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MIT's Walter Lewin's twice surprises the EE professors! (fun)

by Cyberkatru
Tags: lewin, professors, surprises, walter
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Cyberkatru
#1
Jan9-07, 05:00 AM
P: n/a
MIT physicist/astronomer is quite fun to what lecture. In lectures 20
and 16 of his online basic E&M lectures he makes some interesting
points. I will tell you about both of them and ask your opinions but
notice that the second one I mention (which actually comes first in
lecture 18) is the most interesting and maybe a bit puzzling so be sure
to get through this whole post.

First from lecture 20:
At some point a little past the 8 minute mark in Walter Lewin's basic
E&M video lecture #20 he makes the following statement:
"Almost every college physics book does this wrong!"
He is referring to the use of Kirchoff's voltage rule for loops in
cicuits with self inductors.
I think he is right about this.
In a way he is denying the assumptions of the so-called "lumped matter
discipline" in electrical engineering--with regard to inductors
anyway.

It is all quite entertaining. You can see the video here starting at
about the 8 minute mark of lecture number 20.
..http://ocw.mit.edu/OcwWeb/Physics/8-...ures/index.htm

He also has a write up of this issue here:

http://ocw.mit.edu/NR/rdonlyres/Phys...0/lecsup41.pdf

So do we agree with him on this bit of basic physics 101?

Now on to the more interesting case of "nonconservative" fields and
wacky voltmeters:
This starts at about minute 38 in video lecture 16 when he commands to
"hold onto your hats!". A few minutes later gives a demonstation of
this where he says professors visiting in the audience refused to
believe what they were seeing.

Do you agree with Walter's interpretation of what is happening?

I think I basically agree but frankly I am not sure I like his
insistance that this is a case of what should be called
"nonconservative" fields (except in some unusual sense of
nonconservative). Here is why:

Mathematically, the field depends on time and so is now properly a
field on spacetime. Now a closed loop in spacetime should return not
only to the same place but also the same time. So he has not really
demonstrated that the field is nonconservative in the usual
mathematical sense (cohomology). He has just used a path that didn't
really return to the same point in the manifold (spacetime).

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William R. Frensley
#2
Jan11-07, 05:00 AM
P: n/a
Cyberkatru wrote:
> MIT physicist/astronomer is quite fun to what lecture. In lectures 20
> and 16 of his online basic E&M lectures he makes some interesting
> points. ...
>
> http://ocw.mit.edu/NR/rdonlyres/Phys...0/lecsup41.pdf
>
> So do we agree with him on this bit of basic physics 101?
>

There is most certainly a widespead misconception here, but unfortunately
this work does not find the real problem.

The fundamental misconception is the identification between the electrostatic
potential of Maxwell's theory and the circuit-theory concept of voltage, which
for reasons explained below should always be called "node voltage."
We all understand that the Maxwell quantity is defined as limit of U(r)/Q
as Q approaches zero, and U(r) is the potential energy of an infinitesmal test
charge Q. The circuit-theory voltage is actually (-) the electrochemical
potential for mobile electrons within a chunk of matter that contains enough
electrons to be considered an electron reservior, and that is in a local
quasi-equilibrium. This chunk of (almost always metallic) matter is represented
in circuit space as a "node" in the circuit graph.

The minus sign in the above definition of voltage is a result of the negative
charge on the electron. (Benjamin Franklin had a even chance, but unfortunately
he made the wrong choice.) The electrochemical potential for electrons is
known in semiconductor device theory as the (quasi)Fermi level, not to be
confused with other meanings of "Fermi energy" in solid-state physics. This is
the quantity that voltmeters measure, as pointed out by William Shockley
(Electrons and Holes in Semicnoductors, 1950, p. 305). To understand this,
let's assume that the voltmeter is a classic Wheatstone Bridge, consisting of
an ammeter connected between the unknown voltage node and an adjustable voltage
standard (which can be just a battery connected to a potentiometer). The
measurement procedure is to adjust the voltage standard until the current
through the ammeter reads zero. The condition for zero (electron) current is
that the electrochemical potentials for electrons on the two sides of the meter
must be equal.

