# Changing magnetic field and a point charge, seems unresolvable

## Main Question or Discussion Point

Consider a current loop with a switch and an AC source in space, there is a point charge(electron) inside this loop, my question is,

Will the point charge feel any force when we close the switch, i.e. will the point charge feel any force in the presence of changing magnetic field ?

If yes, then, in which direction the force will be and of what magnitude ? Since, all sorts of electric line integral pass through this point charge, Does the force depend on the shape of the loop too ?

If no, then, I think there is a problem with in the Feynman's Lecture on physics, vol. 2, ch. 17, section. 4, A paradox.

And don't forget the betatrons work almost on the same principle, except the electrons are already moving in a well defined loop themselves when the changing magnetic field is applied to increase their energy.

Thanks

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mfb
Mentor
It depends on the position of the charge. If the loop is a perfect circle, the magnetic field is symmetric around the center, and changing, this gives rise to a circular electric field.
If the point charge is not in the center, it will feel a force along these electric field lines, perpendicular to the line "point charge <-> center". You can find its magnitude by integrating along the loop to get the magnetic field as a function of time and integrating over its time-derivative in the area around the center. In other words: It can be done, but unless you need the result for something practical I would not recommend it.
If the loop is not a perfect circle, it becomes more complicated to evaluate the forces, but the basic concept is the same.

It depends on the position of the charge. If the loop is a perfect circle, the magnetic field is symmetric around the center, and changing, this gives rise to a circular electric field.
If the point charge is not in the center, it will feel a force along these electric field lines, perpendicular to the line "point charge <-> center". You can find its magnitude by integrating along the loop to get the magnetic field as a function of time and integrating over its time-derivative in the area around the center. In other words: It can be done, but unless you need the result for something practical I would not recommend it.
If the loop is not a perfect circle, it becomes more complicated to evaluate the forces, but the basic concept is the same.
Since you say there will be a force, and for a perfect circle you even suggested the direction.

Does that mean that a point charge standing near a straight long current carrying wire which is also connected to an AC source, will also feel the force ?

If yes then in which direction ? Because among the possible line integral for electric field, are at both sides of the point charge, and therefore both line integral will give different direction for the force, and there are many more direction if we start considering all the line integrals !

So, my question is, are there any known solutions or experiments(other than betatron) which shows how we should proceed about it.

Thanks

mfb
Mentor
The AC source is a source of a variable electric field - not only in the wire, but also in the environment. This can accelerate electrons, and the direction is simply the "+" side.

So, my question is, are there any known solutions or experiments(other than betatron) which shows how we should proceed about it.
Solve the Maxwell equations?

The AC source is a source of a variable electric field - not only in the wire, but also in the environment. This can accelerate electrons, and the direction is simply the "+" side.

Solve the Maxwell equations?

I see you are going around about the problem instead of tackling it directly.

I think I know that there will be an electric field due to changing magnetic filed from a changing current, the problem is how am I supposed to apply the maxwell's equation for a point charge ?

Since $∇ × E = - ∂B/∂t$ , the curl of E is only definable for a given metal loop or a given orbit of electrons, how can we define the same for a stationary point charge ?

mfb
Mentor
Maxwells equations give you the external field, given by the AC source and wire - and your point charge is affected by this field. Unless you have a very large point charge, you can ignore the effects of this charge on the charge distribution in the wire (and the corresponding changes in the electromagnetic field of the wire).

Maxwells equations give you the external field, given by the AC source and wire - and your point charge is affected by this field.
Let's suppose I calculate the magnetic field vector as a function of time, around the wire.

Now, the problem is how am I supposed to know about the direction and magnitude of the electric field induced, for a point charge ?

For simplicity, let's assume that magnetic field is given by the equation $B = B_0 sin(\omega t)$, all over the place. Now, tell me how and why does the electron should or would move in a particular direction.

