Problem with differential form of Maxwell's third equation?

[leright]

Why does it seem as if the standard differential form of Maxwell's third equation (Faraday's Law) for time varying fields not take into account motional EMF. The differential form simply says that the curl of E is equal to minus the time rate of change of B field. However, there could be a curl of E even in cases where B is constant with a time varying loop in a static B field, so the differential form fails in motional emf, correct? Why isn't the differential motional emf term added to the right hand side of Maxwell's third equation (in differential form)? If it were added in, Maxwell's third equation would work in all cases.

The integral form of Maxwell's third equation works fine in all cases, as long as the derivative is kept OUTSIDE of the flux integral. When it is brought in then there is trouble, and clearly two different answers will be obtained for the emf when comparing the derivative being inside of outside the integral (when the derivative is only applied to the B-field, and not ds). I feel as if this is a flaw in the logic in the derivation of the differential form of the third equation.

This is not independent research. I am just looking for answers to a question.

Thanks.

 

[leright]

Any opinions?

He said that the differential motional emf term that I added was just an approximation at low velocities. Also, when Maxwell formulated the third equation he assumed the moving charge to be the reference frame. However, it seems that it wouldn't be too difficult to add in a term that describes the motional emf at a reference frame other than the bar.

The motional EMF term I added was integral(u x B)*dl. Stokes theorem can be applied to obtain integral(curl(u x B))*dS Therefore, the term curl(u x B) can be added to the right hand side of the differential form of the third equation.

 

[vanesch]

However, there could be a curl of E even in cases where B is constant with a time varying loop in a static B field, so the differential form fails in motional emf, correct?

No, the differential form doesn't fail, because BY DEFINITION, you have to calculate the curl (or the circular integral in the integral formula) in a time-fixed loop, that is, a curve which is time-independent wrt the coordinate frame in which you are expressing the E and B fields.

Why this apparent paradox ? The reason is that the E and B fields do not transform as vectors when you switch from a coordinate frame to another, moving frame (in Lorentz speak, a boost).
E and B aren't really vectors in fact (they only behave as vectors under translations and rotations of coordinates, but not under boosts). They are really elements of a 2-tensor.
So if you want to consider a "moving curve" in frame A, then you should first go to the frame B that is static wrt the curve, and then transform your E and B field from frame A into frame B. And now comes the miracle:

A pure B field in frame A transforms into a B field PLUS an E field in frame B. The extra E field is exactly what corresponds to the EMF.
Indeed, the B-field will be "moving" but that is just "changing" in the B frame, and hence there will be a changing B field and thus a curl of the E-field in the frame B.

 

[leright]

I know that the curl is only calculated in a time fixed frame and I realize that if the derivative of B field in a time fixed frame is 0 then the B field does not induce an E field, and if the derivative of B field in a time fixed frame is NOT 0 then the B field DOES induce an E field. My issue is that an e-field CAN be induced by a constant B-field in the event of the motion of free charge (non-free space conditions) in the field.

I think that the commonly used differential form of Maxwell's equation assumes free space conditions where there is no moving free charge.

I also understand that the commonly used differential form of Maxwell's equation assumes that if there IS moving free charge the reference frame should be taken to be the free charge itself, and this is why Maxwell's equation in differential form "works", but it simply does not convey all possible imformation.

 

[leright]

message has been deleted.

 

[Hurkyl]

My issue is that an e-field CAN be induced by a constant B-field in the event of the motion of free charge (non-free space conditions) in the field.

That's your mistake.

EMF is being induced -- not an E-field.


In the case of the time-varying magnetic field, the reason EMF gets induced is because the time-varying magnetic field induces an electric field, and that electric field starts pushing your (stationary) charges around.


In the case of motional emf, what happens is that the magnetic field itself starts pushing your (moving) charges around. It has absolutely nothing to do with the electric field.


In fact, my textbook specifically says that the integral form:

<br /> \oint \vec{E} \cdot d\vec{l} = - \frac{d \Phi_B}{dt}<br />

is "valid only if the path around which we integrate is stationary."

 

[leright]

That's your mistake.

EMF is being induced -- not an E-field.In the case of the time-varying magnetic field, the reason EMF gets induced is because the time-varying magnetic field induces an electric field, and that electric field starts pushing your (stationary) charges around.In the case of motional emf, what happens is that the magnetic field itself starts pushing your (moving) charges around. It has absolutely nothing to do with the electric field.In fact, my textbook specifically says that the integral form:

<br /> \oint \vec{E} \cdot d\vec{l} = - \frac{d \Phi_B}{dt}<br />

is "valid only if the path around which we integrate is stationary."

In order for there to be an induced EMF (capacity to do work on charges) there MUST be an corresponding e-field. In order for there to be an induced EMF that can drive current around a closed path there MUST also be a "non-conservative" e-field produced, and the only way a non conservative e-field can be produced is from the energy stored in a b-field. The b-fields themselves do not do work. The energy in the b-fields INDUCE e-fields that can do work, and work over a closed path at that (conservation of energy is not violated since the energy allowing the e-field to do work over the closed path is coming from the b-field)

And in motional EMF, the magnetic field is not doing the work. The magnetic field is storing the ENERGY to produce an e-field that does the work. The force on a moving charge or current element is due to an induced e-field which comes from the energy in the b-field.

This is all getting more and more clear to me, and the more I think about it, the more things make sense. Before, the more I thought about it and the deeper I went, the more confused I got. I've reached the point where the deeper I get and the more I think about it, the more clearly things become.

Just because something has energy in some form does not mean that it itself can do work, even though energy is defined as the capacity to do work. The energy has to be converted to other forms of energy that can do the work. For instance, b-fields themselves cannot do work because the force is ALWAYS perpendicular to the direction of motion, and the b-field itself cannot increase the kinetic energy of a charge. However, the energy in the b-field induces an e-field which does the work for it. So, even if the energy itself cannot do work that doesn't imply that it isn't really energy. It can be converted to other forms of energy which can the work. So b-fields aren't DOING work, but they are facilitating work by inducing e-fields.

 

[Hurkyl]

Work it out yourself:

(1) Electric field that is zero everywhere.
(2) Constant magnetic field.
(3) Conducting wire.

When the wire starts to move, you have moving charges in a magnetic field. Thus, those charges start to move along the wire, and you get current, even though the electric field is zero, and the magnetic field is constant.


Your entire line of thought is based on the fact that magnetic forces cannot do work, correct? So all that means is that something else is doing the work. Why do you assume it's the electric field? There's a much simpler explanation that doesn't involve rejecting established physical laws!

I wonder how that wire started moving...

 

[leright]

Work it out yourself:

(1) Electric field that is zero everywhere.
(2) Constant magnetic field.
(3) Conducting wire.

When the wire starts to move, you have moving charges in a magnetic field. Thus, those charges start to move along the wire, and you get current, even though the electric field is zero, and the magnetic field is constant.Your entire line of thought is based on the fact that magnetic forces cannot do work, correct? So all that means is that something else is doing the work. Why do you assume it's the electric field? There's a much simpler explanation that doesn't involve rejecting established physical laws!

I wonder how that wire started moving...

ok, the b-field still is not doing WORK on the charges in the rod. When the rod moves in the constant magnetic field motional emf occurs. Motional emf is caused by an e-field that pushes the charges and that e-field is induced by the magnetic field. I am not rejecting any physical laws.

It is an electric field that is induced perpendicular to the magnetic field that causes the emf. The force is perpendicular to the magnetic field so it is not doing work...the induced e-field is doing the work, but the energy to produce the e-field is coming from the b-field.

Why is the force due to a magnetic field called Fsubm and when you divide Fsubm by Q it's called Esubm? Perhaps because it is an e-field doing the work. Also, take a look at the lorentz transformations (relationships between E and B fields with relativistic considerations) and you'll notice a UxB term when getting the e-field from b-field. The UxB is the force due to the B-field.

http://farside.ph.utexas.edu/teaching/em/lectures/node123.html - look at the determination of the perpendicular e-field due to the b-field.

 

[Galileo]

In order for there to be an induced EMF (capacity to do work on charges) there MUST be an corresponding e-field. In order for there to be an induced EMF that can drive current around a closed path there MUST also be a "non-conservative" e-field produced, and the only way a non conservative e-field can be produced is from the energy stored in a b-field. The b-fields themselves do not do work. The energy in the b-fields INDUCE e-fields that can do work, and work over a closed path at that (conservation of energy is not violated since the energy allowing the e-field to do work over the closed path is coming from the b-field)

And in motional EMF, the magnetic field is not doing the work. The magnetic field is storing the ENERGY to produce an e-field that does the work. The force on a moving charge or current element is due to an induced e-field which comes from the energy in the b-field.

A moving (conducting) loop in a constant magnetic field (like a generator) will induce an EMF, but not because of Faraday's law. It's simply because in the loop you now have moving charges in a magnetic field which are pushed along by the Lorentz force and this causes the current. The magnetic force is not doing work ofcourse, but the person pulling the loop is.

