Is the nature of a magnetic field always conservative or is it case dependent?

[Blogical]

Is Magnetic field conservative or non-conservative in nature, I have searched online regarding this, it seems to be a divided house, MIT professor Walter Lewin demonstrated that it is non-conservative using an experiment, but still many arent convinced with the way experiment was conducted raising the question whether Lumped Matter Discipline is indeed valid!

 

[andrien]

A vector field is conservative if it can be represented by the gradient of a scalar field.that means
B=-∇∅,but this implies that curl of magnetic field is zero.which is true only if (according to maxwell eqn) when there are no current sources and no time varying electric field is present.however it is still possible to define a scalar potential but that is just another thing.

 

[vanhees71]

No, in the general case there's no scalar potential for the magnetic components of the electromagnetic field but a vector potential. From the Maxwell equation
$$\vec{\nabla} \cdot \vec{B}=0$$
follows, provided sufficiently well behaved fields, the existence of a vector potential
$$\vec{B}=\vec{\nabla} \times \vec{A}.$$

The force of the electromagnetic field on a test charge, however is given by the Lorentz-force law
$$\vec{F}=q \left (\vec{E}+\frac{\vec{v}}{c} \times \vec{B} \right).$$
This is only a conservative force for the case of purely electrostatic fields, where there exists a time-independent scalar potential of the electric field and $\vec{B}=0.$

Despite other claims in this forum magnetic fields never do work on charges or currents, because obviously
$$\vec{v} \cdot \vec{F}=q \vec{v} \cdot \vec{E}$$
in the general case!

 

[Q-reeus]

Is Magnetic field conservative or non-conservative in nature, I have searched online regarding this, it seems to be a divided house, MIT professor Walter Lewin demonstrated that it is non-conservative using an experiment, but still many arent convinced with the way experiment was conducted raising the question whether Lumped Matter Discipline is indeed valid!

Lewin lecture referred to has nothing to do with magnetic field's nature, but rather with how to interpret and apply circuit laws affected by presence of solenoidal electric field associated with (not caused by) a time-changing magnetic field.

 

[cabraham]

No, in the general case there's no scalar potential for the magnetic components of the electromagnetic field but a vector potential. From the Maxwell equation
$$\vec{\nabla} \cdot \vec{B}=0$$
follows, provided sufficiently well behaved fields, the existence of a vector potential
$$\vec{B}=\vec{\nabla} \times \vec{A}.$$

The force of the electromagnetic field on a test charge, however is given by the Lorentz-force law
$$\vec{F}=q \left (\vec{E}+\frac{\vec{v}}{c} \times \vec{B} \right).$$
This is only a conservative force for the case of purely electrostatic fields, where there exists a time-independent scalar potential of the electric field and $\vec{B}=0.$

Despite other claims in this forum magnetic fields never do work on charges or currents, because obviously
$$\vec{v} \cdot \vec{F}=q \vec{v} \cdot \vec{E}$$
in the general case!

But magnetic fields do work on magnetic dipoles. A motor is an example where magnetic fields spin the rotor, not electric. This is important to know. Every university motor text cited B fields as the field doing the work on the rotor.

Claude

 

[Blogical]

Lewin lecture referred to has nothing to do with magnetic field's nature, but rather with how to interpret and apply circuit laws affected by presence of solenoidal electric field associated with (not caused by) a time-changing magnetic field.

Your point being?
Conservative or not??
And how??

 

[Q-reeus]

Your point being? Conservative or not?? And how??

It depends on what your real focus and intent was in #1:

Is Magnetic field conservative or non-conservative in nature, I have searched online regarding this, it seems to be a divided house, MIT professor Walter Lewin demonstrated that it is non-conservative using an experiment, but still many arent convinced with the way experiment was conducted raising the question whether Lumped Matter Discipline is indeed valid!