Since, according to the maxwell's third equation that I posted earlier, $∇×E=−∂B/∂t$, by stokes's theorem leads to $\oint E. ds =−∂\phi/∂t , where, \phi = \oint B. da$, Now, since the integral is only meaningful for a closed loop, how should I to use this equation for the force on a stationary point charge ?

mfb
Mentor
The field induced by a point charge? You can approximate this point charge with a small object, or use distributions. However, this field will not create a force on the charge.

Now, tell me how and why does the electron should or would move in a particular direction.
The electromagnetic field you describe does not exist, there has to be some electric component, and B cannot fill the space like that. The force on the point charge is given by $F=q(E + \vec{v} \times \vec{B})$ with fields E and B from your device.

The field induced by a point charge? You can approximate this point charge with a small object, or use distributions. However, this field will not create a force on the charge.
No, the point charge is just the test charge, the self interaction is not the point here.
The electromagnetic field you describe does not exist, there has to be some electric component, and B cannot fill the space like that. The force on the point charge is given by $F=q(E + \vec{v} \times \vec{B})$ with fields E and B from your device.
I won't argue if such electromagnetic field is possible or not, but I know you have not yet told me how can I use Faraday's law for stationary point charges.

But here is another real scenario, consider an infinitely long current carrying wire with current $I = I_0 . t$, i.e it is proportional to the time(for simplicity). Now, a point charge 'q' at a distance 'r' from the current carrying wire should feel a force.

Please tell me in which direction and of what magnitude ?

Thanks

Dale
Mentor
I see you are going around about the problem instead of tackling it directly.

I think I know that there will be an electric field due to changing magnetic filed from a changing current, the problem is how am I supposed to apply the maxwell's equation for a point charge ?

Since $∇ × E = - ∂B/∂t$ , the curl of E is only definable for a given metal loop or a given orbit of electrons, how can we define the same for a stationary point charge ?
The fields for a point charge are given by the Lienard Weichert potential:
http://en.wikipedia.org/wiki/Liénard–Wiechert_potential

The fields for a point charge are given by the Lienard Weichert potential:
http://en.wikipedia.org/wiki/Liénard–Wiechert_potential

Seems like you didn't get my question, my question is not about the field of a point charge, but the force on a point charge due to changing magnetic field.

Seems like you didn't get my question, my question is not about the field of a point charge, but the force on a point charge due to changing magnetic field.
So what you do is apply those field eqn's to the source current distribution - coil or line current etc. One approach is to first evaluate the vector potential A at the charge but owing to summing over all moving charge contributions in the coil, line current etc., which is then differentiated wrt time to give an unambiguously defined E field -dA/dt wherever one wishes to evaluate it. Or compute the E, B fields directly from summing over the actual retarded E, B fields owing to each and every moving source charge (or current density usually) in that coil etc. Latter I would prefer. The Faraday expression is useful for emf calcs in a coil but as you have discovered has limited utility outside of that.

So what you do is apply those field eqn's to the source current distribution - coil or line current etc. One approach is to first evaluate the vector potential A at the charge but owing to summing over all moving charge contributions in the coil, line current etc., which is then differentiated wrt time to give an unambiguously defined E field -dA/dt wherever one wishes to evaluate it. Or compute the E, B fields directly from summing over the actual retarded E, B fields owing to each and every moving source charge (or current density usually) in that coil etc. Latter I would prefer. The Faraday expression is useful for emf calcs in a coil but as you have discovered has limited utility outside of that.
Thankyou very much for the clear cut explanation.

And if one uses the vector potential A to calculate electric field as $\vec{E}=-∂\vec{A}/∂t$ (since scalar potential is zero for currents in a wire), does it mean that the effect is due to retardation of electric field of the moving electrons in the current carrying wire ?

Since, I thought that Lienard and wiechert potentials are retarded, i.e. A in above equation works only if we consider the retardation effects while formulating the A itself.