 

[arunbg]

I believe Galileo is right, the phenomenon is similar to Hall's effect which is attributed to Lorentz's forces.

However what I don't understand is that in many cases such as a rod rotating in a uniform magnetic field the induced emf across the ends of the rod can also be calculated by the eqn for motional "induced" emf
E = Blv (it is done so in my text) even though there is no change in net flux.I calculated the answer independently using lorentz forces and got the exact same answer which seemed the right approach to me.

Does this imply some kind of relation or is this simply a freak coincidence?

 

[leright]

A moving (conducting) loop in a constant magnetic field (like a generator) will induce an EMF, but not because of Faraday's law. It's simply because in the loop you now have moving charges in a magnetic field which are pushed along by the Lorentz force and this causes the current. The magnetic force is not doing work ofcourse, but the person pulling the loop is.

It is because of Faraday's law. Faraday's law says that the induced EMF is equal to the time rate of change in the flux through a conducting loop. This is what Faraday's law says. You can produce a change in the magnetic flux by changing the area of the loop, even in a constant B field.

The induced EMF can be because of a number of physical phenomena, but all of the physical phenomena is contained in Faraday's law.

 

[arunbg]

It is because of Faraday's law. Faraday's law says that the induced EMF is equal to the time rate of change in the flux through a conducting loop.

There is no time rate of change of flux through a fixed conducting loop moving in a uniform magnetic field.

 

[Hurkyl]

Why is the force due to a magnetic field called Fsubm and when you divide Fsubm by Q it's called Esubm?

My book doesn't use that notation, so I don't know what you mean by F_m and E_m.

Also, take a look at the lorentz transformations (relationships between E and B fields with relativistic considerations) and you'll notice a UxB term when getting the e-field from b-field. The UxB is the force due to the B-field.

But we're not doing a change of reference frame, so the Lorentz transformations are irrelevant.


The Lorentz transformations say that the EMF induced by motion through a pure magnetic field in one frame has to have the same effect as the EMF induced by the electric field in some other reference frame where there is no motion.

But that does not mean that an electric field spontaneously appears in the original reference frame!

The force is perpendicular to the magnetic field so it is not doing work...the induced e-field is doing the work, but the energy to produce the e-field is coming from the b-field.

No -- there is no work being done in the production motional EMF. The energy to create the current comes from the kinetic energy of the wire. Something caused the conducting wire to start moving -- the magnetic field acts to change that motion into a current. If there was no friction, then given enough time and space, the magnetic field will convert all of the wire's kinetic energy into current, and the wire will stop moving.

I am not rejecting any physical laws.

You're rejecting the third of Maxwell's laws. And, it seems, the mathematical theorem that the integral form is true if and only if the differential form is true.

 

[leright]

My book doesn't use that notation, so I don't know what you mean by F_m and E_m.
But we're not doing a change of reference frame, so the Lorentz transformations are irrelevant. The Lorentz transformations say that the EMF induced by motion through a pure magnetic field in one frame has to have the same effect as the EMF induced by the electric field in some other reference frame where there is no motion.

But that does not mean that an electric field spontaneously appears in the original reference frame!
No -- there is no work being done in the production motional EMF. The energy to create the current comes from the kinetic energy of the wire. Something caused the conducting wire to start moving -- the magnetic field acts to change that motion into a current. If there was no friction, then given enough time and space, the magnetic field will convert all of the wire's kinetic energy into current, and the wire will stop moving.
You're rejecting the third of Maxwell's laws. And, it seems, the mathematical theorem that the integral form is true if and only if the differential form is true.

no no no...my whole point in this thread is that the differential form of the third equation FAILS to take into consideration motional EMF (and yes, motional EMF induces an E field with a curl), which is where there is a change in reference frame. That is the correction to the equation I am making, and for this you would need Lorentz transformations.

 

[jtbell]

There is no time rate of change of flux through a fixed conducting loop moving in a uniform magnetic field.

If the loop rotates (as in a simple generator), the flux through the loop does change. For a fixed-area loop in a uniform magnetic field:

\Phi = AB \sin \theta

\frac{d \Phi}{dt} = AB \frac {d}{dt} (\sin \theta) = AB \cos \theta \frac {d\theta}{dt}

 

[leright]

There is no time rate of change of flux through a fixed conducting loop moving in a uniform magnetic field.

ok, well then take a time varying conducting loop!

 

[Hurkyl]

no no no...my whole point in this thread is that the differential form of the third equation FAILS to take into consideration motional EMF (and yes, motional EMF induces an E field with a curl)

And my answer is that your point is based upon flawed assumptions.


The third equation does not fail to take into consideration motional EMF -- motional EMF does not act as a source of the electric field, and thus (correctly) does not appear in the equation..

This is true both in the differential and integral forms. The equation:

\oint_C \vec{E} \cdot d\vec{l} = - \frac{d \Phi_B}{dt}

is only valid for a fixed contour C. This equation cannot tell you anything about motional EMF.


If you do not understand this, then allow me to demonstrate some of the many problems that you will encounter, both mathematical and physical.

Suppose that

\oint_S (\nabla \times \vec{E}) \cdot d\vec{s} = - \frac{d \Phi_B}{dt}

is true for a time-varying surface S. Then we can consider two different time-varying surfaces, S_1 and S_2. Suppose that at t = 0 that S_1 = S_2. Then, the above equation tells us that the time derivative of the flux through S_1 is equal to the time derivative of the flux through S_2 -- but that can't be right, because those surfaces can be varying in vastly different ways!


Some physical problems: you want a velocity to appear in Maxwell's third equation. Well, velocity of what?

Let's suppose you're right -- some electric field does spontaneously appear due to the velocity (of something), and that's what causes motional EMF.

Well, what if our moving conducting wire passes right next to a stationary conducting wire. Well, since the moving conducting wire (somehow) causes there to be an electric field, then that should induce an EMF in the other wire in the same direction. Does that happen? No! In fact, once we account for all effects, don't we see the exact opposite?

If we had two conducting wires passing by each other moving at equal speeds, your argument would say that electric fields are induced... but since they're in opposite directions they cancel, and we see no motional EMF at all. But that's not what happens: we will see a current in both wires.



For the sake of completion, I will admit that you are right (but as far as I can tell, for the wrong reasons): motional EMF does induce an E-field with curl, but for reasons that can be entirely explained via Maxwells equations, differential or otherwise.


Motional EMF is, at the lowest level, nothing more than the acceleration of moving charged particles.
Accelerating charged particles (generally) result in nonstationary currents.
Because current acts as a source of the magnetic field, a nonstationary current gives rise to a time-varying magnetic field.
A time-varying magnetic field acts as a source of the electric field.
And in this way, motional EMF does induce an electric field.


But this is an electric field caused by motional EMF. The electric field has absolutely nothing to do with causing the motional EMF. Motional EMF is caused entirely by the magnetic field: moving charges in a magnetic field are deflected in a direction perpendicular to their motion.

(Of course, the electric field could indirectly cause motional EMF... say, by having the time variations of the electric field induce a magnetic field)

 

[leright]

For the sake of completion, I will admit that you are right (but as far as I can tell, for the wrong reasons): motional EMF does induce an E-field with curl, but for reasons that can be entirely explained via Maxwells equations, differential or otherwise.Motional EMF is, at the lowest level, nothing more than the acceleration of moving charged particles.
Accelerating charged particles (generally) result in nonstationary currents.
Because current acts as a source of the magnetic field, a nonstationary current gives rise to a time-varying magnetic field.
A time-varying magnetic field acts as a source of the electric field.
And in this way, motional EMF does induce an electric field.But this is an electric field caused by motional EMF. The electric field has absolutely nothing to do with causing the motional EMF. Motional EMF is caused entirely by the magnetic field: moving charges in a magnetic field are deflected in a direction perpendicular to their motion.

(Of course, the electric field could indirectly cause motional EMF... say, by having the time variations of the electric field induce a magnetic field)

OK, I see your point regarding point charges moving (non stationary current) which in turn produces a time variant magnetic field, which in turn produces an E field when in the presence of a B field. You may be correct on this statement, but I feel this only holds for moving CHARGES. Let us now consider a loop of conducting wire (but no current) that is rotated in free space with absolutely no B field present. This wire has no net charge OR current so as it moves it is not producing any b-field or changing b field. We know that no E field or EMF will be induced (or do we?) Let us assume for a moment that it doesn't produce a B field. It would be nice to have some experimental results handy claiming this is true. Someone should rotate an electrically neutral loop of wire somehow in a controlled environment (no B field, no E field, and in a vacuum) to and see if a B field is produced.

