Coulomb's law and a voltage frame of reference

[Zeranin]

Well, here it is, a perfectly constructed  ‘magnetic monopole cube’, each face magnetized such that the South pole is on the exterior surface. The attached plot is a 2D cross section through the cube, so you can see 4 of the faces in cross section. Each face is plotted in a different colour, signifying that the direction of magnetization is different for each face. The FEA program chugged away and found the solution, meaning it solved for the magnetic flux density, B, at every point in space. No drama of any sort. To see the image in sufficient detail, you may need to click on the filename and open it and view it full size in a proper jpeg viewer, rather than just clicking on the small icon of the image.

Overlaid on the same attached plot are the magnetic field direction vectors at every point in space on the 2D slice that is being viewed. In my previous posting I attached an identical plot showing the field vectors when only one face (the top orange face) is present, so you already know from that plot that each side is a powerful and effective magnet, that really pushes the magnetic flux through the space around the magnet, achieving a flux density of 0.12 tesla a few mm from the surface.

OK. Looking at the attached plot of the cube, we straight away we can see that something odd is going on here, as predicted, because we are attempting to build something that Maxwell says cannot exist. Rather than plot field vectors, the program has instead plotted a matrix of little crosses, actually ‘+’ signs. Why so? Why has the program refused to plot the field vectors? It’s a fairly smart program, and it will not (indeed cannot) plot something that does not exist. The FEA program has correctly calculated the Bfield at every point in space, no problems, and found that the said field strength is precisely zero, everywhere. You can’t plot the direction of a zero length vector, so the program instead plots little ‘+’ signs, informing the user that the field strength is zero. Although we are viewing a single 2D slice through the center of the model, I can choose any and every 2D slice within the model and the magnetic field strength, B, is zero everywhere, both outside the cube, inside the cube, and inside the NDFeB magnetic material.

If you were to actually build such a cube of magnets, or a sphere, it would behave as if made from unmagnetized NdFeB. It would not stick to your fridge, and two such ‘monopoles’ would neither attract nor repel each other, any more than two plastic golf balls. You could get your best magnetometer probe and wave it around everywhere, including touching the surface of the magnets, and measure nothing at all. Truth stranger than fiction.

Maybe we should not be surprised at the result, we did after all attempt to break the Law, but it is nonetheless an unusual and unintuitive result. Anyone who has ever played with NdFeB magnets knows how incredibly powerful they are, to the point where it can be very difficult to separate such a magnet from a piece of steel, once attached. And yet, if you take 6 such magnets and arrange them into a ‘monopole’ cube, they will become completely ‘dead’. Very cute, I reckon.  If you were to build such a cube, demonstrate to your mate that your metal cube is magnetically as dead as a dodo, and then tell him it was made from powerful NdfeB magnets, he probably wouldn’t believe you. On a practical note, if you were to actually build such a cube, then if you later disassembled the cube, you may find that the magnets are permanently demagnetized to some extent, because when assembled in this configuration, each magnet experiences a massive demagnetizing field provided by the other five. Actually, if you were to build the cube with one side missing, then in terms of the field produced, this ensemble of 5 magnets would behave identically to the single missing side. Very cool.

For anyone that feels tempted to build such a cube, be aware that the commonly available rectangular shaped magnets won’t do the trick, because they won’t properly fit together at the edges of the cube, you would need properly beveled edges. Also, for total field cancellation, all 6 magnets would need to be of identical strength, and in the real world they won’t be. The FEA result is correct, and tells us that the more ‘perfectly’ we build such a ‘monopole’, the more it will converge to producing zero magnetic field at all points in space. Of course, if we build it imperfectly, then it will behave as a (weak) conventional magnet, not as a monopole. The magnetic monopole remains as elusive as ever.

In the next posting I’ll include a plot of the magnetic potential produced by our ‘monopole’ cube, and discuss why this cube behaves in the way that it does, using a number of independent arguments.

[John Heath]

Surprise and fascinating. I have a question.

Will the cube be stressed outward trying to be a sphere rather than a square? There are many more questions but that is the important one.

[Zeranin]

Surprise and fascinating. I have a question.

Will the cube be stressed outward trying to be a sphere rather than a square? There are many more questions but that is the important one.

Hmm. Good question. Intuition tells me 'Yes', and we know that the first few magnets will repel one another like crazy as we bring the North poles together to form the interior surface. So I'll say yes, when the cube is assembled, all faces will experience an outward force. If I change my mind, I'll let you know. 

[Zeranin]

I have attached a plot of ‘magnetic potential’ for our ‘monopole cube’ just for completeness. Boring plot, isn’t it. Outside of the cube is black, meaning the magnetic potential is uniformly zero. Inside of the cube, the magnetic potential is very high at 1782 ampere-turns (yellow colour), but equally uniform. The uniformity of magnetic potential tells us that there will be no flux ‘flowing’ from one place to another, and thus the flux density will be zero. There is, however, a large difference in magnetic potential (AKA magneto-motive-force) between the inside and outside of the cube. The magnets can be seen as a sort of ‘magnetic battery’ that provides this difference in magnetic potential from one end of the magnet to the other, so we certainly expect to see the inside of the cube at a different magnetic potential to the outside. If there was a hole in the cube, this considerable difference in potential would drive flux through the hole, creating a ‘fringing leakage flux’ from one pole back to the other. However, as there are no such holes in this cube, or a sphere, there is simply no path for flux to return from outside to inside, and the flux density is zero. The coercive force of the NdFeB material that I have modelled is 891000 ampere-turns per meter length of magnet, in the direction of magnetization. Therefore, as these magnets are 2mm thick in their direction of magnetization, we should expect them to produce a magneto-motive-force of 891000 x 0.002 = 1782 AT. When I took this screen shot, the cursor was in the center of the cube, and at the bottom of the screen it shows that the magnetic potential at this point (or anywhere within the cube) is indeed 1782 AT. This FEA stuff really does work, and I hope some have found it of interest.