Taken as stated, you want to know if magnetic field is conservative, but I assume the lecture you were referring to was this one , and the 'divided house' bit referred to here

Well as per #4, static or quasi-static magnetic fields, as against the properties of solenoidal 'transformer action' electric fields, are two quite different topics. In one sense a magnetic field is non-conservative in that it is solenoidal in nature (field lines always forming closed loops), being defined as B = ∇×A. Thus if magnetic monopoles exist (none found so far), the energy exchange with a magnetic field would be entirely path dependent, hence non-conservative. However in an important sense magnetic fields are conservative in that if you move two electromagnets relative to each other, the change in electrical work done in the electromagnet conductor windings, (induced transformer action E fields acting on the currents flowing), plus mechanical work extracted, is exactly compensated for in the change in net magnetic field energy. The same sort of thing applies in the case of permanent magnet relative motions, though in that case the accurate statement is that net change in mechanical+heat energy is exactly compensated in the net change in magnetic potential energy ∫m.Bdv. The latter also equal to the net change in magnetic field energy - if computed on the basis of the Gilbert model using fictitious magnetic poles. [That entails working from a scalar potential formulation of H field, see here and here]

If your focus was on Lewin's lecture, it is simply not about magnetic fields as such, though the ∂B/∂t mentioned is closely association with the electric fields of real interest there.

 

[A Dhingra]

For the case of Regular time- varying Electric field, the magnetic field is non-conservative.
But can i please ask, Does a time-varying Electric field (whose curl is non-zero or the induced electric) produces a magnetic field? If yes, what is the nature of this field?
Is it conservative or non conservative??

 

[Q-reeus]

For the case of Regular time- varying Electric field, the magnetic field is non-conservative.
But can i please ask, Does a time-varying Electric field (whose curl is non-zero or the induced electric) produces a magnetic field? If yes, what is the nature of this field?
Is it conservative or non conservative??

You seem to have essentially answered at least part of your own question, but in a reverse-order fashion! Anyway, although it's often stated as cause-effect relation, E and B fields do not 'cause' each other. Both are caused by source charges, free or bound, plus any magnetic media, in general motion.

 

[Dickfore]

Fields are not conservative, forces are. And the Lorentz force is gyroscopic.

 

[A Dhingra]

You seem to have essentially answered at least part of your own question, but in a reverse-order fashion! Anyway, although it's often stated as cause-effect relation, E and B fields do not 'cause' each other. Both are caused by source charges, free or bound, plus any magnetic media, in general motion.

Ahh... So it is like saying that the time varying magnetic field caused the charged particles of the system to move, generating their Induced Electric field, which is just an effect of the motion of the charged particles; then these charged particles produce the magnetic field which (i think) has to be like the original one,i.e., non conservative in nature... right?

 

[Q-reeus]

Ahh... So it is like saying that the time varying magnetic field caused the charged particles of the system to move, generating their Induced Electric field, which is just an effect of the motion of the charged particles; then these charged particles produce the magnetic field which (i think) has to be like the original one,i.e., non conservative in nature... right?

My earlier statement needs some qualification. The ultimate source of any EM field is charge - as per Lienard-Wiechert expression http://www.google.com.au/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&ved=0CCMQFjAA&url=http%3A%2F%2Fen.wikipedia.org%2Fwiki%2FLi%25C3%25A9nard%25E2%2580%2593Wiechert_potential&ei=pCBbUN_WCOTnmAXFm4DYAg&usg=AFQjCNEzvXiF4oKT7yRiMQFzcD7jok57cw
However it's also fair to say that an impinging EM field causes motion of charge which then sets up new fields. Hence e.g. the interplay between broadcasting and receiving antennas, photosynthesis from sunlight, etc. So in that sense yes to your question. Radiation fields are commonly referred to as 'source-free' in that they are free to propagate far from the original source charges - but they always owe existence to such source charges.

Equally it's the case that the emf induced in the secondary windings of a transformer is not caused by the time changing magnetic field, even though that is commonly how it is said. The cause is primarily the time-changing magnetization within the 'iron' core, plus to a smaller degree the time-changing current flowing in the primary windings. Both induced emf (circulating E field) and magnetic field are results, not primary causes one of the other. Hope this helps more than confuses.