Now, let us move this same conducting loop of wire into a time invariant B field. We can rotate it in this B field or do whatever we want and produce a changing flux (not due to time variance of the B field, but time variance of the surface area of the loop...as we change the angle of the loop in the B field we are changing the magnetic flux in the B field even if the scalar area of the loop stays constant, since the angle between the surface vector of the loop and the time invariant B field changes we are changing the magnetic flux). We know from experiment that this changing FLUX induces a curl in E which produces EMF (or that this induces an EMF which in turn produces a curl in E, depending on how you look at it). However, there is no current or moving charges or anything, so that originally static field is still static. In this case we are changing ONLY the magnetic flux through the loop and not the B field. However, we know that there is an EMF induced by experiment, and if there is an EMF there is an E-field, PERIOD! NOWHERE in the differential form of Maxwell's third equation (and a changing B field doesn't pop out in any of the other equations in differential form, so your previous agument fails in this case) is this scenario accounted for, but the integral form covers it. The error (or assumption, perhaps) Maxwell made (or did Heaviside do this, I can't remember) when going from the integral form to the differential form was when he brought a partial derivative into the integral of B over a surface and didn't account for a changing area over time (ds). I don't know how many electrodynamics problems you've worked through, but there are COUNTLESS problems where the diffential form fails to yield the same result as the integral form. You can attempt some problems by computing the curl of the E field using the differential form and then integrating the curl over a surface to get the EMF and you DO NOT get the same answer as the integral form. This is a FACT, and is clear proof of the failure of the differential form of Maxwell's equation. Maxwell assumed that no motional EMF was taking place when he went from differential form to integral form. In fact, he assumed a lot of things. The equations are not wrong if this assumption is made clear whenever the equations are written down, but most textbooks FAIL to make this clear. In fact there are tons of websites out there with modified differential forms of maxwell's equations taking things like motional EMF, material space, etc into account (and even quantum effects if you want to get into QED). Trust me, the differential form of Maxwell's equation (3rd and 4th, really) has plenty of errors (or assumption that were made, so I don't insult Maxwell. )

Your comment about about how the EMF causes the electric field, but an electric field does not cause the EMF I feel is incorrect, simply because of the way EMF is defined. I don't feel one necessarily causes the other, but I simply feel like they COMPLEMENT each other and occur simultaneously (an E field doesn't appear by itself in a time invariant frame and the EMF due to that E field appears in the next frame). If you have an EMF it is because of energy in a magnetic field allowing a non conservative E field to be produced. Remember, EMF is the closed path integral of a an E field. And I think it is pointless to argue over this (if the e-field produces emf, or if the emf produces the e-field). Under maxwell's equations, the condition of the field is taken in a time invariant frame so they are both really occurring simultaneously. This isn't what I am arguing about, really.

 

[leright]

http://www.wolfram-stanek.de/maxwell_equations.htm

here's a great website with all of the extensions on Maxwell's equations. Notice the ones on the top are the equations in question, and they fail in the case of moving objects, according to the website.

 

[Hurkyl]

Remember, EMF is the closed path integral of a an E field. And I think it is pointless to argue over this

I don't think it is pointless, because as far as I know, you're wrong. And I'm optimistic enough to think that whichever one of us is wrong will be convinced in the end.


You're confusing an equation with a definition -- and trying to apply the equation in circumstances in which it doesn't apply.

EMF is the "force" that causes a current to be induced -- nothing more, nothing less. It can be viewed as the integral of an electromotive force field. The term "EMF field" is not synonymous with "electric field"; it's just that they just happen to be equal in the case of a stationary loop.

I hope our disagreements are just this confusion over terminology. When I speak of the electric field, I'm talking about the fundamental field, and not some "effective" field that describes the amalgamation of multiple effects.


IMO, this entire thread is summed up in one statement:

Maxwell's third equation is not about the EMF field.​

You seem primarily interested in the EMF field -- and because Maxwell's equations are only equations for the electric and magnetic fields, you find them inadequate for your interests.

But I do not see that as a problem with Maxwell's equations. They (correctly) do what they were meant to do.

(Incidentally, I know very little about the macroscopic H and D fields -- I have only been interested in learning about the microscopic fields)




Bo Thidé's book derives a differential expression for the EMF field felt by a moving conductor:

\mathbf{E}^{\text{EMF}} = \mathbf{E} + \mathbf{v} \times \mathbf{B}



He derives it through a computation involving Faraday's law and a moving loop. (In particular, he does not use "Maxwell's generalized form" of Faraday's law, and even explicitly states that equation is only valid for a "generic stationary 'loop'")


It can be more quickly derived from the force law -- this is what I have been saying when I say that the magnetic field directly induces motional EMF. In the nonrelativistic case, the force on a moving charge is q(E+vxB), so the EMF field felt by that charge must be E+vxB.

(But, Bo had not yet introduced the force law when he did his derivation, so he couldn't have done it this way)

 

[leright]

I don't think it is pointless, because as far as I know, you're wrong. And I'm optimistic enough to think that whichever one of us is wrong will be convinced in the end. You're confusing an equation with a definition -- and trying to apply the equation in circumstances in which it doesn't apply.

EMF is the "force" that causes a current to be induced -- nothing more, nothing less. It can be viewed as the integral of an electromotive force field. The term "EMF field" is not synonymous with "electric field"; it's just that they just happen to be equal in the case of a stationary loop.

I hope our disagreements are just this confusion over terminology. When I speak of the electric field, I'm talking about the fundamental field, and not some "effective" field that describes the amalgamation of multiple effects.IMO, this entire thread is summed up in one statement:

Maxwell's third equation is not about the EMF field.​

You seem primarily interested in the EMF field -- and because Maxwell's equations are only equations for the electric and magnetic fields, you find them inadequate for your interests.

But I do not see that as a problem with Maxwell's equations. They (correctly) do what they were meant to do.

(Incidentally, I know very little about the macroscopic H and D fields -- I have only been interested in learning about the microscopic fields)

Bo Thidé's book derives a differential expression for the EMF field felt by a moving conductor:

\mathbf{E}^{\text{EMF}} = \mathbf{E} + \mathbf{v} \times \mathbf{B}
He derives it through a computation involving Faraday's law and a moving loop. (In particular, he does not use "Maxwell's generalized form" of Faraday's law, and even explicitly states that equation is only valid for a "generic stationary 'loop'")It can be more quickly derived from the force law -- this is what I have been saying when I say that the magnetic field directly induces motional EMF. In the nonrelativistic case, the force on a moving charge is q(E+vxB), so the EMF field felt by that charge must be E+vxB.

(But, Bo had not yet introduced the force law when he did his derivation, so he couldn't have done it this way)

EVERYWHERE I look on the internet and all the literature claims that EMF is produced by a special type of E-field (but it is still an E-field...it is just nonconservative E-field). I've never heard of this thing called an EMF field.

And I don't understand when you say Maxwell's third equation is not about an EMF field (if there is such a thing). Of course it is! It describes the curl of the E-field due to changing B fields (or with the motional EMF extension, movement of objects in the field). If you can consider something such as an "EMF field" it would be the total E-field (conservative + non-conservative parts), I would think, and Maxwell's equation gives the non-conservative part of the picture. Whenever change in magnetic flux induces an E-field it is a non-conservative E-field, which is what causes EMF

And I've used Maxwell's third integral equation for TONS of scenarios where the loop is time varying in many different applications (and so do textbooks when they work examples). the integral form DOES work in cases where the flux through a conducting loop is time varying but due to constant B field in a time varying loop. The differential form FAILS because it doesn't take into account a time varying loop. Did you look at the link I provided and the difference between the regular Maxwell's equations and the modified equations taking motion into account? The equation that takes motion into account has a total derivative in front of the B-field, and the equation that doesn't take motion into account has a partial derivative in front of the B-field.

Maxwell's original equations contained the partial derivative and not a total derivative because he assumed motion was not occurring.

Also, what is the E in the EMF field equation? It is the conservative part of the E-field. the VxB part is the non-conservative part. The EMF-field is the static E field + the time variant E field. That is all. It's all an E-field.

EDIT: OK, I admit that I made a terrible mistake. The EMF-field is the conservative E-field + the non-conservative E-field (if you want to call it an EMF-field). Maxwell's third equation describes the non-conservative part of the E-field and how it is produced. When you say it isn't about an EMF field then what IS it about? f you consider the EMF field to just be the non-conservative part of the E-field then OF COURSE the third equation is about the EMF field.

 

[Hurkyl]

EVERYWHERE I look on the internet and all the literature claims that EMF is produced by a special type of E-field (but it is still an E-field...it is just nonconservative E-field).

Yes -- and everything I found describes EMF in the setting of a stationary loop! If motional EMF is described, it was more-or-less as an addendum.


By definition, the electric field E is the vector value associated with a point in space that would cause a particle with charge q to experience a force qE.


I haven't succeeded in tracking down a general definition of the EMF field. But, it's essentially the field with the property that the net force on the charges in your circuit is qE^EMF. Or, more accurately, the field such that integrating it around the loop gives you the EMF.


From the definitions of the fields, we see that there's no automatic reason they should be equal. In fact, there are some sources of EMF that are clearly not electromagnetic: for example, if I physically grab a charged ring and start rotating it, I have generated EMF in a mechanical way.