The simplest explanation of why a ‘monopole’ cannot exist is to examine one of Maxwell’s Equations, sometimes known as Gauss’s Law for magnetics, which states that the total net flux leaving a closed volume must equal zero. By definition, a magnetic monopole would have a net flux emanating from it’s surface, so a magnetic monopole would violate this Law. FEA programs make use of this Law, so it is impossible to create an FEA magnetic model of a structure that violates this law, and so it is impossible right from the outset to create a magnetic monopole with my FEA program. That said, I wanted to see what happened when I tried, so I went ahead and modelled it anyway.

However,it’s quite OK to have equal amounts of flux leaving and entering the volume, as the Law in no way says that the flux density on any given part of a closed surface must be zero. What my FEA program actually predicts for the cube is that the flux density, B, will be zero at all points in space, but the Law does not in itself require that to be true, for the Law would still be satisfied if the flux left the cube at some places, such as in the centre of the faces, and returned at other places such as near the corners. In the simpler case of a permanent magnet sphere, it MUST be the case that B is zero everywhere, because symmetry dictates that there can be no place(s) on the sphere where the flux would preferentially choose to leave or enter, but you can’t use that argument for the cube.
 
Here is an elegant explanation of why our cube produces zero magnetic field everywhere. You will need to take my word for it that a thin, planar permanent magnet can be modelled as a current-carrying loop around the perimeter. Assume for the moment that the permanent magnet faces of the cube are ‘very thin’, so we replace each of them with a square loop of very-small-cross-section current-carrying wire, a single-turn square winding if you like. So the cube is formed by 6 such square loops of wire. Each side of the cube will therefore consist of 2 lengths of wire, one from each adjacent loop, but here is the trick. If the direction of current in each loop is chosen so that the resultant magnetic fields all point outward, then you will find (easily drawn as a sketch on a piece of paper) that on every edge, the current flows in opposite direction in the two wires, and thus no magnetic field is produced, because we all know that the net magnetic field from 2 counter-current , very thin, co-located parallel conductors is zero.

In practice, the sides of the cube are not infinitely ‘thin’ actually 2mm thick in my FEA model, but this does not change the argument. All you need to do is stack a large number of ‘very thin’ magnets together to get the thickness you want, and the field from each and every one of them still cancels, as already described. To get the sides to fit exactly you need to bevel the edges, but that does not alter the argument either, just means that each ‘thin’ square face is a slightly different dimension. Thus we see that our square ‘monopole’ cube is expected to produce zero field everywhere, exactly as shown by the FEA, even though the Law would have permitted flux to leave the square faces in one place and return in another. This ‘square-winding-model’ also explains why if you remove one face, then the remaining 5 behave identically to the one that was removed, except of opposite sign. Try it on a sketch. It’s true, and cute as.

I find the above ‘square-winding’ argument to be rather beautiful, and it gives me great pleasure to find that the FEA model produces the same result, of zero field everywhere.

Interestingly, the same argument predicts that we can make our cube of different width, length and height, and still it will produce zero magnetic field everywhere. Anyone care to agree or disagree, or do I need to ‘stretch’ my FEA modelled cube to find out.  ?

Taking it a step even further, I’ll stick my neck out and say we can build a permanent magnet ‘monopole’ of any shape at all, and in all cases it will produce zero field at all points in space, just provided that the magnetization vector is always at right angles to the surface, and every face is the same thickness. You agree with that, John Heath?

[John Heath]

I am getting a sense you want myself to disagree with you. You have a nagging doubt. Hey join the club. I will do my best to make a counter argument for what its worth. In an earlier screen shot of a square magnet north top south bottom the corners have the highest magnetic density becoming weaker towards the center of the square magnet. This is sounding somewhat similar but opposite to Mr Monopole. I will now use your symmetry argument as my counter argument. Your elegant analogy of a thin wire square loop is indeed exactly the opposite to the current edge of the other monopole edge but this ignores the center of the monopole plate. The thin wire square loop should spiral round and round inward to the center of the monopplole wall to justify a uniform magnetic density. However the center of the monopole wall is far away from the edge where it's opposite counterpart lives. For this reason the magnetic field in the center of the wall should express itself unlike the edge of the wall that can not as the other walls cancel it out. This would be my symmetry counter argument.

[Zeranin]

I am getting a sense you want myself to disagree with you. You have a nagging doubt. Hey join the club. I will do my best to make a counter argument for what its worth. In an earlier screen shot of a square magnet north top south bottom the corners have the highest magnetic density becoming weaker towards the center of the square magnet. This is sounding somewhat similar but opposite to Mr Monopole. I will now use your symmetry argument as my counter argument. Your elegant analogy of a thin wire square loop is indeed exactly the opposite to the current edge of the other monopole edge but this ignores the center of the monopole plate. The thin wire square loop should spiral round and round inward to the center of the monopplole wall to justify a uniform magnetic density. However the center of the monopole wall is far away from the edge where it's opposite counterpart lives. For this reason the magnetic field in the center of the wall should express itself unlike the edge of the wall that can not as the other walls cancel it out. This would be my symmetry counter argument.

