Do the fields extend to infinity in a solenoid and for a wire?

[PainterGuy]

Hi,

I wanted to clarify a point about the magnetic fields of a solenoid and wire. Do the fields extend to infinity? In my opinion, they don't but they can assuming the current also goes to infinity. They don't extend to infinity for a limited amount of current because they need to follow a closed path. But in the case of a single charge the field does extend to infinity. In the case of two oppositely charged bodies, again the fields don't extend to infinity.

Could you please confirm if I have it correct? Thank you!

 

[Delta2]

The fields usually extend to infinity even if the sources (current densities sources or charge densities sources) are finite. The fields intensity drops according to $1/r$ or $1/r^2$ or $1/r^3$ where r is the distance from the source. There might be some large value of r for which the fields become too weak, but theoretically $1/r$ never becomes zero for any value of r. So in all four cases you mention in your pictures, the fields theoretically extend to infinity.

The fact that the field lines might form closed curves (which happens in 3 of your pictures, the finite solenoid, the finite wire and the two charges) or might be straight lines (which happen in the case of the single point charge) has nothing to do with whether the fields extend to infinity or not.

The only exception to this is when there is some sort of conducting structure that acts as shielding and prevents the fields from extending to infinity. Such a shielding can be for example for the point charge , a thin spherical conducting shell of any radius that encloses the point charge in its center.

 

[Merlin3189]

...
The only exception to this is when there is some sort of conducting structure that acts as shielding and prevents the fields from extending to infinity. Such a shielding can be for example for the point charge , a thin spherical conducting shell of any radius that encloses the point charge in its center.

This is the only bit I would quibble with. If you have a conducting shell containing charge, then itself is now a charged object with a field extending outwards to infinity.
Maybe a balanced dipole charge inside a perfect conducting shell could have no external field, because there is no net charge inside it. But even there I think the separation of charges in the conductive shell caused by the internal field, would replicate a dipole field externally.

I don't know what relativity says about all this, but as far as I can see, all fields from every charge in the universe (whatever that is) extend to everywhere else, always have and always will. Just that the net sum is near zero, provided there is not a preponderance of one polarity overall*, or a local imbalance.
When we create observable fields, it's just that we've moved cahrges around to create such a local imbalance.
*I don't know, but my guess is, that charges must always be created in opposite pairs, so that there never can be an overall charge on the universe.

 

[Delta2]

This is the only bit I would quibble with. If you have a conducting shell containing charge, then itself is now a charged object with a field extending outwards to infinity.
Maybe a balanced dipole charge inside a perfect conducting shell could have no external field, because there is no net charge inside it. But even there I think the separation of charges in the conductive shell caused by the internal field, would replicate a dipole field externally.

I think you are right (and I was wrong) . We certainly cannot shield the field of the point charge by the way I described, (not even if it was a dipole) for the reasons you say. Gauss's law for the case of the point charge would simply imply that there is net flux in a spherical area that surrounds the conducting shell (and the point charge) because there would be net charge enclosed. But as you say even in the case of dipole where there would be no net charge enclosed, I believe there would be some sort of dipole field externally.

The shielding I describe works probably for shielding electromagnetic waves (which are not electrostastic fields), and I have experimental verification of this, once I shielded my mobile phone by surrounding with aluminium foil and I just couldn't make a call to it :D.

 

[Merlin3189]

I think the Faraday cage idea shields the inside from external fields, rather than shielding the outside from inside fields.
External fields cause mobile charges on the outside surface to move around and cancel the field that would otherwise have gone through the internal space.

(In terms of my vague musings about the field from every charge existing everywhere, I'd have to suppose that the movement of these external surface charges (on any conducting shell) were such in quantity and positioning as to create an exactly equal and opposite field inside the shell, so that there was no net field there. And the idea that there can be no field inside an empty conducting shell, would instead say that the sum of all the field contributions from external charges is always zero inside the conductive shell.)

 

[PainterGuy]

Thank you, both of you!

The fields usually extend to infinity even if the sources (current densities sources or charge densities sources) are finite. The fields intensity drops according to $1/r$ or $1/r^2$ or $1/r^3$ where r is the distance from the source. There might be some large value of r for which the fields become too weak, but theoretically $1/r$ never becomes zero for any value of r. So in all four cases you mention in your pictures, the fields theoretically extend to infinity.