The phenomenon of motional EMF demonstrates that, even if we consider only electromagnetic sources, the EMF field is not a fundamental thing; the value of the EMF field is intimately tied to the motion of our conducting loop. So, unlike the electric field, the EMF field cannot be described as an independent entity.

And I've used Maxwell's third integral equation for TONS of scenarios where the loop is time varying in many different applications (and so do textbooks when they work examples). the integral form DOES work in cases where the flux through a conducting loop is time varying but due to constant B field in a time varying loop.

No, the textbooks do not use Maxwell's third integral equation. They use Faraday's law:

EMF = -(d/dt) {magnetic flux through the loop}


It is a mistake to use Maxwell's third integral equation, because:
(1) Maxwell's third integral equation is only for fixed loops.
(2) The left hand side of Maxwell's third integral equation is not EMF.

Also, what is the E in the EMF field equation? It is the conservative part of the E-field. the VxB part is the non-conservative part.

No, it's not. E is the electric field: the force field that causes the force qE on a particle with charge q. B is the magnetic field: the force field that causes the force qvxB on a particle with charge q and velocity v.

Did you look at the link I provided and the difference between the regular Maxwell's equations and the modified equations taking motion into account?

Yes.

Incidentally, that link provides some advice:

never forget the necessary relations for field transformations
(i.e. Lorentz' transformation) hidden in the left sides of the equations (1a) - (4a)​

if I read the page right, those equations are not for the electromagnetic field, but for a Lorentz transformation of the electromagnetic field -- the third equation is relating the electric field in one reference frame to the magnetic field in another frame.

 

[vanesch]

And I've used Maxwell's third integral equation for TONS of scenarios where the loop is time varying in many different applications (and so do textbooks when they work examples). the integral form DOES work in cases where the flux through a conducting loop is time varying but due to constant B field in a time varying loop.

It is indeed a happy coincidence that the use of the integral form also works for non-time-stationary loops in certain circumstances (such as rotating loops in a homogeneous magnetic field), but in fact this is NOT a direct application of the integral form of Maxwell's third equation - although it surely takes on the same form.

As Hurkyl correctly pointed out, the EMF around a loop is the total force exerted on a charge around the loop integrated over the loop. In order to calculate this, one has to dispose many small test charges all oround the loop (co-moving with the loop) and calculate the force excerted on it. You then multiply with the (infinitesimal) distance, and sum over all the forces. If that loop is stationary in a frame, then the force acting on a test charge "fixed to the loop" is simply q.E, and the integral of this force around the loop is nothing else but the loop integral of the E-field, which is given by the integral form of Maxwell's third equation.
However, when the loop is not time-stationary, but moving or changing, then the (co-moving) testcharges are subjected to the total electromagnetic force, which is q(E + v x B), so THIS (as Hurkyl pointed out) is the quantity entering in the EMF. Here, v is the local velocity of the loop where the test charge is fixed.

And the "miracle" is now the following. The contribution of a test charge in a piece of loop dr is given by q dr.(E + v x B). But dr. (v x B), with v = ds/dt (the perpendicular motion of the piece of conductor) takes on the form: dr ds B / dt, which is dA B / dt, dA being the area whiped out by the piece of conductor dr in a time dt, and B dA being the magnetic flux whiped out by this piece of conductor, during time dt.

So we have that the contribution of the part v x B in the force law gives us, as a contribution of the EMF, simply d phi/ dt, the change of flux seen by the piece of wire due to its local motion.
And the contribution to the EMF due to the E-field is also d phi/ dt, but this time, the change in time of the flux in the stationary loop due to a changing B in time. So BOTH contributions take on the form d phi / dt, and hence the second contribution (from v x B) is AUTOMATICALLY taken into account when you calculate the total EMF due to a moving conductor loop. This was the "coincidence" I talked about.

And now you also understand why this works for the integral form, and not for the differential form: the differential form gives you only E, while the integral form automatically takes with it the contribution of v x B.

 

[leright]

ok, well, I am getting mad at my undergrad electromagnetics textbook.

Tomorrow I am going to order a copy of Jackson's Classical Electrodynamics and start working through it to learn this stuff correctly. I am serious about learning this stuff, and I don't feel satisfied with what I got out of my ugrad coursework. Terminology like "EMF-field" is meaningless to me. If I understand correctly, the EMF-field is simply the field that models the TOTAL force that can act on a charge (not just the electric field), and can be from anything.

First, my textbook takes Faraday's law (Vemf is minus the rate of change in magnetic flux) and replace flux with the integral of B.ds. Then they take the derivative inside of the integral as a partial and proceed to apply stokes theorem on Vemf (rewriting integral(E.dl) as integral(curl(E).ds)) and then say that curl(E) = -partial(B)/partial(t). They DO say that this only holds for time invariant loops.

Also, whether or not the integral form holds for moving bodies or not depends on if you consider the integral form as being faraday's law with the flux simply replaced with integral(B.ds), or if you consider the integral form where they bring in the derivative into the integral. If you consider the integral form of the 3rd equation to be the former then the equation holds for moving objects. If you consider the integral form of the 3rd equation to be the latter then the equation DOES NOT hold. Unfortunately, I have seen the integral form of the 3rd equation being represented both ways on the internet.

Now, my textbook in chapter 8 says the mechanical force due to a magnetic field that acts on a moving charge is qV X B. I understand this. But then, in chapter 9, they went and defined a motional electric field Esubm, which was V X B. They then proceed to take the circulation integral of this around a closed path and call this the motional EMF (the EMF due to moving charge or loop in a constant B-field). They apply Stoke's and get the curl of E is the curl of (V X B). So, for a moving loop in an magnetostatic field curlE = curl(V X B).

The author then proceeds to claim that therefore for moving loops in time variant fields the total curl of E is the sum of the curl of U X B and negative the partial of B wrt t.

Hurkl also claimed that an E-field is not doing the work in the case of motional EMF, but the magnetic field is doing the work. Well, my book calls U X B the motional ELECTRIC field and determines the EMF by dotting this motional electric field over a closed path.

OK, so in the end I was CORRECT in my original post when I said that Maxwell's differential form of the third equation doesn't take into account motional EMF. I know for a fact that it doesn't. I now realize that the integral form also does not take into account motional EMF since the derivative is brought into the integral as a partial derivative and only B is differentiated. I was thinking that the integral form of the third equation the derivative was kept outside the integral, making it work in all cases. However, many sources on the internet show the integral for as having the derivative outside of the integral. In this case, then the integral for is the SAME THING AS FARADAYS LAW, and works in motional EMF cases also.

Another part of my confusion hinges on this motional electric field, which is Fsubm/q, or simply U X B. So should I be thinking about the force on moving charges moving in a magnetic field as being caused directly by the magnetic field, or should I be thinking about the force being caused directly by this so-called motional electric field, where the motional electric field is caused by the magnetic field. I think the latter is incorrect, but it seems as if all the literature disagrees, since they say that in a static magnetic field with a moving charge, there is a curl in E. However, a curl in E implies there is a time varying E field, and then therefore there would be a time varying magnetic field caused, and so on, which would imply an electromagnetic wave. However, we all know that a dc current in a magnetic field does not produce an electromagnetic wave (or does it?)

So, I really think the terminology was developed carelessly and there are lots of flaws.

Another thing I do not understand is the idea of a constant magnetic field in the, for example, x-direction. How can this be? If you have a constant B-field (at all points the magnitude and direction is the same) in the x-direction that would imply that it is a conservative B-field, which is bogus. The only way you could produce a constant B-field is with an infinite conducting place which is purely hypothetical. I see so many problems in textbooks that have constant B-fields in them. Are they just saying that the B-field is only constant in a particular region (center of a huge conducting plane, for instance?) I suppose this is ok, but it kinda bothers me a bit.

Also, one more question, in free space, maxwell's 4th equation says that if there is a changing D-field then there is a curl in H. LEt us now suppose that we have a D field whose derivative is a constant. If there is a constant derivative in the D field then there is a constant curl in H. This means that H only depends on x,y,z, but not on t. Therefore B is constant wrt time. Therefore E does not have a curl according to MAxwell's 3rd equation, correct? Therefore electromagnetic waves to dot propagate.

 

[Maxwell]

What book are using, leright? My book for E&M sucks as well. I'm using "Fundamentals of Applied Electromagnetics" by Fawwaz T. Ulaby.

 

[leright]

What book are using, leright? My book for E&M sucks as well. I'm using "Fundamentals of Applied Electromagnetics" by Fawwaz T. Ulaby.

Well, I wouldn't say it sucks, but I think it simply lacks the rigor I am looking for. It is "Elements of Electromagnetics" by Sadiku. There are also tons of errors everywhere in this text, and this is unacceptable considering it is the third edition. I wonder what the first edition looked like.

 

[leright]

Also, regarding that link I provided with the modified Maxwell's equations for material space, movement of objects, and relativistic effects, is it possible to formulate the wave equation taking all of these other factors into account and then solving for it (I know it would be a horrendous task)? Has this ever been attempted?