When we understand node voltage to be this statistical quantity, all the
Krichoff's Law paradoxes disappear. The potential energy of the electrons in
the node can be raised or lowered by the electrostatic potential phi, or by
dA/dt: the voltage is still a scalar quantity. Its gradient is NOT in general
equal to the electric field E.

Why has the confusion between these quantities persisted for so long? The first
problem is that they are measured in the same units (Volts). The second reason
is that -> within a given metallic material <- the voltage and the electrostatic
potential are coupled by an additive constant, the work function. Connecting
two metals with different work functions produces a structure with a constant
voltage, but with a discontinuity in the electrostatic potential, which is
known as a "contact potential."

The additive linkage between potential and voltage is obviously broken in
vacuum, and also in most semiconductor device structures. A very little known
fact is that a difference in the spatial behavior of the potential and of
the voltage is a necessary condition for the operation of active (i.e. gain-
producing) electron devices.

- Bill Frensley

Chris H. Fleming
#3
Jan11-07, 05:00 AM
P: n/a
Cyberkatru wrote:
> MIT physicist/astronomer is quite fun to what lecture. In lectures 20
> and 16 of his online basic E&M lectures he makes some interesting
> points. I will tell you about both of them and ask your opinions but
> notice that the second one I mention (which actually comes first in
> lecture 18) is the most interesting and maybe a bit puzzling so be sure
> to get through this whole post.
>
> First from lecture 20:
> At some point a little past the 8 minute mark in Walter Lewin's basic
> E&M video lecture #20 he makes the following statement:
> "Almost every college physics book does this wrong!"
> He is referring to the use of Kirchoff's voltage rule for loops in
> cicuits with self inductors.
> I think he is right about this.
> In a way he is denying the assumptions of the so-called "lumped matter
> discipline" in electrical engineering--with regard to inductors
> anyway.
>
> It is all quite entertaining. You can see the video here starting at
> about the 8 minute mark of lecture number 20.
> .http://ocw.mit.edu/OcwWeb/Physics/8-...ures/index.htm
>
> He also has a write up of this issue here:
>
> http://ocw.mit.edu/NR/rdonlyres/Phys...0/lecsup41.pdf
>
> So do we agree with him on this bit of basic physics 101?



I don't get the last paragraph of the first page.

"There is no electric field in this loop if the resistance of the wire
making up the loop is zero."

Then he says.

"(this may bother you - if so, see the next section)"

Well I read on and it didn't stop bothering me. When he is explaining
this on page 4, in paragraph 3 he sets all the currents at every point
in the loop to be the same, but in the very next paragraph he
contradicts himself with charge buildups at certain points on the loop.


> Now on to the more interesting case of "nonconservative" fields and
> wacky voltmeters:
> This starts at about minute 38 in video lecture 16 when he commands to
> "hold onto your hats!". A few minutes later gives a demonstation of
> this where he says professors visiting in the audience refused to
> believe what they were seeing.
>
> Do you agree with Walter's interpretation of what is happening?
>
> I think I basically agree but frankly I am not sure I like his
> insistance that this is a case of what should be called
> "nonconservative" fields (except in some unusual sense of
> nonconservative). Here is why:
>
> Mathematically, the field depends on time and so is now properly a
> field on spacetime. Now a closed loop in spacetime should return not
> only to the same place but also the same time. So he has not really
> demonstrated that the field is nonconservative in the usual
> mathematical sense (cohomology). He has just used a path that didn't
> really return to the same point in the manifold (spacetime).



Cyberkatru
#4
Jan12-07, 05:00 AM
P: n/a
MIT's Walter Lewin's twice surprises the EE professors! (fun)

But Bill, he actually uses the voltmeter to make the point. He moves
the voltmeter without changing where it is connected and gets a
different reading? This is puzzling if there is a real scalar function
(node voltage) along the wire whose changes around a loop add to zero.
This demonstration is in lecture 16. He actually used the physical
voltmeter to show that the changes around the loop don't add to zero.

Another thing that bothers me with your explanation is that you said
that the node voltage and the electrostatic potential are related by a
constant. How could an additive constant make a difference here? KVL
wouldn't be affected by an additive constant would it?

You might have the right explanation but it still not clear to me. It
is almost as if you are saying that the line integral \int E \dot dl is
not the relevant quantity for circuites. But this seems inconsitent
with Jackson's book.