The fact that the field lines might form closed curves (which happens in 3 of your pictures, the finite solenoid, the finite wire and the two charges) or might be straight lines (which happen in the case of the single point charge) has nothing to do with whether the fields extend to infinity or not.

The only exception to this is when there is some sort of conducting structure that acts as shielding and prevents the fields from extending to infinity. Such a shielding can be for example for the point charge , a thin spherical conducting shell of any radius that encloses the point charge in its center.

Now I understand that the fields extend to infinity. But I'd say that one should rather say that the fields could extend to infinity if allowed enough time. The magnetic fields, for example from a solenoid/inductor or wire, propagate outward at the speed of light so just saying that the field extend to infinity is not just correct, in my opinion. The same goes for electric field from a point charge.

I have another question about the magnetic field of solenoid and wire. A solenoid is an inductor which stores its energy in the form of magnetic field; a wire also acts like an inductor with very small inductance. When the current stops flowing through the inductor, its magnetic field collapses and energy is released. If DC current has been running through inductor for some time, its field might have extended beyond the reaches of solar system. So, shouldn't it take some noticeable time for the collapsing field to return energy to the inductor? Practically, the energy is returned instantaneously. I think a big portion of energy is returned instantaneously and remaining some unnoticeable infinitesimal portion might take longer.

Thanks a lot for you help!

 

[Delta2]

Thank you, both of you!
Now I understand that the fields extend to infinity. But I'd say that one should rather say that the fields could extend to infinity if allowed enough time. The magnetic fields, for example from a solenoid/inductor or wire, propagate outward at the speed of light so just saying that the field extend to infinity is not just correct, in my opinion. The same goes for electric field from a point charge.

Yes you are very right on this, as the fields propagate with the speed of light from the source, it takes time to reach the space that surrounds the source, from the time moment we "switch on" the source.

I have another question about the magnetic field of solenoid and wire. A solenoid is an inductor which stores its energy in the form of magnetic field; a wire also acts like an inductor with very small inductance. When the current stops flowing through the inductor, its magnetic field collapses and energy is released. If DC current has been running through inductor for some time, its field might have extended beyond the reaches of solar system. So, shouldn't it take some noticeable time for the collapsing field to return energy to the inductor? Practically, the energy is returned instantaneously. I think a big portion of energy is returned instantaneously and remaining some unnoticeable infinitesimal portion might take longer.

Thanks a lot for you help!

As the fields intensity drop according to the inverse square law $\frac{1}{r^2}$, the farther we are from the source, the less portion of energy is stored in the field there cause the field gets weaker and weaker the farther we are from source. I believe within the sphere with radius the first 1km from the source (let the source be a solenoid or a straight wire) something like 99% of the total energy is stored, and the remaining 1% energy is stored in the remaining space (distance bigger than 1km). 1km is nothing when we travel with the speed of light, so yes practically 99% of the energy is returned instantaneously.

 

[PainterGuy]

Thank you!

As the fields intensity drop according to the inverse square law 1r21r2\frac{1}{r^2}, the farther we are from the source, the less portion of energy is stored in the field there cause the field gets weaker and weaker the farther we are from source. I believe within the sphere with radius the first 1km from the source (let the source be a solenoid or a straight wire) something like 99% of the total energy is stored, and the remaining 1% energy is stored in the remaining space (distance bigger than 1km). 1km is nothing when we travel with the speed of light, so yes practically 99% of the energy is returned instantaneously.

I'm still confused about this. Okay, but that 99% stored energy which lies within the radius of 1 km would have flown away 3x10⁸ meters the very next second! It would mean that the energy which returns to solenoid/inductor instantaneously after it is turned off is only a (small) portion of total magnetic field energy generated while the solenoid/inductor was running; and we could say that that returned 'instantaneous' field energy only corresponds to the field which lies, say, within 1 km radius. Do I make any sense?

Thanks a lot!