Also, I know that when you formulate the wave equation for free space conditions you take the curl of both sides of the third equation and then sub in the 4th equation for the curl of B. When the curl of the curl of E is re-written as the laplacian of E plus the divergence of E, the divergence of E is assumed to be zero (free space conditions, so therefore no charge). What does the wave equation solution looks like when this is not zero, and it is instead the volume charge density? I know this would make the problem a pita, but I am curious if anyone solved the the wave equation for conditions other than free space.

Thanks.

 

[leright]

Below is the modified differential form of Maxwell's 3rd equation taking into account motion of bodies. There are more extensions to correct MAxwell's equations for material media that are not necessarily linear, homogeneous, and isotropic. The 4th equation has similar modifications.

curl E' = - d B / dt = - ∂ B / ∂ t + curl (v x B)

 

[vanesch]

Tomorrow I am going to order a copy of Jackson's Classical Electrodynamics and start working through it to learn this stuff correctly. I am serious about learning this stuff, and I don't feel satisfied with what I got out of my ugrad coursework. Terminology like "EMF-field" is meaningless to me.

I think we are in agreement here. EMF only makes sense for a specific setup (a loop, moving or not). It doesn't make sense as an independently existing field in space. It would be like talking about the field of friction force. The force of friction exists, but depends upon the state of motion of the thing undergoing the friction, and is not a pre-determined quantity in space (as a field would be).

If I understand correctly, the EMF-field is simply the field that models the TOTAL force that can act on a charge (not just the electric field), and can be from anything.

Yes, that's right.

First, my textbook takes Faraday's law (Vemf is minus the rate of change in magnetic flux) and replace flux with the integral of B.ds. Then they take the derivative inside of the integral as a partial and proceed to apply stokes theorem on Vemf (rewriting integral(E.dl) as integral(curl(E).ds)) and then say that curl(E) = -partial(B)/partial(t). They DO say that this only holds for time invariant loops.

Yes, commuting integral and time derivative only work when the loop of the integral is not time-dependent of course.

Now, my textbook in chapter 8 says the mechanical force due to a magnetic field that acts on a moving charge is qV X B. I understand this. But then, in chapter 9, they went and defined a motional electric field Esubm, which was V X B. They then proceed to take the circulation integral of this around a closed path and call this the motional EMF (the EMF due to moving charge or loop in a constant B-field).

The problem of course is the "V", which is situation-dependent. It is a property of a characteristic loop. But they are right in calculating the circulation integral, because now you're just summing the forces on charges along the loop, which is what EMF of the loop was defined to be.

They apply Stoke's and get the curl of E is the curl of (V X B). So, for a moving loop in an magnetostatic field curlE = curl(V X B).

This, however, sounds like bull to me. Maybe I'm misunderstanding what the author wants to do, but this is rather strange.

The author then proceeds to claim that therefore for moving loops in time variant fields the total curl of E is the sum of the curl of U X B and negative the partial of B wrt t.

This sounds very messy to me.

I think it is much better to consider a moving loop and the total force E+vxB, and then to calculate the vxB part due to the moving piece of loop (as I sketched).

Hurkl also claimed that an E-field is not doing the work in the case of motional EMF, but the magnetic field is doing the work. Well, my book calls U X B the motional ELECTRIC field and determines the EMF by dotting this motional electric field over a closed path.

This sounds utterly messy, to call U x B a "motional electric field".

OK, so in the end I was CORRECT in my original post when I said that Maxwell's differential form of the third equation doesn't take into account motional EMF.

Maxwell's differential form relates the electric field to the magnetic field, and that's all it is supposed to do. Motional EMF is NOT DUE TO THE ELECTRIC FIELD, so you shouldn't be surprised that it doesn't show up in an equation that deals with the E-field.
Calling vxB an "electric field" is wrong (as I understand your textbook wants to do). It's probably because one wants to think of "force on charge" as "electric field" but that is not true: the force on a charge is q(E + vxB).

So of course the equation giving you the E-field will not include a term that depends only on the B-field !

I know for a fact that it doesn't. I now realize that the integral form also does not take into account motional EMF since the derivative is brought into the integral as a partial derivative and only B is differentiated. I was thinking that the integral form of the third equation the derivative was kept outside the integral, making it work in all cases. However, many sources on the internet show the integral for as having the derivative outside of the integral. In this case, then the integral for is the SAME THING AS FARADAYS LAW, and works in motional EMF cases also.

Yes. Let's summarise:

A) the ELECTRIC FIELD is described by curl E = - dB/dt
As this is a differential relation, it has nothing to do with loops.

Going to a stationary loop, we can apply Stokes' theorem, and find that:

Loop integral of E around loop L = - d phi / dt, phi the flux of B in L.

Stokes' theorem only works for stationary loops of course.

B) the EMF around a loop K (not stationary) is given by:

Loop integral of (E + v x B) over K

The non-stationary part is given by the v x B

Now, this loop integral can be written as:

Loop integral of E over K + Loop integral of v x B over K

The first integral reduces to the loop integral of stokes' theorem, and is thus the change of flux in the *stationary* loop.

The second integral can (happy coincidence) be shown to be equal to the flux of B whiped out by the moving parts of the loop.

So the sum of both is the CHANGE OF TOTAL FLUX of B calculated over the moving loop: d/dt integral B dS

Bringing the d/dt INSIDE makes the loop stationary, and gives you the first part (Maxwell's 3rd equation). The remaining part is given by the loop integral of v x B, which corresponds to the second term in the EMF.

If the B-field is stationary, then only the second part will give a non-zero contribution (generator).
If the loop is stationary, then only the first part will give a contribution (transformer).

In the general case, both can contribute.

Another part of my confusion hinges on this motional electric field, which is Fsubm/q, or simply U X B. So should I be thinking about the force on moving charges moving in a magnetic field as being caused directly by the magnetic field, or should I be thinking about the force being caused directly by this so-called motional electric field, where the motional electric field is caused by the magnetic field. I think the latter is incorrect, but it seems as if all the literature disagrees, since they say that in a static magnetic field with a moving charge, there is a curl in E. However, a curl in E implies there is a time varying E field, and then therefore there would be a time varying magnetic field caused, and so on, which would imply an electromagnetic wave. However, we all know that a dc current in a magnetic field does not produce an electromagnetic wave (or does it?)

I think that the concept of "motional electric field" is bull. The force on a moving charge is q (E + v x B), and it doesn't make sense to call v x B a kind of "electric" field.

So, I really think the terminology was developed carelessly and there are lots of flaws.

Yes. I agree.

Another thing I do not understand is the idea of a constant magnetic field in the, for example, x-direction. How can this be? If you have a constant B-field (at all points the magnitude and direction is the same) in the x-direction that would imply that it is a conservative B-field, which is bogus. The only way you could produce a constant B-field is with an infinite conducting place which is purely hypothetical. I see so many problems in textbooks that have constant B-fields in them. Are they just saying that the B-field is only constant in a particular region (center of a huge conducting plane, for instance?) I suppose this is ok, but it kinda bothers me a bit.

Well, strictly speaking, a constant B-field through all of space is a possibility, just as a constant E-field is. Both are solutions to the Maxwell equations. But in practice, it means that you can limit yourself to a chunk of space where this is approximately valid.

Also, one more question, in free space, maxwell's 4th equation says that if there is a changing D-field then there is a curl in H. LEt us now suppose that we have a D field whose derivative is a constant. If there is a constant derivative in the D field then there is a constant curl in H. This means that H only depends on x,y,z, but not on t. Therefore B is constant wrt time. Therefore E does not have a curl according to MAxwell's 3rd equation, correct? Therefore electromagnetic waves to dot propagate.

I don't see how you arrive at your last statement.
The case you cite is the case of a capacitor charging with constant current (D rises linearly). So this gives indeed rise to a stationary magnetic field H, as would a current in a wire, and no waves.

 

[leright]

I think it is much better to consider a moving loop and the total force E+vxB, and then to calculate the vxB part due to the moving piece of loop (as I sketched).

yes, this is much clearer to me. BUT, if this is the case then textbooks need to stop saying things like Vemf = integral(E.dl) = integral(E + VxB).dl, which would imply that curlE = -(partial of B wrt t) + curl(U x B). The U x B should be considered a separate force. A better way to write this would be integral[(F/q).dl] = integral(E + VxB).dl, or curl(F/q) = -(partial of B wrt t) + curl(U x B)

It's funny, my text goes on about this motional electric field (exact phrasing) for a whole page and says stuff like curlE = -(partial of B wrt t) + curl(U x B) and they go and mention the Lorentz force equation q(E + U x B) three pages later, like U x B is now NOT an electric field. When something is not an electric field, you shouldn't call it "motional electric field" The book is elements of electromagnetics, by Sadiku, if you want to check it out.