Is your explanation written in and advanced E&M texts?

On Jan 10, 4:07 pm, "William R. Frensley" <frens...@utdallas.edu>
wrote:
> Cyberkatru wrote:
> > MIT physicist/astronomer is quite fun to what lecture. In lectures 20
> > and 16 of his online basic E&M lectures he makes some interesting
> > points. ...

>
> >http://ocw.mit.edu/NR/rdonlyres/Phys...-and-Magnetism...

>
> > So do we agree with him on this bit of basic physics 101?There is most certainly a widespead misconception here, but unfortunately

> this work does not find the real problem.
>
> The fundamental misconception is the identification between the electrostatic
> potential of Maxwell's theory and the circuit-theory concept of voltage, which
> for reasons explained below should always be called "node voltage."
> We all understand that the Maxwell quantity is defined as limit of U(r)/Q
> as Q approaches zero, and U(r) is the potential energy of an infinitesmal test
> charge Q. The circuit-theory voltage is actually (-) the electrochemical
> potential for mobile electrons within a chunk of matter that contains enough
> electrons to be considered an electron reservior, and that is in a local
> quasi-equilibrium. This chunk of (almost always metallic) matter is represented
> in circuit space as a "node" in the circuit graph.
>
> The minus sign in the above definition of voltage is a result of the negative
> charge on the electron. (Benjamin Franklin had a even chance, but unfortunately
> he made the wrong choice.) The electrochemical potential for electrons is
> known in semiconductor device theory as the (quasi)Fermi level, not to be
> confused with other meanings of "Fermi energy" in solid-state physics. This is
> the quantity that voltmeters measure, as pointed out by William Shockley
> (Electrons and Holes in Semicnoductors, 1950, p. 305). To understand this,
> let's assume that the voltmeter is a classic Wheatstone Bridge, consisting of
> an ammeter connected between the unknown voltage node and an adjustable voltage
> standard (which can be just a battery connected to a potentiometer). The
> measurement procedure is to adjust the voltage standard until the current
> through the ammeter reads zero. The condition for zero (electron) current is
> that the electrochemical potentials for electrons on the two sides of the meter
> must be equal.
>
> When we understand node voltage to be this statistical quantity, all the
> Krichoff's Law paradoxes disappear. The potential energy of the electrons in
> the node can be raised or lowered by the electrostatic potential phi, or by
> dA/dt: the voltage is still a scalar quantity. Its gradient is NOT in general
> equal to the electric field E.
>
> Why has the confusion between these quantities persisted for so long? The first
> problem is that they are measured in the same units (Volts). The second reason
> is that -> within a given metallic material <- the voltage and the electrostatic
> potential are coupled by an additive constant, the work function. Connecting
> two metals with different work functions produces a structure with a constant
> voltage, but with a discontinuity in the electrostatic potential, which is
> known as a "contact potential."
>
> The additive linkage between potential and voltage is obviously broken in
> vacuum, and also in most semiconductor device structures. A very little known
> fact is that a difference in the spatial behavior of the potential and of
> the voltage is a necessary condition for the operation of active (i.e. gain-
> producing) electron devices.
>
> - Bill Frensley


billb@eskimo.com
#5
Jan12-07, 05:00 AM
P: n/a

Chris H. Fleming wrote:
>
>
> I don't get the last paragraph of the first page.
>
> "There is no electric field in this loop if the resistance of the wire
> making up the loop is zero."
>
> Then he says.
>
> "(this may bother you - if so, see the next section)"
>
> Well I read on and it didn't stop bothering me.


Remember that ideal capacitors have zero resistance in their plates and
terminals, just as ideal inductors have zero resistance in their
windings. If you apply a voltage across the terminals of an ideal
inductor, the value of current will start changing continuously, and it
would go to infinity if you wanted to wait for infinite time to pass.
A perfectly conductive loop is not a short circuit, instead it is a
small-value inductor.



> When he is explaining
> this on page 4, in paragraph 3 he sets all the currents at every point
> in the loop to be the same, but in the very next paragraph he
> contradicts himself with charge buildups at certain points on the loop.