I have no 'doubts', nagging or otherwise, just find the topic interesting. It is true that I've never modelled a system of magnets that produces zero magnetic field everywhere, but then again, I never tried to model an apparent monopole either.

My method of modelling permanent magnets is correct, but I understand what you are saying as to why it does not seem intuitively right. The following argument is both elegant and intuitive.

In reality, you would expect that a thin permanent magnet is in effect composed of a very large number of tiny current-carrying coils uniformly occupying the entire area of the magnet, not just on the outside as my model appears to be. Agreed.

OK, get a piece of paper and sketch it. Draw a square, say 100x100mm representing a thin, square magnet, magnetized through the thickness. Now draw a matrix of small, square current-carrying coils, for example a 5x5 matrix, so each of the 25 small squares will be 20 x 20mm. Of course, the current around each square loop will be in the same direction, say clockwise. Draw arrows showing the direction of current, on each side of each square. About now, a light bulb will be turning on inside your head. All of the current-carrying conductors inside the square cancel out, except for the sides that form the outside of the 100x100 mm square, and this will always be true regardless of how small we make each current loop, right down to the size of atoms. Nature is very beautiful, is she not? So now you can see why it is valid to model a permanent magnet as a thin sheet of current travelling around the outside surface.  Quite literally, if you were to build the matrix of 25 squares and pump a current I through all of them, then the field produced would in every way be identical to the field produced by passing the same current through a single square 100 x 100 loop.

The FEA model and result are correct, I guarantee it. Just for fun, I might punch a small hole through the centre of one face on our FEA 'monopole' cube, and we should see a torrent of flux squirting through the hole, from the inside of the cube to the outside, as a result of the large magneto-motive-force between inside and outside, as shown on one of my previous plots. That will convince you that the permanent magnet faces are well and truly alive and kicking, including in the center of the face, and yet plug up the hole and you get zilch, nothing, no flux density anywhere and everywhere, even for shapes that lack the perfect symmetry of a sphere. Nature is beautiful.

[Zeranin]

Just for fun, I might punch a small hole through the centre of one face on our FEA 'monopole' cube, and we should see a torrent of flux squirting through the hole, from the inside of the cube to the outside, as a result of the large magneto-motive-force between inside and outside, as shown on one of my previous plots. That will convince you that the permanent magnet faces are well and truly alive and kicking, including in the center of the face, and yet plug up the hole and you get zilch, nothing, no flux density anywhere and everywhere, even for shapes that lack the perfect symmetry of a sphere. Nature is beautiful.

And here it is, a real 'flux cannon', sort of like punching a hole in a hollow cube with high pressure gas inside.  If you remember back to where I modelled the top face in isolation, the flux density a couple of millimetres above the face (pole piece) was 0.15 Tesla.

When this screen shot was taken I had the cursor right in the middle of the hole, so as to read out the flux density at that point. If you look at the bottom of the plot you can see that the flux density inside the hole is 0.49 Tesla, much higher than 0.15 Tesla the top face was able to produce on it's own. Very cute. The flux density does fall off very rapidly beyond the hole though.

[John Heath]

I am getting a sense you want myself to disagree with you. You have a nagging doubt. Hey join the club. I will do my best to make a counter argument for what its worth. In an earlier screen shot of a square magnet north top south bottom the corners have the highest magnetic density becoming weaker towards the center of the square magnet. This is sounding somewhat similar but opposite to Mr Monopole. I will now use your symmetry argument as my counter argument. Your elegant analogy of a thin wire square loop is indeed exactly the opposite to the current edge of the other monopole edge but this ignores the center of the monopole plate. The thin wire square loop should spiral round and round inward to the center of the monopplole wall to justify a uniform magnetic density. However the center of the monopole wall is far away from the edge where it's opposite counterpart lives. For this reason the magnetic field in the center of the wall should express itself unlike the edge of the wall that can not as the other walls cancel it out. This would be my symmetry counter argument.

You misunderstand. There can be no doubt to the screen shot model as that is based on known laws of physics. The doubt comes from the counter intuitive picture the known laws of physics paints. You seemed frustrated so I presented a counter argument as a possible way out of this frustration.

Your rebuttal that every square in the screen shot may be cionsidered 1 atom with electron rotating clock wise

 

I have no 'doubts', nagging or otherwise, just find the topic interesting. It is true that I've never modelled a system of magnets that produces zero magnetic field everywhere, but then again, I never tried to model an apparent monopole either.

My method of modelling permanent magnets is correct, but I understand what you are saying as to why it does not seem intuitively right. The following argument is both elegant and intuitive.

In reality, you would expect that a thin permanent magnet is in effect composed of a very large number of tiny current-carrying coils uniformly occupying the entire area of the magnet, not just on the outside as my model appears to be. Agreed.

OK, get a piece of paper and sketch it. Draw a square, say 100x100mm representing a thin, square magnet, magnetized through the thickness. Now draw a matrix of small, square current-carrying coils, for example a 5x5 matrix, so each of the 25 small squares will be 20 x 20mm. Of course, the current around each square loop will be in the same direction, say clockwise. Draw arrows showing the direction of current, on each side of each square. About now, a light bulb will be turning on inside your head. All of the current-carrying conductors inside the square cancel out, except for the sides that form the outside of the 100x100 mm square, and this will always be true regardless of how small we make each current loop, right down to the size of atoms. Nature is very beautiful, is she not? So now you can see why it is valid to model a permanent magnet as a thin sheet of current travelling around the outside surface.  Quite literally, if you were to build the matrix of 25 squares and pump a current I through all of them, then the field produced would in every way be identical to the field produced by passing the same current through a single square 100 x 100 loop.