What frightens me about this most of all is that there's TONS of literature with nonsense like this, considering U X B an electric field. If this is the case, then, as you mentioned, the Lorentz force equation would simply be qE, where E would encompass everything, including the electrostatic field, the field caused by changing B-field, and the field caused by U x B. IT might be a good publishable paper topic to shed some light on this subtle confusion that is rampant everywhere...hmmm...

Stokes' theorem only works for stationary loops of course.

Well, whenever you find the circulation integral (or the curl) you are finding it in a time-invariant frame (snapshot). In this sense, you can determine the curl (or circulation integral) when in a time varying loop, and just treat t as a constant. Then when you've found the curl of circulation you'll have a variable t in it, and you can find the curl (or circulation) at any point in time.

I think that the concept of "motional electric field" is bull. The force on a moving charge is q (E + v x B), and it doesn't make sense to call v x B a kind of "electric" field.

agreed.

I don't see how you arrive at your last statement.
The case you cite is the case of a capacitor charging with constant current (D rises linearly). So this gives indeed rise to a stationary magnetic field H, as would a current in a wire, and no waves.

Well, people say that a changing D-field produces a changing B-field, which produces a changing D-field, etc, which is wave propagation. However, According to the 4th equation a LINEARLY changing D-field (constant second derivative) produces a stationary H-field, which doesn't allow wave propagation. What am I missing here?

THANK! All of this confusion over semantics! It's not only my book, but tons of literature that seem to make these mistakes. I am hoping books like Jackson and Griffith do not make these errors.

 

[leright]

Check out this website. This site goes on and on about errors in Maxwell's equations because they do not take into account motion of charges in a magnetic field, claiming that U x B contributes to an E-field...hmmm...

But, one question, if U x B isn't to be considered an E-field caused by a b-field in the presence of moving charges, then where does this expression (Fsubm/q) = U x B fit into Maxwell's equations if we want to model systems of moving bodies in material space. The beginning of my book claims that the 4 differential expressions of maxwell's equations summarize everything in the book, which I think is BS. Nowhere is the motional EMF part of the Lorentz force mentioned in Maxwell's equations. http://www.angelfire.com/sc3/elmag/Seriously, type in "corrections to Maxwell's equations" and you may be shocked at what you see. So many websites saying stuff like curlE = -partial of B wrt t + curl(U x B), and then making claims like curlE = -dB/dt is correct, but having partial derivatives is incorrect. There's some mathematical theorem that says having a total derivative of B takes care of both the cross product and the partial derivative.

It seems the whole world is saying this "motional electric field" is indeed an electric field and should be treated the same as the electric field generated by changing B-field.

 

[vanesch]

yes, this is much clearer to me. BUT, if this is the case then textbooks need to stop saying things like Vemf = integral(E.dl) = integral(E + VxB).dl, which would imply that curlE = -(partial of B wrt t) + curl(U x B). The U x B should be considered a separate force. A better way to write this would be integral[(F/q).dl] = integral(E + VxB).dl, or curl(F/q) = -(partial of B wrt t) + curl(U x B)

I agree with everything here, except maybe the very last equation. It doesn't make sense to talk about "curl F", because F is not a vector FIELD over an open domain in space. It has only an existence ON THE MOVING LINE and depends on the precise motion. So you cannot take, for instance, partial derivatives of F to x, y and z, because it is not defined over a patch of space, but only on your specific loop.

It's funny, my text goes on about this motional electric field (exact phrasing) for a whole page and says stuff like curlE = -(partial of B wrt t) + curl(U x B) and they go and mention the Lorentz force equation q(E + U x B) three pages later, like U x B is now NOT an electric field. When something is not an electric field, you shouldn't call it "motional electric field" The book is elements of electromagnetics, by Sadiku, if you want to check it out.

I don't know the book, but indeed, I find this highly confusing "pedagology". There are tons of bad books out there who introduce silly concepts with the excuse of making things "easier" or "by analogy".

What frightens me about this most of all is that there's TONS of literature with nonsense like this, considering U X B an electric field. If this is the case, then, as you mentioned, the Lorentz force equation would simply be qE, where E would encompass everything, including the electrostatic field, the field caused by changing B-field, and the field caused by U x B. IT might be a good publishable paper topic to shed some light on this subtle confusion that is rampant everywhere...hmmm...

It wouldn't get accepted, because it is "well known" of course :-) You can't publish papers over confusions in the literature. I tried that myself a few times and got rejected each time ! And indeed, it is scary what bull is published. It's even worse on the internet.

Well, whenever you find the circulation integral (or the curl) you are finding it in a time-invariant frame (snapshot). In this sense, you can determine the curl (or circulation integral) when in a time varying loop, and just treat t as a constant. Then when you've found the curl of circulation you'll have a variable t in it, and you can find the curl (or circulation) at any point in time.

Yes, that is correct. But you now see that the time derivative remains OUTSIDE of the integral. Stokes' theorem applied to Maxwell's third equation needs the derivative INSIDE the integral. And you can only commute both when the loop is time-independent.

The "miracle" (or happy coincidence) is that this v x B term in the Lorentz force exactly makes for the difference between the d/dt outside and inside of the integral.

Well, people say that a changing D-field produces a changing B-field, which produces a changing D-field, etc, which is wave propagation.

Well, the correct statement is that a changing D-field produces A B FIELD. Whether that one is changing or not is not given. If D changes linearly, then B doesn't change (but is non-zero).
The statement is however correct if the change is sinusoidal, for instance.

 

[vanesch]

Check out this website. This site goes on and on about errors in Maxwell's equations because they do not take into account motion of charges in a magnetic field, claiming that U x B contributes to an E-field...hmmm...

I just skimmed through it but I'd be tempted to classify this site as a "crackpot" site, honestly...
Be careful with websites which are not endorsed by a university or other scientific institute. That doesn't mean that some personal sites are not of high scientific quality ! PF being one for instance There are many out there who are. But first refer to "standard" works, and find out for yourself (or ask here: PF is known for its "crackpot hunting policy" ), but always remain sceptical when reading something like "Maxwell equations are erroneous"

 

[leright]

I just skimmed through it but I'd be tempted to classify this site as a "crackpot" site, honestly...
Be careful with websites which are not endorsed by a university or other scientific institute. That doesn't mean that some personal sites are not of high scientific quality ! PF being one for instance There are many out there who are. But first refer to "standard" works, and find out for yourself (or ask here: PF is known for its "crackpot hunting policy" ), but always remain sceptical when reading something like "Maxwell equations are erroneous"

Well, according to that website there was a passage from Griffiths also supporting the idea that the force on a moving charge in a magnetic field is due to a "motional electric field", and Griffiths incorporates this into his text. Check this paper out.

this document has me convinced that there is at least lots of confusion out there over this issue (even in the respected classical electrodynamics literature) - http://www.angelfire.com/sc3/elmag/files/EM05FL.pdf Tomorrow I am going to take it to some professors and discuss it with them.

What about this website?

http://www.wolfram-stanek.de/maxwell_equations.htm

 

[vanesch]

literature) - http://www.angelfire.com/sc3/elmag/files/EM05FL.pdf Tomorrow I am going to take it to some professors and discuss it with them.

Well, the first equation is already wrong, so I stopped reading. You cannot permute the integral and the d/dt if the surface is moving.

This is easily verified: take the vector field a to be equal to (3,0,0) when (x,y,z) is within the cube (0<x<1, 0<y<1, 0<z<1) and consider a unit square parallel to the yz plane, which is in the plane x = 0.5 and which touches the Y and Z axis at t = 0, and is moving with a velocity = 5 in the z-direction.
Clearly, d a / dt = 0 because it is a static field, independent of t, so the right hand is 0. However, the left hand isn't: the flux over this moving surface is 3 at t=0, and decreases linearly to 0 at t = 0.2, so d phi/dt = -15.

 

[vanesch]

What about this website?

http://www.wolfram-stanek.de/maxwell_equations.htm

Sad.

There is a way to interpret this confusing story, and the hint of where this comes from is given here:

2. IF we are using this replacement ( ∂ / ∂ t ) by (d / dt) in case of moved bodies
never forget the necessary relations for field transformations
(i.e. Lorentz' transformation) hidden in the left sides of the equations (1a) - (4a)

The ' fields are the fields in the locally co-moving frame. THEN of course all this is correct: the E' field is the E-field AS SEEN BY A LOCALLY CO-MOVING OBSERVER, and not by the observer in the coordinate frame where we are "working from" (in which, for instance, the velocity is specified). So all this is correct IN A NON-INERTIAL FRAME.
This is what I hinted at in the beginning of this thread:
a constant B field for observer A is seen as an E-field plus a B field for a moving observer (the E field being v x B with v the velocity of B wrt A, and B the same, if the transformation is non-relativistic).
BUT IT IS EXTREMELY CONFUSING to use an E' field, which is defined for DIFFERENT OBSERVERS in different places !
Of course, for this co-moving observer, the seen E-field is the only one acting upon the test charge there, because relative to this observer, this test-charge is at rest (so the vxB term for the test charge is 0, given that the v OF THE CHARGE WRT THE LOCAL CO-MOVING OBSERVER is 0).