Thankyou very much for the clear cut explanation.

And if one uses the vector potential A to calculate electric field as $\vec{E}=-∂\vec{A}/∂t$ (since scalar potential is zero for currents in a wire), does it mean that the effect is due to retardation of electric field of the moving electrons in the current carrying wire ?

Since, I thought that Lienard and wiechert potentials are retarded, i.e. A in above equation works only if we consider the retardation effects while formulating the A itself.
Well by examination of those potential and field eqn's it's seen that A itself, evaluated at the field point (where the test charge is) depends on just charge velocity and direction, but evaluated at the retarded time computed via radial distance between source charge/current element and field point. If the source is 'slowly time-varying' (wavelength >> radial distance involved) one can mostly safely ignore differences in retarded time and simply sum for the total A first assuming no retardation, then time differentiate in one go. Handy for nice symmetrical structures like a circular coil or straight wire. When high frequencies are involved (say for electrically large antenna field calcs) this approximation fails and it's best to work by summing over the source currents using the field equations directly, which requires knowledge of retarded acceleration of source charge/currents in addition to velocity, at each source location. There are plenty of internet ham radio sites and the like, and of course textbooks giving worked examples.
[I left out discussing the scalar potential phi. It certainly contributes whenever there are non-uniform phase variations involved - say in dipole antennas, but except for near-field problems, it can be factored out in many cases. You will get examples here: http://www.ece.rutgers.edu/~orfanidi/ewa/] [Broken]

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Well by examination of those potential and field eqn's it's seen that A itself, evaluated at the field point (where the test charge is) depends on just charge velocity and direction, but evaluated at the retarded time computed via radial distance between source charge/current element and field point. If the source is 'slowly time-varying' (wavelength >> radial distance involved) one can mostly safely ignore differences in retarded time and simply sum for the total A first assuming no retardation, then time differentiate in one go. Handy for nice symmetrical structures like a circular coil or straight wire. When high frequencies are involved (say for electrically large antenna field calcs) this approximation fails and it's best to work by summing over the source currents using the field equations directly, which requires knowledge of retarded acceleration of source charge/currents in addition to velocity, at each source location. There are plenty of internet ham radio sites and the like, and of course textbooks giving worked examples.
[I left out discussing the scalar potential phi. It certainly contributes whenever there are non-uniform phase variations involved - say in dipole antennas, but except for near-field problems, it can be factored out in many cases. You will get examples here: http://www.ece.rutgers.edu/~orfanidi/ewa/] [Broken]
Well you put it completely and nicely, and yes I know just like relativistic effects when β is appreciably less we can ignore retardation of motion ! (i.e we can work in coulomb gauge)

And are you aware of any experiments that shows this very effect, I mean moving a stationary free electron/proton(if something like this exist) by changing the magnetic fields.

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Dale
Mentor
Seems like you didn't get my question, my question is not about the field of a point charge, but the force on a point charge due to changing magnetic field.
That is just given by the Lorentz force.
http://en.wikipedia.org/wiki/Lorentz_force

There is no additional force due to the time derivative of the field.

That is just given by the Lorentz force.
http://en.wikipedia.org/wiki/Lorentz_force

There is no additional force due to the time derivative of the field.
I know Lorentz Force does not contain any time derivative of E or B field, but time derivative of one can produce other according to electromagnetism of Maxwell's equations, which of course then validly comes under the formula of Lorentz Force.

My concern is, if we have an electric field due to changing magnetic field as $\vec{E}=-∂\vec{A}/∂t$, then how is it any different than the Faraday's law $∇×\vec{E}=-∂\vec{B}/∂t$. Just insert $\vec{B}=∇×\vec{A}$

The above problem is there (i.e ambiguity of E field at a point due to changing magnetic field), until we introduce the Green's function for elementary current densities and integrate for the whole current density setup. Then again, why do we need E and B of Maxwell's equations if the potentials(phi and A) are more fundamental and less confusing, that is, the model in which we integrate the spherical effect of currents and charges seems more practical than the abstract concept of E and B fields.