 

[Delta2]

Thank you!
I'm still confused about this. Okay, but that 99% stored energy which lies within the radius of 1 km would have flown away 3x10⁸ meters the very next second! It would mean that the energy which returns to solenoid/inductor instantaneously after it is turned off is only a (small) portion of total magnetic field energy generated while the solenoid/inductor was running; and we could say that that returned 'instantaneous' field energy only corresponds to the field which lies, say, within 1 km radius. Do I make any sense?

Thanks a lot!

The fields have a radiating term or far-field term which is proportional to $1/r$ and this term gives the energy that is radiated away from the source and is never coming back (even when we switch off the source). That's why it is called radiating term. This term is small in the case of a solenoid because a solenoid due to its geometry does not radiate a lot.

There is also the inductive term or near field term. This term is proportional to $1/r^2$ and on half the cycle (we assume a wire or solenoid that has a sinusoidal current) it moves forward that is away from the source so the source gives energy to the space around it, but in the next half cycle it is moving backwards and so the field is returning energy from the space that surrounds the source, back to the source.

I understand that there is a problem here, how the inductive fields move forward to fill the space up to infinity, yet the energy that the inductive fields carry is not moving forward (radiated away ) but it is trapped in the space around the source. Hold on while I am thinking on this...

 

[Delta2]

Sorry I can't find any info on my books on how the inductive fields travel from the source to infinity, in all the books that I have the inductive E and B field seem to occupy the space from the source to infinity like the source has persisted for ever. They don't seem to give any info on the transient period on what happens from the moment t=0 we switch on the source and there are no fields at that moment, till the moment that the inductive fields travel to a specific point in space.

Maybe @vanhees71 can enlighten us on how exactly the inductive fields (the $1/r^2$ terms ) from an electric or magnetic dipole travel up to infinity from the time t=0 we switch on the dipole.

 

[PainterGuy]

Thank you!

But I think that the major confusion, as you stated earlier, was about how the inductive fields, fields from inductor, wire, etc., move forward to fill the space up to infinity assuming the source had been running since forever, yet the energy that the inductive fields carry is not moving forward (radiated away) but is trapped in the space around the source in a very small radius. Therefore, when the source is turned off, the energy stored in field is returned to source almost instantaneously.

 

[Delta2]

Seems i forgot something important in my analysis that it is about the phase of the E-field and the B-field around the source.

What happens in the near field region (in that small radius around source) is that it is dominated by the components of E-field and the B-field that are almost at phase difference $\frac{\pi}{2}$ which makes the time averaged poynting vector almost equal to zero, which means that almost no energy is flowing into or out of this region. The energy seems to be trapped in that small radius around the source, and this is just because of the phase difference between the E-field and the B-field. There is still some energy that is escaping this region and it is the energy with which the fields expand to infinity but this energy is small compared to the "trapped" energy around the source.

In the far field region (far away from source, in a radius that is many multiples, typical greater than 10x wavelength) we have the components of E-field and B-field that are almost in phase agreement (phase difference equals almost zero) and this makes the poynting vector to be radially outwards which means that this is the energy which which the E-field and the B-field travel up to infinity.

So to wrap it up
The E-field and the B-field around the source have components that are in phase agreement but also some other components that are in phase difference $\pi/2$.

In a small radius around the source (maybe up to afew multiples of the wavelength) the components that are in phase difference $\pi/2$ dominate (they are much greater than the other components) and so the energy seems to be stationary (to be more accurate in half the cycle the majority of energy flows from source to this space, and in the next half cycle the majority of energy flows from the space back to source, there is still some energy that is escaping this region and it is the energy of the far-field)

Far away from the source (radius 10xwavelength and even greater) the components that are in phase agreement dominate and this makes the energy to flow up to infinity.

 

[PainterGuy]

Thank you, @Delta2!

I really appreciate your effort to help me. I seem to understand those near and far fields around an antenna. In my opinion, all that is described really well in this video from Youtube.

But I was thinking of the magnetic field from an inductor. I don't think an inductor would involve any electric field like a dipole antenna as shown in the video. Even when you have an inductor running on DC, it still stores energy around it in form of magnetic field which is given back once the source is switched off. As long as an inductor is running, it will keep on 'emanating' magnetic field from it. The field would move away from it at the speed of light. Now again the question is if the field is moving away from the source at speed of light then how come all of the stored energy in field is returned back instantaneously.