That is why normally, when talking about the E-field, we assume it to be a field *as seen by one and the same observer everywhere*.

 

[leright]

Well, the first equation is already wrong, so I stopped reading. You cannot permute the integral and the d/dt if the surface is moving.

This is easily verified: take the vector field a to be equal to (3,0,0) when (x,y,z) is within the cube (0<x<1, 0<y<1, 0<z<1) and consider a unit square parallel to the yz plane, which is in the plane x = 0.5 and which touches the Y and Z axis at t = 0, and is moving with a velocity = 5 in the z-direction.
Clearly, d a / dt = 0 because it is a static field, independent of t, so the right hand is 0. However, the left hand isn't: the flux over this moving surface is 3 at t=0, and decreases linearly to 0 at t = 0.2, so d phi/dt = -15.

no no no...there are direct quotes from texts like Jackson in that article that claim otherwise. Remember, total derivatives are different than partial derivatives and total derivatives CAN be commuted into the integral when there is a time varying loop. This is known as the convective derivative. http://mathworld.wolfram.com/ConvectiveDerivative.html

 

[leright]

Well, the first equation is already wrong, so I stopped reading. You cannot permute the integral and the d/dt if the surface is moving.

This is easily verified: take the vector field a to be equal to (3,0,0) when (x,y,z) is within the cube (0<x<1, 0<y<1, 0<z<1) and consider a unit square parallel to the yz plane, which is in the plane x = 0.5 and which touches the Y and Z axis at t = 0, and is moving with a velocity = 5 in the z-direction.
Clearly, d a / dt = 0 because it is a static field, independent of t, so the right hand is 0. However, the left hand isn't: the flux over this moving surface is 3 at t=0, and decreases linearly to 0 at t = 0.2, so d phi/dt = -15.

you're missing the difference between partial time derivatives and total time derivatives.

 

[leright]

Sad.

There is a way to interpret this confusing story, and the hint of where this comes from is given here:The ' fields are the fields in the locally co-moving frame. THEN of course all this is correct: the E' field is the E-field AS SEEN BY A LOCALLY CO-MOVING OBSERVER, and not by the observer in the coordinate frame where we are "working from" (in which, for instance, the velocity is specified). So all this is correct IN A NON-INERTIAL FRAME.
This is what I hinted at in the beginning of this thread:
a constant B field for observer A is seen as an E-field plus a B field for a moving observer (the E field being v x B with v the velocity of B wrt A, and B the same, if the transformation is non-relativistic).
BUT IT IS EXTREMELY CONFUSING to use an E' field, which is defined for DIFFERENT OBSERVERS in different places !
Of course, for this co-moving observer, the seen E-field is the only one acting upon the test charge there, because relative to this observer, this test-charge is at rest (so the vxB term for the test charge is 0, given that the v OF THE CHARGE WRT THE LOCAL CO-MOVING OBSERVER is 0).

That is why normally, when talking about the E-field, we assume it to be a field *as seen by one and the same observer everywhere*.

I am pretty confident that the expressions on that website work even when the inertial reference frame is not the same velocity as the moving charge. If this is the case then the original -partial of B wrt t will work just fine and there's no need to this mumbo jumbo.

Also, I keep thinking back to the Lorentz force equation, that most all textbook state, complementing the above 'modified' maxwell equations. IF it is really an E-field resulting from this motional EMF then the v x B term in should not be there in the Lorentz force equation, and the total force on a charge be reduced to qE. This is what happens when you call every damn field an E-field.

Also, the Vemf is DEFINED as the integral (E.dl) (work that is done on a charge, per unit charge, and the force per unit charge they HAPPEN to use is the E-field, so ANY EMF is caused by some kind of E-field, BY DEFINITION. So, Lorentz force equation CONTRADICTS the definition of EMF. I am so angry at the fellow that created this nonsense (not these websites but the original creators, failing to carefully formulate meaningfulterminology) right now it isn't even funny (this is including maxwell and hertz)

THis confusion alone has really shaken my confidence in the framework of physics, and I am afraid most every other facet of physics is the same way. It's a shame really. People (even the greatest physicists of all time) get too lost in the math, and forget to use their intuition, and think beyond the damn greek letters and equations. It is just a damned mess. Communicating the information in a sensible way is just as important as being able to calculate things to high degrees of accuracy.

I wouldn't be so dissappointed if it were only a few testbooks with this nonsensical garble, but when the most well respected textbooks of the field display the same nonsense, it makes me hurl.

 

[leright]

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

look at the identity here, regarding to convective derivative

 

[leright]

http://www.andrijar.com/phipps/index.html

ok, I take this back...this website clarifies almost all of my confusion, thank god. Here, in Hertzian electrodynamics, F = qE . Hertz calls everything an electric field.

So there are two camps you can lie in when you talk about electrodynamics. You can talk in terms of Hertzian electrodynamics, or Maxwellian electrodynamics. You cannot mix the two ideas. The Lorentz force is different in Herzian electrodynamics, and is actually called Hertzian force.

Personally, I like the Herzian theory much better, since it describes much more ideas in the same amount of equations, but it also expands the meaning of the electric field, claiming that all forces on charges are due to E-fields, and the B-fields present only produce E-fields that exert the forces. I think this is the best way to do things since EMF is defined as E integrated over a path. Wonderful. I wish textbooks would make this distinction more clear though.

I have learned a lot from this discussion. Vanesch, I appreciate you help, and I think we both gained from the struggles in this thread.

 

[vanesch]

http://www.andrijar.com/phipps/index.html
ok, I take this back...this website clarifies almost all of my confusion, thank god. Here, in Hertzian electrodynamics, F = qE . Hertz calls everything an electric field.

Yes, you can do that, it is what I said: you can define the "electric field" E' the electric field in the locally co-moving frame, which means you have a different coordinate frame in each point of space. In that co-moving frame, of course, the v of your test charge is 0, and the Lorentz force E + vxB reduces to E.

But the problem with that approach is that it only works for ONE velocity in a point. What do you do when you have two velocities in a point (like in a plasma, where you may have two species of particles with same or different charge, and which have different hydrodynamical velocities) ? Which frame are you now going to take to calculate E' ? You now have TWO E' in a single point, because you have two velocities.
In other words, E' is not a vector field (mapping from space into a vector space).

It is much, much easier to say that in the observer frame (whatever that is, but defined everywhere), there are two fields, E and B, and that the force on a moving charge with velocity v in that frame equals q (E + vxB).
E and B are then well-defined vector fields for this observer, independent of the motion of any test charge.

 

[jackiefrost]

Thanks to everyone participating in this discussion. This is a facinating subject!

As a sidelight, if any of you have access to the Feynman Lectures (vol 2, 17-1 and 17-2), he delves into this very topic though on a fairly elementary level.

 

[vanesch]

Also, I keep thinking back to the Lorentz force equation, that most all textbook state, complementing the above 'modified' maxwell equations. IF it is really an E-field resulting from this motional EMF then the v x B term in should not be there in the Lorentz force equation, and the total force on a charge be reduced to qE. This is what happens when you call every damn field an E-field.

Yes, of course, you'll have to choose. Or you define the E-field to be the E-field in the frame which is co-moving with the charge (so that the charge is stationary in this frame) and then of course the total force on the charge is always q.E ;
OR you define the E and B field wrt to an observer, and define the motion of a charge wrt that same observer (velocity v), and then the force on that charge is q(E + vxB).

But the first approach misses something important which the second one includes. The first approach needs to define a new E field for every moving charge around, even if they are in the same space point (cfr plasma physics for instance).
The second (Maxwell) approach is much more elegant, for the following reason. Maxwell says that, for a given observer (independent of any state of motion of a particle), to each point in space you can attach two vectors, called E and B. And if a particle is moving through that point with a velocity v, then it will experience a force q(E(x) + v x B(x)). It is the same E(x) field and the same B(x) field, no matter what the state of motion of the particle. We only need position and velocity. It doesn't matter, for instance, what its acceleration is. The force is a function of position, and velocity. You give me position and velocity and I tell you the force.
Newtonian gravity, for instance, is only dependent on position: m.g(x). It doesn't depend on velocity. But the electromagnetic force DOES depend on position and velocity.

The Hertzian approach needs to assign a field to each different state of motion, without the elegance of seeing that the contributions can be written in the form E + v x B.

EMF is simply the integral of the total force on charge carriers on the loop (which, hence, co-move with the loop). This force is then given by E + v x B of course. If the loop is stationary, it reduces to E, otherwise it includes the v x B contribution.

Also, the Vemf is DEFINED as the integral (E.dl) (work that is done on a charge, per unit charge, and the force per unit charge they HAPPEN to use is the E-field, so ANY EMF is caused by some kind of E-field, BY DEFINITION. So, Lorentz force equation CONTRADICTS the definition of EMF.