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By defining B=Curl(A) and E=-(Del(V)+dA/dt) we automatically satisfy Faradays law and gauss' law of magnetism.
V (or Phi, the scalar potential) and A are not more fundamental, they are not directly observable and do not have a one to one relation with their electromagnetic fields.
I do not see what is so unresolvable here.

Edit:After re-reading the original post I am confused what you are asking, first you say you are confused about the potential formulation of EM, and then you go on to say something about the "ambiguity of E field..", without finishing that sentence you continue on to say that the potential formulation is more fundamental then the E and B field one. Could you restate your question please?

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Dale
Mentor
My concern is, if we have an electric field due to changing magnetic field as $\vec{E}=-∂\vec{A}/∂t$, then how is it any different than the Faraday's law $∇×\vec{E}=-∂\vec{B}/∂t$. Just insert $\vec{B}=∇×\vec{A}$
It isn't any different. When you work with potentials instead of fields then Faraday's law is an identity (as is Gauss' law for magnetism). I.e. you can write down fields E and B such that Faraday's law is violated, but you cannot write down potentials V and A such that Faraday's law is violated.

Well you put it completely and nicely, and yes I know just like relativistic effects when β is appreciably less we can ignore retardation of motion ! (i.e we can work in coulomb gauge)
And are you aware of any experiments that shows this very effect, I mean moving a stationary free electron/proton(if something like this exist) by changing the magnetic fields.
None off hand, but especially with toroidal transformers no other explanation really fits the behavior. If I have read between the lines right here, your chief interest is in resolving the Feynman disk paradox referred to in #1. There, a mechanical imbalance of angular momentum arises when the central superconducting coil 'thaws' and a decaying dI/dt -> -dA/dt solenoidal E field acting on the peripheral charges, yet with no commensurate back-reaction on the coil. The usual explanation (Feynman doesn't provide it there) posits a compensating physically real angular momentum stored in the initial distribution of crossed static E and B fields mostly exterior to the physical structure. A variant with analysis is at http://au.search.yahoo.com/r/_ylt=A...n.edu/~mcdonald/examples/feynman_cylinder.pdf
Am I right in thinking your main concern is with the mechanical imbalance, as the notion that the superposition of purely static and independently generated fields creates real field momentum seems fishy? Then you are far from alone but the book-keeping works out in such scenarios. I have looked at certain situations involving precessional motions where the book-keeping balance sure seems hard to find, but won't elaborate here.

It isn't any different. When you work with potentials instead of fields then Faraday's law is an identity (as is Gauss' law for magnetism). I.e. you can write down fields E and B such that Faraday's law is violated, but you cannot write down potentials V and A such that Faraday's law is violated.
Thanks Dalespam, for the insights, on the two ways to deal with electrodynamics. But it seems the original Faraday's law(changing magnetic fields produces currents) as modified for Maxwell's equations by Maxwell(changing magnetic fields produces electric field) is not a logically consistent modification, abiding with the definition of electric field.

However, the assumption that changing magnetic fields produces electric field again encounters another problem(even after resolving the ambiguity issue), when we try to conserve the momentum of the system, as is nicely pointed out by Q-reeus in the above post.