The FEA model and result are correct, I guarantee it. Just for fun, I might punch a small hole through the centre of one face on our FEA 'monopole' cube, and we should see a torrent of flux squirting through the hole, from the inside of the cube to the outside, as a result of the large magneto-motive-force between inside and outside, as shown on one of my previous plots. That will convince you that the permanent magnet faces are well and truly alive and kicking, including in the center of the face, and yet plug up the hole and you get zilch, nothing, no flux density anywhere and everywhere, even for shapes that lack the perfect symmetry of a sphere. Nature is beautiful.

You misunderstand. There can be no doubt to the screen shot model as that is based on known laws of physics. The doubt comes from the counter intuitive picture the known laws of physics paints. You seemed frustrated so I presented a counter argument as a possible way out that being Q X Q / R^2 with focus R^2.

Your rebuttal that every square in the screen shot may be considered 1 atom with electron rotating clockwise is more precise than my wire current spiraling towards the center of a monopole wall. However there is a price to be payed for this analogy. If I have an infinitely large 2 dimensional wall where every atom is oriented the same way with all electron rotating clockwise there can not be a magnetic field as all edges of all atom electron obits cancel out the neighboring atoms as per your argument. There in is the rub ,, I think.

A hole in a monopole " flux cannon " , well named

[John Heath]

Just happened to have a few neodymium iron boron magnets around ear marked for a Faraday rotator. In the spirit of Dave's " don't talk about it , build it and measure " it would suffice to say I now have a 2 inch , 50 mm , square monopole cube. If I open the front door to the cube the outward force is great enough to keep the door open under it's own gravity weight when facing up. Your guess that there is an outward force is confirmed. However it is not a perfect monopole as the magnets are smaller 20 mm on a 50 mm perf board. I think the real monopole would only be a small 5 m cube in the center of the box. Would like to FEA model this box to see how close it comes to a real monopole. Will take pictures and specifications for modeling , if you have to time.   

[IanB]

Great to have you in on the discussions. Just to be clear about what you are saying, do you mean the electric field gradient, in units of V/m/m, or do you mean the electric potential gradient, AKA the electric field, in units of V/m?

I think I mean the electric field, or potential gradient.

Regardless, I maintain that fundamentally a gold-leaf electroscope measures the excess charge in Coulombs on each leaf, and that the repulsive force between the leaves is proportional to Q1xQ2/R^2, as per Coulomb's Law, where Q1 and Q2 are the excess (equal) charge on each leaf, and R is the leaf separation.

The question now becomes excess charge relative to what? After all, both leaves are at the same potential and so neither can have excess charge relative to the other. I think it is only possible for the whole instrument to have an excess of charge relative to something outside of it, essentially "the surroundings".

That said, your way of looking at it is really just the same thing, for you cannot have excess charge on the leaves without an electric field extending outward from the leaves. We are talking about 2 sides of the same coin.

This for me is the crucial point. In a sense, Coulumb's Law is an outcome, not a first principle. The first principle is that a charged particle within an electric field experiences a force. If two charged bodies are close to each other, then body A produces an electric field and body B within this field experiences a force. Likewise, body B produces a field and body A within B's field experiences a force. If the two bodies are in free space with no interference from other charged bodies or external fields nearby then Coloumb's Law describes what happens. However, if there is a third charged body nearby, or if there are external electric fields, then the situation becomes more complicated.

You refer to the electroscope being at a different potential to it's surroundings in order to deflect, but in one sense that is not true. If the electroscope was the only object in the Universe, in which case there are no physical 'surroundings', the electroscope will still operate correctly, and it will indicate excess charge on the leaves, just as it always does. That said, there will also be an electric field surrounding the electroscope, dropping off as 1/R^2 at distances large compared to the dimensions of the electroscope, but most people (me included) would say that this field is a consequence of the electroscope having excess charge.

I think that if the electroscope were the only thing in the universe, then the "surroundings" are infinite space, which has a potential of zero at infinity. Then the field drops off towards zero as it extends away from the electroscope. Since both leaves are at the some potential there is no potential gradient and thus no field between them, only a field around them dropping away towards infinity. Therefore the leaves don't respond to each other, but each responds independently to the unsymmetrical field surrounding them which they both have created.

If we were to bring the surroundings to the same potential as the electroscope then there would be a flat field all around it with no potential gradients. In this case the leaves would experience no force and would not separate.

If there are physical surroundings at a particular potential (volts) with respect to the electroscope, then the excess charge on the electroscope is given by Q=CV, where C is the capacitance between the electroscope and the surroundings. Then, the force on the leaves can be calculated from Coulomb's law, knowing this excess charge. Yet again, it seems to me that what is fundamentally creating the force on the leaves is the excess charge on the leaves, which in this case has been 'pushed' there by the potential difference between electroscope and surroundings.

Keep in mind that we can create electric fields and field gradients by all manner of electrostatic apparatus, such as the accelerating stack in a CRT, or a particle accelerator. You can place an electroscope smack in the middle of such a field or field gradient, but it won't register anything unless there is excess charge on the leaves, adding weight to my argument that fundamentally the force on the leaves is as a result of excess charge on the leaves.

Mostly, you and I are looking at 2 sides of the same coin, but you may prefer looking at the other side of the coin. Are we in agreement?   

I guess I do prefer looking at the other side of the coin. I think this is because in any arbitrarily complex system the best way to solve it will be to compute the shape of the electric field as a vector field and then compute the force vector acting upon each charged body within this field.