 

[A Dhingra]

Okay.. Thanks

 

[Blogical]

It depends on what your real focus and intent was in #1:

Taken as stated, you want to know if magnetic field is conservative, but I assume the lecture you were referring to was this one , and the 'divided house' bit referred to here

Well as per #4, static or quasi-static magnetic fields, as against the properties of solenoidal 'transformer action' electric fields, are two quite different topics. In one sense a magnetic field is non-conservative in that it is solenoidal in nature (field lines always forming closed loops), being defined as B = ∇×A. Thus if magnetic monopoles exist (none found so far), the energy exchange with a magnetic field would be entirely path dependent, hence non-conservative. However in an important sense magnetic fields are conservative in that if you move two electromagnets relative to each other, the change in electrical work done in the electromagnet conductor windings, (induced transformer action E fields acting on the currents flowing), plus mechanical work extracted, is exactly compensated for in the change in net magnetic field energy. The same sort of thing applies in the case of permanent magnet relative motions, though in that case the accurate statement is that net change in mechanical+heat energy is exactly compensated in the net change in magnetic potential energy ∫m.Bdv. The latter also equal to the net change in magnetic field energy - if computed on the basis of the Gilbert model using fictitious magnetic poles. [That entails working from a scalar potential formulation of H field, see here and here]

If your focus was on Lewin's lecture, it is simply not about magnetic fields as such, though the ∂B/∂t mentioned is closely association with the electric fields of real interest there.

So do u mean to say that conservative or non-conservative nature is case dependent??
If so, is there a clear-cut way to recognize this like in the case of an induced electric field due to varying magnetic field is always non-conservative...

 

[Q-reeus]

So do u mean to say that conservative or non-conservative nature is case dependent??

What I was getting at is there has to be a clear idea of what one means by 'conservative vs non-conservative', and in the broadest sense that is context sensitive. Charged particles moving through a magnetic field always experiences a vxB force normal to the particles motion, hence the interaction with such field is entirely conservative as no particle energy change is involved. That does not define the field as conservative though. Magnetic field has the potential to be non-conservative owing to it being inherently solenoidal - the curl of a vector potential B =∇×A. In practice that is never exhibited since there are no magnetic monopoles to take advantage of inherent path dependence which characterizes nature of a solenoidal field. When magnetized media interacts with a magnetic field, it is customary to treat such media as either composed of microscopic current loops or true magnetic dipoles. In either case it turns out energy exchanges are dependent only on the initial and final states, not the path taken. Hence in all physically realizable cases, interactions with B field are considered path independent thus conservative. This is not to be confused with the behavior of say electric motors and alternators, where a B field can certainly indirectly facilitate overall path dependent energy exchanges. But frictionless gears and pulleys can do the same in a mechanical setting.
[Edit: Above applies for any static B field case. In time-changing B situations an in vacuo non-zero curl B exists, permitting magnetized media to be subject to non-conservative forces. Usually far too weak to be of any consequence however.]

If so, is there a clear-cut way to recognize this like in the case of an induced electric field due to varying magnetic field is always non-conservative...

Unlike magnetic monopoles, electric charges exist so for inherently solenoidal E field linked to a time-changing B, it's non-conservative nature is manifested in e.g. transformer action. Integral form of Maxwell-Faraday law makes non-conservative nature of such an E field evident:

There is no beginning or end to such looping lines (see last three illustrations here: http://en.wikipedia.org/wiki/Toroidal_inductors_and_transformers for depiction of lines of A and E = -∂A/∂t around a toroidal inductor/transformer) While the line-integral relation above is defined for a single-loop closed path, net path length in any given situation can be arbitrary. Hence the net potential difference can be of arbitrary value - for N turns of a transformer secondary winding, multiply above by N. Which is why transformer voltage step-up/step-down is possible. Potential difference between two points depends on the path taken, unlike in electrostatic case where E = -∇V guarantees strictly conservative path independence - e.g. field between charged capacitor plates, which always begins and ends on charges.