By the way, one might think, as I was in my first post, that the magnetic field lines need to follow a closed path, from north to south in case of an inductor or wire, therefore the field doesn't move outward to infinity. Well, I was wrong because magnetic field lines, compasses, iron fillings etc. are indicative of the nature of field and a way to make sense of it. In actuality the field does move outward at the speed of light.

I'm sorry if I'm being redundant.

 

[Delta2]

I am glad you found a youtube video that explains the dipole antenna. I admit I didn't do so good job on explaining how the near field energy remains stationary while the fields propagate up to infinity.

Regarding the case of inductor that is "charged" by a DC voltage via a resistor: There is electric field involved during the transient period which the inductor is charging and the current is $I(t)=\frac{V}{R}(1-e^{-\frac{R}{L}t})$, and because we have a current that is changing, which means the magnetic field will also change and changing magnetic field means we ll have an electric field. This transient period theoretically lasts infinite time but practically lasts 5x to 10x $\tau=\frac{R}{L}$.

Any circuit element, let it be resistor or inductor or capacitor , or even transistors and diodes, that is running with a time varying current will behave at least partially as a dipole antenna. So even in the case of the inductor with a DC voltage source , it will behave as a dipole antenna and have near field energy and far field energy.

 

[tech99]

Hertz carried out an experiment to observe the phase of the fields coming from a dipole. He did this by arranging a race between the fields on a transmission line and those in free space.
He observed that for the first half wavelength from the dipole, there was almost no phase shift in free space, which indicated that the velocity was infinite. Beyond half a wavelength, the phase lagged as expected due to propagation delay.
The electric induction field, in extending to half a wavelength in this way, and flowing out and back, is just being a standing wave, with no net flow of energy.
As far as I can see the electric and magnetic induction fields of an antenna are always of this standing wave nature. A standing wave does not have a uniform phase progression.
However, I cannot see how to explain the quasi-static situation when we switch on a solenoid, as we do not seem to observe a continuing energy loss as the fields expand, nor the converse.

 

[PainterGuy]

Thank you, @Delta2, @tech99!

So, in view of post #14 above, an inductor could function like a dipole. I'd assume that it's about the inductor running on AC.

I do accept that the fact that far field carries away the energy to infinity (or, away from the dipole at the speed of light) and for this field both magnetic and electric are in phase and constitute an electromagnetic wave. This carried away energy never returns to the dipole.

So, when an inductor is switched off, the instantaneously returned energy only comes from collapsing near field which has electric and magnetic fields 90 degrees out phase with each other.

What happens in the near field region (in that small radius around source) is that it is dominated by the components of E-field and the B-field that are almost at phase difference π2π2\frac{\pi}{2} which makes the time averaged poynting vector almost equal to zero, which means that almost no energy is flowing into or out of this region. The energy seems to be trapped in that small radius around the source, and this is just because of the phase difference between the E-field and the B-field. There is still some energy that is escaping this region and it is the energy with which the fields expand to infinity but this energy is small compared to the "trapped" energy around the source.

I understand that near field becomes a source of far field so one can say that far field extends to the infinity but the near field remains confined in a smaller radius.

In a small radius around the source (maybe up to afew multiples of the wavelength) the components that are in phase difference π/2π/2\pi/2 dominate (they are much greater than the other components) and so the energy seems to be stationary (to be more accurate in half the cycle the majority of energy flows from source to this space, and in the next half cycle the majority of energy flows from the space back to source, there is still some energy that is escaping this region and it is the energy of the far-field)

Far away from the source (radius 10xwavelength and even greater) the components that are in phase agreement dominate and this makes the energy to flow up to infinity.

Agreed. The important point is that it involves AC so energy flows out in half cycle and flows back in the next half cycle.

However, I cannot see how to explain the quasi-static situation when we switch on a solenoid, as we do not seem to observe a continuing energy loss as the fields expand, nor the converse.

Exactly that's what troubling me. The operation of a solenoid involves DC and field(s) keeps expanding toward infinity and there is no back and forth transfer of energy in this case, but as soon as it is switched off all the energy is returned back instantaneously by the field(s) which had expanded to the infinity.

I better take some time and hopefully it'll make sense after some months. Anyway, I once again appreciate your help. Thanks.