[Zeranin]

Regardless, I maintain that fundamentally a gold-leaf electroscope measures the excess charge in Coulombs on each leaf, and that the repulsive force between the leaves is proportional to Q1xQ2/R^2, as per Coulomb's Law, where Q1 and Q2 are the excess (equal) charge on each leaf, and R is the leaf separation.

The question now becomes excess charge relative to what? After all, both leaves are at the same potential and so neither can have excess charge relative to the other. I think it is only possible for the whole instrument to have an excess of charge relative to something outside of it, essentially "the surroundings".

No!

Excess charge is excess charge, and is not relative to anything. An object with no excess charge (net charge is you prefer) has equal numbers of +ve and -ve charges, for example, equal numbers of electrons and protons. This is purely an accounting exercise, and is not relative to anything. Similarly, an electron has an absolute net charge, that can be expressed in Coulombs. I can address your other points as necessary, but some may now have been 'neutralized', if you'll pardon the pun. 

[IanB]

No!

Excess charge is excess charge, and is not relative to anything. An object with no excess charge (net charge is you prefer) has equal numbers of +ve and -ve charges, for example, equal numbers of electrons and protons. This is purely an accounting exercise, and is not relative to anything. Similarly, an electron has an absolute net charge, that can be expressed in Coulombs. I can address your other points as necessary, but some may now have been 'neutralized', if you'll pardon the pun. 

Yes, an electron has a charge of -1 unit. I do not dispute physics.

But I still believe that the electrometer does not measure absolute charge, it measures relative charge when compared to its surroundings. If we charge up an electrometer to an absolute potential of 100 V and then place it in surroundings which are uniformly at a potential of 100 V, then I do not think the electrometer will show any deflection. If the electrometer showed a deflection then I think we would have a way of measuring absolute potential, and I do not think this is possible.

(Similarly, if we consider any two adjacent electrons in a sea of electrons, where all electrons are equidistant from each other, than no adjacent pair of electrons will experience any repulsive force.)

[Zeranin]

You misunderstand. There can be no doubt to the screen shot model as that is based on known laws of physics. The doubt comes from the counter intuitive picture the known laws of physics paints. You seemed frustrated so I presented a counter argument as a possible way out that being Q X Q / R^2 with focus R^2.

Your rebuttal that every square in the screen shot may be considered 1 atom with electron rotating clockwise is more precise than my wire current spiraling towards the center of a monopole wall. However there is a price to be payed for this analogy. If I have an infinitely large 2 dimensional wall where every atom is oriented the same way with all electron rotating clockwise there can not be a magnetic field as all edges of all atom electron obits cancel out the neighboring atoms as per your argument. There in is the rub ,, I think.

A hole in a monopole " flux cannon " , well named

The real problem is that you are constantly trying to break the Laws, but far from being 'frustrated' by this I regard it as a good thing. Good scientists are always questioning the Laws and trying to break them, that is how science progresses. It does seem unlikely that very well established Laws such as Maxwell's Equations could be broken by playing around with a few permanent magnets on the kitchen table, but it sure is fun trying.

If I have an infinitely large 2 dimensional wall where every atom is oriented the same way with all electron rotating clockwise there can not be a magnetic field as all edges of all atom electron obits cancel out the neighboring atoms as per your argument. There in is the rub ,, I think.

There is no problem here. What I actually said, is that the net effect of all those current loops in the centre is the same as a single current loop around the outside, and therefore you are wrong to say there can not be a magnetic field. The large current loop around the outside most certainly does create a magnetic field, and it does so at all points within the loop. FEA is one method of calculating the field produced by a current carrying loop, but the traditional and more fundamental method is to use the Biot Law. Basically, all magnetic fields are produced by moving charge. The Biot Law describes the field produced by a single moving charge, or if you prefer, an infinitely short length of current carrying conductor. If you want to know the field produced by a current carrying loop of any shape, then all you need to do is add up the field contributions from all the short lengths of conductor that make up the loop. Sometimes the resultant integrals are easily soluble, and sometimes not. I also have a program that numerically calculates the magnetic field in 3D space from any arrangement of coils and current carrying conductors, using the Biot Law, and it always gives the same results as my FEA program. However, the Biot Law is no longer applicable when permeable magnetic materials are present, whereas FEA can handle the more general case of any structure consisting of current carrying coils and permeable magnetic materials. 

[Zeranin]

Just happened to have a few neodymium iron boron magnets around ear marked for a Faraday rotator. In the spirit of Dave's " don't talk about it , build it and measure " it would suffice to say I now have a 2 inch , 50 mm , square monopole cube. If I open the front door to the cube the outward force is great enough to keep the door open under it's own gravity weight when facing up. Your guess that there is an outward force is confirmed. However it is not a perfect monopole as the magnets are smaller 20 mm on a 50 mm perf board. I think the real monopole would only be a small 5 m cube in the center of the box. Would like to FEA model this box to see how close it comes to a real monopole. Will take pictures and specifications for modeling , if you have to time.   

Sure, I'm happy to FEA model it. Presumably it's constructed from rectangular shaped NdFeB magnets? The great thing about FEA is that you can build something without actually building it, the best of both worlds.

My 'guess' of an outward force on each face was based on good physics. If we use the 'square-current-carrying-loop' model for our monopole cube, then we agree that on every edge we have a pair of currents flowing in opposite directions. When you have 2 parallel current carrying conductors, they exert a force on one another, and this force is repulsive if the currents are in opposite directions, thus the force on each face must be outwards repulsive. Nature, beautiful one day, perfect the next.   