He's just simplifying things by ignoring the charge-buildup process.
The circuit will "adjust itself" over a very brief time, creating
surface charges and patterns of potential. After this "transient" has
occurred, things will work as he describes. In other words, he might
be describing what happens over a period of many milliseconds, while
ignoring the changes which occur in a range of nanoseconds. A more
complete explanation would have to include ALL changes at ALL time
scales, including the currents which led to the charge buildups.

Here's another cool article which goes into similar problems, similar
in that most books ignore it (but it's about flashlight circuits, not
about induced currents in conductor loops:)

Chabay/Sherwood, A unified treatment of electrostatics and circuits
http://www4.ncsu.edu/%7Erwchabay/mi/circuit.pdf


((((((((((((((((((((((( ( ( (o) ) ) )))))))))))))))))))))))
William J. Beaty Research Engineer
beaty a chem washington edu UW Chem Dept, Bagley Hall RM74
billb a eskimo com Box 351700, Seattle, WA 98195-1700
ph425-222-5066 http://staff.washington.edu/wbeaty/

Cyberkatru
#6
Jan14-07, 05:00 AM
P: n/a
I don't get the last paragraph of the first page.
>
> "There is no electric field in this loop if the resistance of the wire
> making up the loop is zero."
>


He is idealizing isn't he. If the wire is really a perfect conductor
why would force be needed to keep them moving along. It is like no
friction. Not the case for a real wire but so what? Its an adealization
where all the resitance is put into the resistor.

Grouchy
#7
Jan14-07, 05:00 AM
P: n/a
Cyberkatru wrote:
> MIT physicist/astronomer is quite fun to what lecture. In lectures 20
> and 16 of his online basic E&M lectures he makes some interesting
> points. I will tell you about both of them and ask your opinions but
> notice that the second one I mention (which actually comes first in
> lecture 18) is the most interesting and maybe a bit puzzling so be sure
> to get through this whole post.
>
> First from lecture 20:
> At some point a little past the 8 minute mark in Walter Lewin's basic
> E&M video lecture #20 he makes the following statement:
> "Almost every college physics book does this wrong!"
> He is referring to the use of Kirchoff's voltage rule for loops in
> cicuits with self inductors.
> I think he is right about this.
> In a way he is denying the assumptions of the so-called "lumped matter
> discipline" in electrical engineering--with regard to inductors
> anyway.
>
> It is all quite entertaining. You can see the video here starting at
> about the 8 minute mark of lecture number 20.
> .http://ocw.mit.edu/OcwWeb/Physics/8-...ures/index.htm
>
> He also has a write up of this issue here:
>
> http://ocw.mit.edu/NR/rdonlyres/Phys...0/lecsup41.pdf
>
> So do we agree with him on this bit of basic physics 101?
>
> Now on to the more interesting case of "nonconservative" fields and
> wacky voltmeters:
> This starts at about minute 38 in video lecture 16 when he commands to
> "hold onto your hats!". A few minutes later gives a demonstation of
> this where he says professors visiting in the audience refused to
> believe what they were seeing.
>
> Do you agree with Walter's interpretation of what is happening?
>
> I think I basically agree but frankly I am not sure I like his
> insistance that this is a case of what should be called
> "nonconservative" fields (except in some unusual sense of
> nonconservative). Here is why:
>
> Mathematically, the field depends on time and so is now properly a
> field on spacetime. Now a closed loop in spacetime should return not
> only to the same place but also the same time. So he has not really
> demonstrated that the field is nonconservative in the usual
> mathematical sense (cohomology). He has just used a path that didn't
> really return to the same point in the manifold (spacetime).


Electrical engineers would note that "inductive" reactance isn't the
same as "capacitive" reactance. When an antenna is resonant, that is,
its length equals the fomula electrical length, only ohmic and
radiation resistance are present. Only when the antenna is NOT
resonant is reactance present. If an antenna is too long for a given
frequency, it is daid to have "inductive" reactance... if it's too
short the reactance is "capacitive". Power lost in reactance isn't
lost in the same way it's lost in ohmic resistance.

Re the "coupling" matter... technically a radiating antenna is
"coupled" to its surrounding media when it's working correctly (i.e. is
resonant).

