the Biot-Savart law. This is also an integral law telling how to sum contributions from different portions of a circuit so that it is usually written as dB=const*I dl sin(theta)/r^2.
In the limit of very long conductor the contributions from the ends will progressively vanish because of sin(theta) which goes to zero, therefore B is NOT proportional to the conductor length.

Thinking about 2-51 and theta again and again before I went to bed last night, without resorting to any textbook (proudly!), I reached the same thing (though only in the form of speculation)-
1) both "B" and "l" should actually be differentials, i.e. dB and dl; and
2) we must take the vector (referred to as "radius vector" in the book) as starting from the differential segment of conductor, dl, to the point of consideration.

Without this, the inclusion of theta in the formula is meaningless - would always equal to 90 degrees otherwise. (And the term "radius vector" is very inaccurate and misleading, if not incorrect!)

With the above understanding, I now perfectly understand that every unit length of conductor does not play an equal role in contributing to the flux density measured at the given point. It's only proportional to sin(theta) which approaches sin(0)=0 when dl goes to infinity (relative to the point of measurement). This now makes sense to me.

I need more time to digest the rest of your reply. Thanks a lot. I cannot overstate how helpful this is to me.

[max.wwwang]

I had more thinking about the experiment at the end of the MIT course when I was running this early morning. I've become reasonably confident now that the professor was misled by and hence misinterpreting the observations of the experiment. Of course, I don't believe he intended to cheat but, unfortunately, he was only cheated (by the experiment). Will follow up when I have the thoughts tidied up.

[pwlps]

2) we must take the vector (referred to as "radius vector" in the book) as starting from the differential segment of conductor, dl, to the point of consideration.

Yes, the Biot-Savart law is more clearly expressed in its vector form:
https://www.quora.com/What-is-the-vector-form-of-Biot-Savart-law

With below in mind, I daresay both 2-51 and 2-52 are not only misleading; they are incorrect.

I agree, I wouldn't recommend this book if you want to understand the underlying physics - notwithstanding its appealing title it seems lacking basic scientific rigour (Written by an EE ?  )

[max.wwwang]

I agree, I wouldn't recommend this book if you want to understand the underlying physics - notwithstanding its appealing title it seems lacking basic scientific rigour (Written by an EE ?  )

Doh! Although I'm not an EE I have great respect for them, especially after I had a revisit, recently, of the achievements made by them during last century (especially the lower half) until today and the impact on the life of all of us. I couldn't help thinking, at least at one point, perhaps most of the smartest people were working in the field of EE.

It's a pity the book I'm reading is so short of scientific rigour! 

Very keen to have your thoughts on the experiment in the MIT course.

[max.wwwang]

The Biot-Savart law is ONLY correct (equivalent to Ampere law) if integrated over either an infinite or a closed loop conductor. Otherwise it would obviously violate the charge conservation.

I have been looking for the theoretical formula for the inductance of a straight conductor but have not found one by Google search. According to what you are saying here, it seems the inductance of a  conductor (integral) is meaningful only if either it's straight but infinitely long or it's a loop?

[pwlps]

Doh! Although I'm not an EE I have great respect for them, especially after I had a revisit, recently, of the achievements made by them during last century (especially the lower half) until today and the impact on the life of all of us. I couldn't help thinking, at least at one point, perhaps most of the smartest people were working in the field of EE.

Ah sorry for this bad joke (but I couldn't resist ), of course I have great admiration for EE too. I remember when trying to understand some RF devices I was wondering how they can have such a great insight into the working principles without the use of advanced mathematical calculations.

I have been looking for the theoretical formula for the inductance of a straight conductor but have not found one by Google search. According to what you are saying here, it seems the inductance of a  conductor (integral) is meaningful only if either it's straight but infinitely long or it's a loop?

As already pointed out by T3sl4co1l there is always a loop and pure straight wires don't exist.
That said straight infinite wire concept can be a good approximation in some circumstances. Take a loop of very thin wire (wire radius much smaller than loop dimensions). Then the majority of the flux comes from the field very close to the wire which varies as 1/r so that it can be approximated by a straight wire: as the contributions of remote parts of the loop are negligible we can integrate to infinity getting an approximate inductance per unit length proportional to log(1/radius). For intermediate cases there are approximate formulas combining together different regimes, see e.g.

https://en.wikipedia.org/wiki/Inductance#Inductance_of_a_straight_single_wire

Very keen to have your thoughts on the experiment in the MIT course.

Didn't have time to look at the video yet, maybe this evening.

[gcewing]

I don't think you can trust this book.

The first equation is not Ampere's Law. Ampere's Law is a relationship between the line integral of the magnetic field around a closed loop surrounding a conductor, and the current in the conductor.

I found a similar equation in my university physics textbook (University Physics, 5th Edition, by Sears, Zemansky & Young) on page 552. It's for the field at a point on the centre line through a circular loop of wire; r is the distance from a point on the loop to the measurement point, and theta is the angle between this line and the centre line.

I can't make any sense of the second equation. If it's supposed to be for the total flux through the loop, then there shouldn't be an r or theta in it at all.