[Zeranin]

But I still believe that the electrometer does not measure absolute charge, it measures relative charge when compared to its surroundings. If we charge up an electrometer to an absolute potential of 100 V and then place it in surroundings which are uniformly at a potential of 100 V, then I do not think the electrometer will show any deflection. If the electrometer showed a deflection then I think we would have a way of measuring absolute potential, and I do not think this is possible.

Here you are appear to be mixing up charge and potential. An electroscope does indeed measure the absolute excess charge on the leaves. There is no such thing as 'relative charge compared to surroundings', though it's fine to talk about the electrical potential of something relative to something else.

When you talk of 'charging an electroscope to an absolute potential of 100V', then you need to be clear about exactly what you mean. This refers to the situation where the electroscope is the only object in the universe, and charging it to an absolute 100V means adding excess charge such that that 100 Joules of energy would be either gained or lost when bringing 1 Coulomb of charge from an infinite distance, up to the electroscope, noting that Volts are in units of J/C. Sure, you can charge your electroscope in that way, OR you can charge it to 100V relative to it's surroundings, which would typically mean connecting a 100V battery between mother earth and the electroscope terminal, let's call this Case B. In this case, it would take 100 Joules per Coulomb to bring additional charge from earth to the electroscope terminal. In both charging scenarios, the electroscope will receive excess charge, but not by the same amount. In Case B, the amount of excess charge received is given by Q=CV, where C is the capacitance between electroscope and surroundings.

Either way, your electroscope will receive excess charge, AKA it will be charged, and the leaves will be deflected. Now remove any connection from the electroscope terminal, so that the amount of excess charge is trapped on the electroscope. Now place the electroscope in a Faraday cage that is at a potential of 100V, ie, one side of a 100V battery is connected to Earth, and the other to the cage. The electroscope deflection will remain unchanged, because the electroscope fundamentally responds to the excess charge on it's leaves, and this has not changed.

Now to really confuse you, charge up your electroscope by whatever means you choose, and remove any connection from it's terminal. Now, by whatever means you choose, charge up a sphere on the end of an insulating rod (ie give it excess charge), and bring the sphere close to the electroscope terminal, but not touching it. The electrometer will respond, with the leaves falling or rising higher, depending on the sign of the excess charge on the sphere, although no charge has moved from the sphere to the electrometer. Aha, I hear you say, gotcha there Zeranin, no change in electroscope charge, and yet it responds! But what is actually happening here is called induced charge. If the sphere is positively charged, then it will attract electrons to the electroscope terminal, from the gold leaves. Thus the excess charge on the gold leaves is changed, and the leaves respond accordingly. In every case, the electrometer leaves rise or fall depending on the amount of excess charge on the leaves. That is fundamentally what an electroscope measures, excess charge on the leaves, and everything else gets down to a discussion about different methods of producing an excess charge on the said leaves.
 

(Similarly, if we consider any two adjacent electrons in a sea of electrons, where all electrons are equidistant from each other, than no adjacent pair of electrons will experience any repulsive force.)

Not true. If you have a dense bunch of electrons in empty space, then they are indeed repelled from each other, and fly apart. This effect is known as 'space charge', and it is a pain in the butt for electrostatic designers such as me that design electron guns and lenses and the like, because it means that when you have a high-current electron beam at low energy (low speed), the beam 'blows apart' rather than staying nicely focussed as you would like. Standard FEA is concerned only with voltages between electrodes, and therefore does not account for space charge effects, but space charge can be accounted for separately in the modelling.

[IanB]

Here you are appear to be mixing up charge and potential.

A change in potential is produced by a spacial displacement of charge. The two are inextricably linked, are they not?

An electroscope does indeed measure the absolute excess charge on the leaves. There is no such thing as 'relative charge compared to surroundings', though it's fine to talk about the electrical potential of something relative to something else.

OK. I can't do the theoretical derivation to prove this for myself, so I will accept it as correct.

When you talk of 'charging an electroscope to an absolute potential of 100V', then you need to be clear about exactly what you mean.

Yes, you are right. I was thinking of the electroscope being in free space and far from anything else.

Either way, your electroscope will receive excess charge, AKA it will be charged, and the leaves will be deflected. Now remove any connection from the electroscope terminal, so that the amount of excess charge is trapped on the electroscope. Now place the electroscope in a Faraday cage that is at a potential of 100V, ie, one side of a 100V battery is connected to Earth, and the other to the cage. The electroscope deflection will remain unchanged, because the electroscope fundamentally responds to the excess charge on it's leaves, and this has not changed.

I think this is because inside the Faraday cage there is a uniform electric field, and therefore no external field gradients to influence the electroscope.

However, believe that the electroscope inside the Faraday cage forms one plate of a capacitor, the Faraday cage the other. If we raise or lower the potential of the Faraday cage from outside then the potential of the electroscope will change by the same amount, since the potential difference between plates of a capacitor fixed in space remains unchanged unless there is a displacement of charge between them. In order to achieve the desired goal of bringing the electroscope and the Faraday cage to the same relative potential we need to transfer charge between them until their potentials are equalized. Once we do this the electroscope leaves will no longer deflect, even if the system of Faraday cage and electroscope has a huge excess charge and has a high potential difference relative to the surroundings.

Now to really confuse you, charge up your electroscope by whatever means you choose, and remove any connection from it's terminal. Now, by whatever means you choose, charge up a sphere on the end of an insulating rod (ie give it excess charge), and bring the sphere close to the electroscope terminal, but not touching it. The electrometer will respond, with the leaves falling or rising higher, depending on the sign of the excess charge on the sphere, although no charge has moved from the sphere to the electrometer.