William R. Frensley
#8
Jan14-07, 05:00 AM
P: n/a
Cyberkatru wrote:
> But Bill, he actually uses the voltmeter to make the point. He moves
> the voltmeter without changing where it is connected and gets a
> different reading? This is puzzling if there is a real scalar function
> (node voltage) along the wire whose changes around a loop add to zero.
> This demonstration is in lecture 16. He actually used the physical
> voltmeter to show that the changes around the loop don't add to zero.


(I looked at the video -- despite the best efforts of RealPlayer to keep
me from doing so. I see what you are talking about.) As with any magic
trick, there is a misdirection right at the start. In this case, the
misdirection is in leading you to believe that different points along a
wire represent the same circuit node. In the absence of time-varying
magnetic fields, they should be, but the point of Faraday induction is
that this is no longer true when the B field changes. To treat this
case with circuit theory, we have to make it a distributed circuit, and
two points along the wire are no more the same circuit node than two
points along the center conductor of a coaxial cable (transmission line)
are. Or, for that matter, the two ends of the wire coming out of an
inductor coil.

We would analyze the situation the same way we analyze a transmission
line: find the equivalent circuit model for a differential length along
the wire. In this case there will a resistive element R0 dx where R0 is
the resistance per unit length. In series with this will be the Faraday
generator, a voltage source of magnitude -dA/dt (dot) dx, A being the
vector potential. To include the possibility that the wires are moving
in the magnetic field, we should probably take the time derivative of
the whole dot product. If we're simply looking at a closed wire loop
with induced eddy curent, the Faraday generator raises the voltage a dV
and the resistor immediately drops it -dV, leading to V(x) looking like
a differential sawtooth pattern until we take the limit dx to 0, where
it smooths out into a constant.

For the case Lewin does, the series resistors limit the current in the
loop to a value much lower than the full eddy current allowed by the
wire. Thus, the Faraday generators can add up to a macroscopic voltage
along the length of the wires connecting the two resistors. Despite
what the circuit schematic might imply, the two resistors are in no way
connected to the same two circuit nodes. If you model the circuit
correctly by taking into account the Faraday generation in the wires,
there is no violation of Kirchoff's Voltage Law.

Note that you may also need to include induction in the voltmeter leads
too.

This explanation in no way contradicts what Lewin writes in his
supplement, about charge redistribution in the circuit. The nice thing
about circuit theory is that it automatically takes this into account,
incorporating such things as the build-up of charges on the contacts to
a resistor as required to create the voltage drop across that resistor.
>
> Another thing that bothers me with your explanation is that you said
> that the node voltage and the electrostatic potential are related by a
> constant. How could an additive constant make a difference here? KVL
> wouldn't be affected by an additive constant would it?


Note that I said they were connected by a constant within a metal of a
given composition. In other conductive media, there are more
complicated correction factors. (Again, multiply everything that
follows by that pesky minus sign due to the electron charge.) In
semiconductors, voltage (Fermi level) is equal to the potential energy
for an electron at rest (the conduction band edge energy) + kT ln
(n/Nc), where n is the electron concentration and Nc is a constant whose
origin we don't need to worry about. Now, -k ln (n/Nc) is just the
entropy of the electron gas, so we see that the voltage can be
interpreted as the Helmholtz free energy per electron. In dilute
electrolytes, you have a similar situation, resulting in the Nernst
equation. In metals, the electrons are so highly degenerate that we can
ignore the entropy correction.

Since the voltage is a measure of free energy, the normal relations like
P=VI really do measure do measure the available energy (or power).
Thermodynamics have already been taken into account.
>
> You might have the right explanation but it still not clear to me. It
> is almost as if you are saying that the line integral \int E \dot dl is
> not the relevant quantity for circuites. But this seems inconsitent
> with Jackson's book. I think I ansered that above. E dot dl is very
> relevant, but it's not the only thing that needs to be considered.
> Is your explanation written in and advanced E&M texts?


In my experience E&M authors have very little interest in the grubby
details of what happens within technological materials. It will
certainly be in a textbook in the future, since I am working on one on
the topic of electron devices. This is obviously sufficiently confusing
that it is worth a section or perhaps an appendix.

(By the way, my mail server seems to be having problems, because I have not
seen my post show up yet. If anyone else responded to my post please email
me directly.)