Yes, and no, because you have to consider each piece of loop in its own reference frame, in which it is at rest, so in which the velocity is 0 (although that velocity is not 0 of course wrt the original observer). So even if you add a v x B term, it doesn't contribute because v is by definition always 0. In this Hertzian approach, the velocity of a charged particle is always 0 wrt to the frame in which the field is defined, and hence the lorentz force always reduces to q E.



I am so angry at the fellow that created this nonsense (not these websites but the original creators, failing to carefully formulate meaningfulterminology) right now it isn't even funny (this is including maxwell and hertz)

THis confusion alone has really shaken my confidence in the framework of physics, and I am afraid most every other facet of physics is the same way. It's a shame really. People (even the greatest physicists of all time) get too lost in the math, and forget to use their intuition, and think beyond the damn greek letters and equations. It is just a damned mess. Communicating the information in a sensible way is just as important as being able to calculate things to high degrees of accuracy.

I wouldn't be so dissappointed if it were only a few testbooks with this nonsensical garble, but when the most well respected textbooks of the field display the same nonsense, it makes me hurl.

You don't find this kind of confusion in standard graduate texts. Most of these confusions are "pedagology" to make things "easier".

 

[leright]

Yes, you can do that, it is what I said: you can define the "electric field" E' the electric field in the locally co-moving frame, which means you have a different coordinate frame in each point of space. In that co-moving frame, of course, the v of your test charge is 0, and the Lorentz force E + vxB reduces to E.

But the problem with that approach is that it only works for ONE velocity in a point. What do you do when you have two velocities in a point (like in a plasma, where you may have two species of particles with same or different charge, and which have different hydrodynamical velocities) ? Which frame are you now going to take to calculate E' ? You now have TWO E' in a single point, because you have two velocities.
In other words, E' is not a vector field (mapping from space into a vector space).

It is much, much easier to say that in the observer frame (whatever that is, but defined everywhere), there are two fields, E and B, and that the force on a moving charge with velocity v in that frame equals q (E + vxB).
E and B are then well-defined vector fields for this observer, independent of the motion of any test charge.

No no no, F = qE also works when a charge is moving relative to a inertial reference frame when you are using the hertzian definition of the E-field, which claims that the magnetic force should really be considered an electric force due to an electric field, and the magnetic field simply induces the E-field that exerts the force. This, to me anyway, makes more sense simply because EMF is defined as the integral of E over a closed path the work that an E-field does on a charge). This definition of EMF claims the E-field is a mathematical description of anything that causes a force on a charged object, and this is how the Hertzian description of the E-field treats the subject.

Also, I would think F = qE still holds in Hertzian electrodynamics when there are multiple charges moving at different relative velocities wrt an inertial reference frame due to the superposition principle. The "E-field" (as it is defined by Hertz) due to charges q1, q2, q3, etc, moving in a B-field with different velocities would simply be the sums of the forces due to the individual charges, and each force would be divided by the respective charge that caused it.

If you want to use the idea that EMF is the integral of E-field over a closed loop (whcih everyone does) then the Hertz school of thought fits better, I would think. If you want to want to stick with Maxwell then you might want to define EMF at least as integral(E + V x B).dl, since EMF is not only caused by an E-field.

All of this really comes down to clear definitions of key terminology. There is no doubt we could correctly solve the problems without clear definitions, but physics isn't just about solving problems.

And the part you mentioned about it be impossible to define the E-field with the V x B term when there are distributions of charge (ring, for example) because all of the differential charges are moving at differnet speeds...well, in this case you would simply integral V x B over the contour path. The E-field would vary with time since the velocity at the different points on the ring varies with time, and also the B-field might vary with time, but this is problem. However, this really doesn't matter (outside of philosophical discussion) since the problems become rather trivial when solved with good ol' faraday's law (minus the rate of change of magnetic flux.)

 

[leright]

Thanks to everyone participating in this discussion. This is a facinating subject!

As a sidelight, if any of you have access to the Feynman Lectures (vol 2, 17-1 and 17-2), he delves into this very topic though on a fairly elementary level.

He delves into the problems with Maxwell's equations, or he delves into the subject of electrodynamics?

 

[leright]

But yes, having an expression that allows the E-field and B-field to be the same regardless of the motion of the reference frame or the velocity of the charge moving through the field may seem to be more elegant. However, how do you know this description is physically accurate.

But then another thought occurred to me. What if the charge is what we might consider stationary, but you are moving, and therefore the charge is moving. How do we think through this? Therefore a force acts on this charge (regardless of whether it's due to an E-field or not)? Maybe this velocity should be interpreted as the velocity relative to the B-field, instead? But what about if the B-field is time-varying? Then it is moving relative to the charge, regardless of the movement of the inertial reference frame. Maybe this idea encompasses the fact that curlE is minus the partial of B wrt t. The changing B-field (even if the inertial reference frame is not changing) causes an E-field that is perpendicular to the motion of the B-field, which is changing (which is what curlE implies...the curl of E points in the opposite direction of the changing B-field, so the E-field "swirls" around the changing B-field).

So, maybe, this should be interpreted as meaning all of these velocities are wrt to the B-field. Faraday did his experiments, I assume, only in a stationary reference frame, and therefore the reference frame of the B-field and the observer is the same speed.

So, maybe the reference frame of the observer means nothing in these problems, and V x B is the velocity of the charge movement wrt the B-field.

So, maybe the lesson here is (which I think is known to most people), is that induced E-fields are produced in such a way as to work against changing B-fields (so the magnetic flux produced by the E-field resists the B-field's changing flux), and induced B-fields are produced in such a way as to work against changing E-fields (so the electric flux produced by the B-field resists the E-field's changing flux). After this is all said and done, it is the resulting E-field that exerts all of the force on test charges.

I think I am getting somewhere, and I bet I can eventually explain special relativity in an intuitive way based on nothing more than the fundamental experimental observations in classical electrodynamics.

 

[vanesch]

Also, I would think F = qE still holds in Hertzian electrodynamics when there are multiple charges moving at different relative velocities wrt an inertial reference frame due to the superposition principle. The "E-field" (as it is defined by Hertz) due to charges q1, q2, q3, etc, moving in a B-field with different velocities would simply be the sums of the forces due to the individual charges, and each force would be divided by the respective charge that caused it.

I think we're talking next to each other here. In order for something to be a field, it needs to take on a value at a certain point in space. Now imagine that there are 3 particles, each with charge 1, moving through point (0,0,0). Particle A has velocity (1,0,0), Particle B is at rest, and Particle C has velocity (0,0,1). The Maxwell B-field is (0,1,0) and the Maxwell E-field is (2,0,0).

What's the value of the Hertzian "E-field" in point (0,0,0) so that I can calculate the forces on particles A,B and C ?
For particle A, it is (2,0,0) + (0,0,1) = (2,0,1)
For particle B, it is (2,0,0)
For particle C, it is (2,0,0) + (-1,0,0) = (1,0,0)

So for each different particle, we'd need a DIFFERENT value of the hertzian E-field in one and the same point (0,0,0).

If you want to use the idea that EMF is the integral of E-field over a closed loop (whcih everyone does) then the Hertz school of thought fits better, I would think.

Of course, because the starting point is wrong of course, and not everybody does that. With this definition of EMF, there would for instance, not be an EMF in a circuit with a battery. The EMF is the total force exerced on charges around the loop. These forces can be of electromagnetic, or of other origin (such as electrochemical).

If you want to want to stick with Maxwell then you might want to define EMF at least as integral(E + V x B).dl, since EMF is not only caused by an E-field.

Yes, and that's the correct way of course.

And the part you mentioned about it be impossible to define the E-field with the V x B term when there are distributions of charge (ring, for example) because all of the differential charges are moving at differnet speeds...well, in this case you would simply integral V x B over the contour path.

Look at my example above. I'm not talking about different V at different points, I' m talking about different V at the same point.

The E-field would vary with time since the velocity at the different points on the ring varies with time, and also the B-field might vary with time, but this is problem. However, this really doesn't matter (outside of philosophical discussion) since the problems become rather trivial when solved with good ol' faraday's law (minus the rate of change of magnetic flux.)

Yes, of course, for electric machines, Faraday's law is entirely correct.

 

[vanesch]

But yes, having an expression that allows the E-field and B-field to be the same regardless of the motion of the reference frame or the velocity of the charge moving through the field may seem to be more elegant. However, how do you know this description is physically accurate.

Well, because it is experimentally confirmed, and hence postulated as a law of nature! As all other parts of physics...

But then another thought occurred to me. What if the charge is what we might consider stationary, but you are moving, and therefore the charge is moving. How do we think through this?

Imagine that in the frame (A) where the charge is stationary, the E field is 0, and the B field is constant. The lorentz force is 0 in this case.
Now, go to a frame B which is moving with velocity V wrt A. In this frame (B), there is an electric and a magnetic field: E = V x B and B remains what it was. Now, the total Lorentz force in this frame, on the charge, which is moving with velocity -V in this frame, is given by:
F = q (E + (-V) x B) = q ( V x B + (-V) x B) = 0 too.