No confusion here.

(Similarly, if we consider any two adjacent electrons in a sea of electrons, where all electrons are equidistant from each other, than no adjacent pair of electrons will experience any repulsive force.)

Not true. If you have a dense bunch of electrons in empty space, then they are indeed repelled from each other, and fly apart. This effect is known as 'space charge', and it is a pain in the butt for electrostatic designers such as me that design electron guns and lenses and the like, because it means that when you have a high-current electron beam at low energy (low speed), the beam 'blows apart' rather than staying nicely focussed as you would like. Standard FEA is concerned only with voltages between electrodes, and therefore does not account for space charge effects, but space charge can be accounted for separately in the modelling.

What I meant was a sea of electrons, uniform and infinite in extent. If there is any boundary then uniformity is lost and the equidistant property no longer holds.

[Rerouter]

Hmm at a basis, the original question could likely be solved through indirect measures, a 10KV charged cage would cause ionizing in the surrounding air, so by measuring the difference in gas composition he could come up with a rough basis for the voltage to the outside reference frame, with nothing but the air moving through the holes in the cage,

Next up, again probably not that cheap of a way to accomplish it, measure the energy input and wavelength spectra out of a laser, with a higher voltage reference frame the atoms already have higher energy levels, so you should see a tiny amount of higher energy wavelengths compared to the outside measurement, (This one is me guessing from previous knowledge)

[Zeranin]

Either way, your electroscope will receive excess charge, AKA it will be charged, and the leaves will be deflected. Now remove any connection from the electroscope terminal, so that the amount of excess charge is trapped on the electroscope. Now place the electroscope in a Faraday cage that is at a potential of 100V, ie, one side of a 100V battery is connected to Earth, and the other to the cage. The electroscope deflection will remain unchanged, because the electroscope fundamentally responds to the excess charge on it's leaves, and this has not changed.

I think this is because inside the Faraday cage there is a uniform electric field, and therefore no external field gradients to influence the electroscope.

However, believe that the electroscope inside the Faraday cage forms one plate of a capacitor, the Faraday cage the other. If we raise or lower the potential of the Faraday cage from outside then the potential of the electroscope will change by the same amount, since the potential difference between plates of a capacitor fixed in space remains unchanged unless there is a displacement of charge between them. In order to achieve the desired goal of bringing the electroscope and the Faraday cage to the same relative potential we need to transfer charge between them until their potentials are equalized. Once we do this the electroscope leaves will no longer deflect, even if the system of Faraday cage and electroscope has a huge excess charge and has a high potential difference relative to the surroundings.

More correctly, inside the Faraday cage is zero electric field, rather than a uniform electric field, but I think you were just being careless here in your wording, rather than in error as such. I think we agree that if the Faraday cage is highly charged (large excess charge), the electroscope sitting inside will not have an excess charge (or deflection) when it's terminal is connected to the cage, because all of the excess charge resides on the outside of the cage/sphere, again illustrating my point that the electroscope responds to excess charge on it's leaves.

A change in potential is produced by a spacial displacement of charge. The two are inextricably linked, are they not?

The two are linked, yes. However, if I connect a capacitor across my favourite power supply, and then crank up the supply voltage, then most people would say that the movement of charge was as a result of changing the potential, not the other way around. In other cases, such as a Van der Graaf generator I agree that a change in potential can be produced by physically displacing charge.

Apart from those minor points, I agree with all your entire posting.

[John Heath]

Hmm at a basis, the original question could likely be solved through indirect measures, a 10KV charged cage would cause ionizing in the surrounding air, so by measuring the difference in gas composition he could come up with a rough basis for the voltage to the outside reference frame, with nothing but the air moving through the holes in the cage,

Next up, again probably not that cheap of a way to accomplish it, measure the energy input and wavelength spectra out of a laser, with a higher voltage reference frame the atoms already have higher energy levels, so you should see a tiny amount of higher energy wavelengths compared to the outside measurement, (This one is me guessing from previous knowledge)

Energy input to spectra out of a laser or frequency = energy / h . Cool idea and thanks for suggesting it. I tried it fell fell right on my face. By trying I mean googled the pieces out of lasers. Turns out laser are not like atomic clocks despite the fact that both are atomic events. What sets the high Q bandwidth of a laser is the resonate optical cavity after the photon source. This being the case the optical resonate cavity is subject to imperfection in the reflecting mirrors and expansion temperature coefficient of the cavity itself. At hundreds of THz just looking at it will cause too much of a change in the 1 KHz range. In short too unstable to measure a relativistic change in time for a 10 K volt change in a voltage frame of reference but thanks for bringing it up.

[John Heath]

Great questions coming up from new blood in the thread. Great stuff. Have the monopole cube pictures.

The magnets are round however if made square the PEA rendering should be in the ballpark considering they are only 1/2 the size of the cube walls.

Spec:

A] Magnets are about 3 Gauss at 2 inches , 50 mm. The 6 magnets are within in 15 percent of each other. North faces in with south on the outside.

B] Your outside single loop current magnet model would be in the range of a few thousand amps with diameter of 20 m as the magnet diameter is 20 m.

When cube door is closed the outside Gauss at 2 inches is 2 Gauss however when door closed the Gauss reading at 2 inches is 1 Gauss +- 20 percent.  Not a problem for you however I will have to eat a little crow and take a second look at your thin wire around the edge as the better magnet model. I suspect a open door and closed door modeling of the cube will verify this. If this is the case then a perfect square magnets the same size as the cube walls will have no magnetic field on the outside. This would mean a monopole can not have a magnetic field. Put another way you can not have a north pole without a south pole. 
A problem popped up. Wghen the c

[Zeranin]

Great questions coming up from new blood in the thread. Great stuff. Have the monopole cube pictures.