- Bill Frensley

billb@eskimo.com
#9
Jan14-07, 05:00 AM
P: n/a
Cyberkatru wrote:

> Now on to the more interesting case of "nonconservative" fields and
> wacky voltmeters:
> This starts at about minute 38 in video lecture 16 when he commands to
> "hold onto your hats!". A few minutes later gives a demonstation of
> this where he says professors visiting in the audience refused to
> believe what they were seeing.
>
> Do you agree with Walter's interpretation of what is happening?


All these phenomena are expected when you're working in an environment
having an intense alternating b-field. If you measure voltages with
your DVM, and short the leads together, it won't read zero unless you
also twist the leads to form a shielded cable. Any closed loop of
conductor will experience an electric current unless the area enclosed
by the loop is zero.

> I think I basically agree but frankly I am not sure I like his
> insistance that this is a case of what should be called
> "nonconservative" fields (except in some unusual sense of
> nonconservative). Here is why:


Rather than "nonconservative," I've heard this described as "circuits
in multiply-connected regions," as is done by the authors here:

Faraday's law in a multiply-connected region
http://adsabs.harvard.edu/abs/1982AmJPh..50.1089R

((((((((((((((((((((((( ( ( (o) ) ) )))))))))))))))))))))))
William J. Beaty Research Engineer
beaty a chem washington edu UW Chem Dept, Bagley Hall RM74
billb a eskimo com Box 351700, Seattle, WA 98195-1700
ph425-222-5066 http://staff.washington.edu/wbeaty/

jmc8197@softhome.net
#10
Jan16-07, 05:00 AM
P: n/a
Cyberkatru wrote:
> MIT physicist/astronomer is quite fun to what lecture. In lectures 20
> and 16 of his online basic E&M lectures he makes some interesting
> points. I will tell you about both of them and ask your opinions but
> notice that the second one I mention (which actually comes first in
> lecture 18) is the most interesting and maybe a bit puzzling so be sure
> to get through this whole post.
>
> First from lecture 20:
> At some point a little past the 8 minute mark in Walter Lewin's basic
> E&M video lecture #20 he makes the following statement:
> "Almost every college physics book does this wrong!"
> He is referring to the use of Kirchoff's voltage rule for loops in
> cicuits with self inductors.
> I think he is right about this.
> In a way he is denying the assumptions of the so-called "lumped matter
> discipline" in electrical engineering--with regard to inductors
> anyway.
>
> It is all quite entertaining. You can see the video here starting at
> about the 8 minute mark of lecture number 20.
> .http://ocw.mit.edu/OcwWeb/Physics/8-...ures/index.htm
>
> He also has a write up of this issue here:
>
> http://ocw.mit.edu/NR/rdonlyres/Phys...0/lecsup41.pdf
>
> So do we agree with him on this bit of basic physics 101?
>
> Now on to the more interesting case of "nonconservative" fields and
> wacky voltmeters:
> This starts at about minute 38 in video lecture 16 when he commands to
> "hold onto your hats!". A few minutes later gives a demonstation of
> this where he says professors visiting in the audience refused to
> believe what they were seeing.
>
> Do you agree with Walter's interpretation of what is happening?
>
> I think I basically agree but frankly I am not sure I like his
> insistance that this is a case of what should be called
> "nonconservative" fields (except in some unusual sense of
> nonconservative). Here is why:
>
> Mathematically, the field depends on time and so is now properly a
> field on spacetime. Now a closed loop in spacetime should return not
> only to the same place but also the same time. So he has not really
> demonstrated that the field is nonconservative in the usual
> mathematical sense (cohomology). He has just used a path that didn't
> really return to the same point in the manifold (spacetime).


Is there is a fundamental problem in the way he has used Faraday's Law?
He traverses the circuit to calculate the total E_dot_dl, even though E
has been modified by the circuit rather than being simply E due just to
the changing B.

Jason Walters.

Cyberkatru
#11
Jan18-07, 05:00 AM
P: n/a


>
> Is there is a fundamental problem in the way he has used Faraday's Law?
> He traverses the circuit to calculate the total E_dot_dl, even though E
> has been modified by the circuit rather than being simply E due just to
> the changing B.
>
> Jason Walters.