None off hand, but especially with toroidal transformers no other explanation really fits the behavior. If I have read between the lines right here, your chief interest is in resolving the Feynman disk paradox referred to in #1. There, a mechanical imbalance of angular momentum arises when the central superconducting coil 'thaws' and a decaying dI/dt -> -dA/dt solenoidal E field acting on the peripheral charges, yet with no commensurate back-reaction on the coil. The usual explanation (Feynman doesn't provide it there) posits a compensating physically real angular momentum stored in the initial distribution of crossed static E and B fields mostly exterior to the physical structure. A variant with analysis is at http://au.search.yahoo.com/r/_ylt=A...n.edu/~mcdonald/examples/feynman_cylinder.pdf
Am I right in thinking your main concern is with the mechanical imbalance, as the notion that the superposition of purely static and independently generated fields creates real field momentum seems fishy? Then you are far from alone but the book-keeping works out in such scenarios. I have looked at certain situations involving precessional motions where the book-keeping balance sure seems hard to find, but won't elaborate here.
Q-reeus you are absolutely right in recognizing my concern, but your explanation raises serious questions which I previously did not have in my mind.

First of all about the part highlighted in bold in your response, how is it possible that the coil does not get back reaction and the disc starts moving ? It would be a simple violation of Newton's third law ! Even the book-keeping requirement of the existence of a real angular momentum should produce the commensurate back-reaction on the coil ! Since one cannot produce angular momentum without the Force that produces it at the first place!

Dale
Mentor
Thanks Dalespam, for the insights, on the two ways to deal with electrodynamics. But it seems the original Faraday's law(changing magnetic fields produces currents) as modified for Maxwell's equations by Maxwell(changing magnetic fields produces electric field) is not a logically consistent modification, abiding with the definition of electric field.
Why not? What specific logical inconsistency arises from the equations when modified in that way? I.e. This is a mathematical claim, can you show it mathematically?

However, the assumption that changing magnetic fields produces electric field again encounters another problem(even after resolving the ambiguity issue), when we try to conserve the momentum of the system, as is nicely pointed out by Q-reeus in the above post.
Momentum is conserved, provided you account for the momentum of the fields as well. The Lagrangian is invariant under both translations and rotations, so both linear and angular momentum are conserved by the fields.

Why not? What specific logical inconsistency arises from the equations when modified in that way? I.e. This is a mathematical claim, can you show it mathematically?
To me mathematics works as one defines it, but here is an insight on the differences of the electric fields from the two laws, the electric field according to Gauss theorem(definition of point charge( $∇.\vec{E_G}=ρ/\epsilon_0$)) definitely has a source of electric field, but the electric field due to Maxwell's equation(Faraday's law) does not have divergence or a point source($∇×\vec{E_M}=-∂\vec{B}/∂t$ ). And this difference remains in SR too, that is one cannot transform one type of field to other without having logical inconsistencies.

That is, one should not replace $\vec{E_M}$ with $\vec{E_G}$ or vice-versa.

Momentum is conserved, provided you account for the momentum of the fields as well. The Lagrangian is invariant under both translations and rotations, so both linear and angular momentum are conserved by the fields.
I think, I don't yet understand the physics behind the momentum of fields, but you should elaborate that part too, since Faraday's law and field momentum seems to be related.

Dale
Mentor
To me mathematics works as one defines it, but here is an insight on the differences of the electric fields from the two laws, the electric field according to Gauss theorem(definition of point charge( $∇.\vec{E_G}=ρ/\epsilon_0$)) definitely has a source of electric field, but the electric field due to Maxwell's equation(Faraday's law) does not have divergence or a point source($∇×\vec{E_M}=-∂\vec{B}/∂t$ ). And this difference remains in SR too, that is one cannot transform one type of field to other without having logical inconsistencies.
There are not different types of E fields here, there is only one kind of E field. Perhaps you do not realize that $\nabla \cdot E$ is not the same as $E$ and similarly with $\nabla \times E$. They are independent conditions that can both be satisfied by one $E$ field.

I think, I don't yet understand the physics behind the momentum of fields, but you should elaborate that part too, since Faraday's law and field momentum seems to be related.
Here are my favorite pages on the momentum of EM fields. The first shows how the momentum of the fields is defined, and the second shows that the momentum is conserved.

http://farside.ph.utexas.edu/teaching/em/lectures/node90.html
http://farside.ph.utexas.edu/teaching/em/lectures/node91.html