The magnets are round however if made square the PEA rendering should be in the ballpark considering they are only 1/2 the size of the cube walls.

Spec:

A] Magnets are about 3 Gauss at 2 inches , 50 mm. The 6 magnets are within in 15 percent of each other. North faces in with south on the outside.

B] Your outside single loop current magnet model would be in the range of a few thousand amps with diameter of 20 m as the magnet diameter is 20 m.

When cube door is closed the outside Gauss at 2 inches is 2 Gauss however when door closed the Gauss reading at 2 inches is 1 Gauss +- 20 percent.  Not a problem for you however I will have to eat a little crow and take a second look at your thin wire around the edge as the better magnet model. I suspect a open door and closed door modeling of the cube will verify this. If this is the case then a perfect square magnets the same size as the cube walls will have no magnetic field on the outside. This would mean a monopole can not have a magnetic field. Put another way you can not have a north pole without a south pole. 
A problem popped up. Wghen the c

Hey, I love the way you are using a clip-on ammeter to measure the equivalent circulating current of the permanent magnet! Did it drive your ammeter off scale? If you put one of your magnets directly in the jaw-gap of your ammeter with no air gap, I’ll bet it will drive it hard off scale, and possibly damage the ammeter as well!

I didn’t know it was possible to build a ‘monopole’ so roughly, but fair enuff just for playing around I suppose. If you really want to approximate a ‘proper’ monopole cube, then you will need to get the magnets much closer together. The magnets are 20mm diameter, right? OK, then make yourself a 20mm cube out of anything convenient, wood would be fine, and then glue or screw one magnet to each face. That would be much better than what you have now, and you would find that the flux density(Gauss) outside of the cube would be quite small, as the closer you get to building the monopole perfectly, the closer the flux produced will approach zero. The conclusion is that you cannot build a monopole, but what you do end up building (if you build it properly) is quite cute in it’s own right, a structure of magnets that produces zero field everywhere, quite novel.

How thick are your magnets, and what is the size of the hole through the centre? I’m happy to FEA model it.

When cube door is closed the outside Gauss at 2 inches is 2 Gauss however when door closed the Gauss reading at 2 inches is 1 Gauss +- 20 percent.

I don’t understand. You are saying that when the door is closed (all 6 faces present) you measure 2 Gauss at 2 inches from the cube, but when the door is closed, you measure 1 Gauss +-20% at 2 inches. Um What!!!

A problem popped up. Wghen the c

Looks like your PC exploded. I did warn you about such things happening when you try to break the Laws of Physics!

[Zeranin]

The attached images are from FEA modelling John's rather 'rough' cube of magnets that he constructed. I didn't have the exact dimensions, but that doesn't really matter. The plots show clearly the kind of magnetic behaviour that such an assembly of 6 permanent magnets will produce.

I have modelled a single side, and the flux density 42mm above the face is approximately 9 Gauss. When the entire cube of magnets is assembled, the flux density 42mm above this same face is only 3.4 Gauss, showing as expected that once you start to assemble a cube of magnets in this way, the fields start to cancel, until in the limiting case of perfectly constructing the cube of magnets, you get zero field everywhere, perfect cancellation.

In the image monopole_rough_vectors, you can see clearly the paths and directions that the magnetic flux is taking. Starting out from any of the outward-facing poles, the flux always eventually curves around and returns to the opposite pole on the inside of the cube, but it is only able to do this because the magnets are smaller than the faces of the cube, providing plenty of space for the said flux to find it's way back to the opposite pole on the inside.

The image monopole_rough_modB is a coloured contour plot, showing the field strength at every point. As you would expect, the flux density is highest near the magnets, rapidly falling away as you get further from the cube.

With the benefit of FEA generated plots such as these, it is actually quite easy to understand and visualize the flow of magnetic flux for any magnetic structure that you wish to dream up. For me, FEA is both powerful and beautiful.

[John Heath]

Ha , yes mother nature does not care for our monopole so she is taking it out on my computer.

The magnet has a 3 m hole and 20 m on the outside with thickness of 3 m .

The amp meter is saturating as you suspected. Need to calibrate at 1 amp current then move away  from the sensor until it reads 10 m amp . By marking that distance as times X100 much like a X10 scope probe. With this in place I should be able to measure the magnetic current.

Electrons moving in opposite directions where the two edges of the monopole cube touch should run into each other , scattering , now and then. If this happen the electrons will have to radiate a photon from a change in velocity or direction. That would be free energy. Can not have free energy so something must be stopping it or if it does happen the magnet is demagnitized to even the score energy wise. I remember you hinting at this saying the edges touching could damage the magnets. If it does happen then it should spit out a photon now and then. I wonder if it could be measured?

Speaking of energy conservation I have an unrelated question. You design then perfect ion accelerator for NASA making sure the ion stream never hits the electrostatic accelerator electrode. If the accelerator electrodes is never hit by an ion then we do not need a power supply to keep it at 10 K volts . All we need is a condenser charged to 10 K volt and our ion accelerator engine will run for years until we run out of ions. Is this not a violation of energy laws??     

[John Heath]

Just received your PEA rendering . Will stew on it . Off to work.

[IanB]

The magnet has a 3 m hole and 20 m on the outside with thickness of 3 m .

That's a bloody big magnet!