I don't think so. He is just using the usual idealized assumtion that
the wires themselves have zero resistence and that the resistance is
concentrated in the resistor.

jmc8197@softhome.net
#12
Jan19-07, 05:00 AM
P: n/a

Cyberkatru wrote:
> >
> > Is there is a fundamental problem in the way he has used Faraday's Law?
> > He traverses the circuit to calculate the total E_dot_dl, even though E
> > has been modified by the circuit rather than being simply E due just to
> > the changing B.
> >
> > Jason Walters.

>
> I don't think so. He is just using the usual idealized assumtion that
> the wires themselves have zero resistence and that the resistance is
> concentrated in the resistor.


Take a region with a resistance R(x,y,z) where there is changing
magnetic flux. How do you know that integral_loop[E.dl] = v is
independent of R(x,y,z)? You seem to be saying that it doesn't matter
if the path is empty space or a resistance in series with a conductor
of 0 resistance.

Bossavit
#13
Jan19-07, 05:00 AM
P: n/a
Cyberkatru:

> [About the quantity integral_loop[E.dl] = v] you seem to be saying
> that it doesn't matter if the path is empty space or a resistance in
> series with a conductor of 0 resistance.


Indeed there is something non-trivial happening there.
This is why so many papers have adressed the question
"What does a voltmeter measure?". The answer is, "the
integral of electric field E along the (open) path
traced out by the leads, from one contact point to
the other". (This path includes the part across the
voltmeter's body proper.) Remarkably, this integral is
*the same* in the two very different situations A
(after connecting the voltmeter and threads), and B
(before doing that, so the path is just a geometric
curve, not yet materialized by the leads-and-voltmeter
apparatus.

The difference between A and B lies in which
parts of the path contribute to the integral: the
whole path in B, only the small part of it inside
the voltmeter in A. Yet the integral is the same
in both cases.

So you are right: as regards this integral, "it
doesn't matter...", etc., surprising as this
may appear. This is why voltmeters are useful:
via this integral, which we may
conveniently call "the emf along p (the path)",
they give information about what the electric field
*was* in situation B (which is of course what one
is interested about) in spite of the considerable
disturbance to the electric field caused by
placing the voltmeter and its connectors.

This equality between the two integrals, emf_A
and emf_B, is easily proved by applying
Faraday's law, under the assumption that the
current derived along path p across the voltmeter
is negligible, which happens because of two things:
(1) The high internal resistance of the voltmeter,
(2) The high conductance of the connectors.
Faraday's law requires a *closed* path p', which
one defines as p itself plus some "return path"
inside the workpiece to which the leads ends are
applied.

The proof is a simple exercise, if one replaces
"high" in (1)(2) by "infinite", and if one assumes
a negligible radius for the connectors (so that p
is well-defined). Of course, "negligible radius"
and "high conductance" are antinomic, so the
mathematically minded will find a rigorous proof
somewhat challenging. It involves an asymptotic
analysis with respect to two competing small
parameters (radius and resistivity of the leads).

A last comment, since the famous paper by Romer,

R.H. Romer: "What do 'voltmeters' measure?
Faraday's law in a multiply connected region",
Am. J. Phys., 50, 12 (1982), pp. 1089-93

has been cited in this thread: It should be
stressed that "multiple connectedness" is an
artefact of the 2D modelling adopted by
Romer for his discussion; topological issues
of this kind play no role in 3D situations.

Gerard Westendorp
#14
Jan22-07, 05:00 AM
P: n/a
Cyberkatru wrote:
[..]


> He is referring to the use of Kirchoff's voltage rule for loops in
> cicuits with self inductors.
> I think he is right about this.
> In a way he is denying the assumptions of the so-called "lumped matter
> discipline" in electrical engineering--with regard to inductors
> anyway.


You might want to check out this circuit diagram on my website:

http://www.xs4all.nl/~westy31/Electr...well_animation

In each closed loop of a circuit, I put a 'mesh inductance'. This
modifies Kirchhoff's law to include the inductance of the loop.

One of the motivations behind this is to understand the transition from
'lumped' models, such as inductors, resistor, and 'continuous' models,
such as wire loops with finite surfaces.

As far as I can see, you could add macroscopic resistors, capacitors and
even (although my confidence is slightly lower for that one) inductors
to this circuit, and continue calculating all quantities in the usual way.

Gerard



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