Mathematics on your fingers: the Mendocino engine and Earnshaw's theorem


DIY Mendocin motor: secrets of the American Kulibin

Date of publication: September 6, 2019
In 1994, all residents of Mendocino County on the California coast vied with each other to discuss the invention of local craftsman Larry Spring. A small motor suspended in the air miraculously rotated by itself and did not require a power connection. Standing on the windowsill of a small shop, the mysterious engine invariably became the subject of close attention of children and adults. Attempts to unravel the master’s secret were unsuccessful until Larry himself admitted to the most persistent visitors what secret he based his invention on.

Everything turned out to be very simple. The ability to adjust the laws of physics to each other and a little ingenuity allowed Spring to design a small engine, the main elements of which are a rotor and a stator - everything, like “real” motors. However, here lies the main secret. A stand with a permanent magnet and a magnetic support is used as a stator. And the role of the rotor is performed by a dielectric frame with a set of solar panels mounted on top of the rotating coils.

The principle of operation of the engine is based on the rotation of the rotor under the influence of magnetic fields arising from the passage of electric current through the coils of the device. The necessary charge is supplied to the motor thanks to the operation of solar panels. Receiving power in turn, the coils are “pushed” away from the emerging magnetic field due to the Ampere force. But, since they are fixed on magnetic supports, the rotation process starts. This is exactly how any low-power magnetic levitation motor operates, such as the Mendocino engine.

Electromagnetic levitation with tracking system

By using a circuit based on an electromagnet and a photo relay, you can make small metal objects levitate. The object will float in the air at some distance from the electromagnet fixedly mounted on the stand. The electromagnet receives power until the photocell mounted in the rack is obscured by a floating object, as long as it receives enough light from a fixed reference source, which means that the object needs to be attracted.

Levitating globe

When the object is sufficiently elevated, the electromagnet turns off, since at this moment the shadow from the object moved in space falls on the photocell, blocking the light of the source. The object begins to fall, but does not have time to fall, as the electromagnet turns on again. So, by adjusting the sensitivity of the photo relay, you can achieve an effect in which the object seems to hang in one place in the air.

In fact, the object continuously falls and then rises again slightly electromagnetically. The result is the illusion of levitation. This principle is the basis for the work of “levitating globes” - rather unusual souvenirs, where a magnetic plate is attached to the globe, with which an electromagnet hidden in the stand interacts.

We assemble a Mendocino motor with our own hands: a detailed examination of the design

The secret of the American inventor opened the opportunity for thousands of home craftsmen to construct a similar device at home in order to impress their relatives and surprise lovers of the mysteries of nature. However, before getting to work, it is worth considering the device in detail. Every centimeter counts here - it is important that all elements are in their place and interact strictly within the framework of physical laws.

The rotor of the Mendocino engine has a square cross-section and is located horizontally in the device. This solution allows solar panels to be placed on its surface. Permanent ring magnets are attached to the ends of the rotor shaft. Thanks to the magnetic field they create, the rotor starts to move, which even the force of mutual friction of the metal elements is unable to stop.

To keep the rotor suspended, magnetic shaft rings are positioned directly above magnetic stands. Another magnet under the rotor is needed to create a magnetic field of the stator, which gives a “start” to the rotation of the rotor.

When sunlight hits one of the solar panels, an electric current is generated. It is sent to the rotor winding, which is located near the magnet directly under the axis. A magnetic field is created at the corresponding pole of the rotor, and the latter begins to rotate, pushing away from the magnetic field of the stator. Sunlight alternately hits each of the solar panels on the four sides of the axis, triggering a similar process in relation to each of the coil windings. This ensures constant rotation of the rotor in its “suspended” state. The device will work properly in the presence of an intense or medium luminous flux.

And the last secret that you need to know before you start manufacturing and assembling a Mendocin motor according to the diagram. Permanent magnets in the rotor suspension are a mandatory design element, thanks to which it is possible to overcome the resulting friction force. Otherwise, the engine power will not be enough, and rotation will stop after the first revolutions.

Mathematics on your fingers: the Mendocino engine and Earnshaw's theorem

The other day I saw an extremely curious thing on the Internet: a Mendocino engine. The rotor is on extremely low friction bearings: the original one had a glass cylinder suspended on two needles, modern ones have a magnetic suspension of the axle. The motor is brushless; solar panels are suspended on the rotor, which provide voltage to coils wound on the rotor. The rotor rotates in the fixed magnetic field of the stator, the solar battery moves away from the directional light, and another one takes its place. An extremely elegant solution that anyone can make at home. This video describes in great detail (in Russian) the principle of operation:

But the following thing seemed even more interesting to me than the engine itself. In the description of this video, Dmitry Korzhevsky wrote the following: “It is IMPOSSIBLE to replace the side support with a magnet!!! Don’t ask this question again!”

Excuse: I am by no means a physicist, I could be very wrong, corrections are welcome.

Oh, that's interesting. Let's take another look at how the magnetic suspension of the rotor works. If we place two magnets, then the potential isoline looks like this depending on the distance between the two magnets:

That is, we place two fixed magnets on the stator. The magnet on the rotor axis will not want to move sideways, because the potential isoline has a certain local minimum. He will want to jump out along the rotor axis. We make two such systems, and we get a rotor axis that is fixed by a magnetic field in the radial direction, but is unstable in the longitudinal direction. We rest the axis against the glass wall and voila, we have a low friction bearing.

But a glass wall is somehow... inelegant, or what? It is quite logical to want to get a rotor completely floating in the air, without any crutches. And obviously Dmitry was bullied with this question, so much so that he was forced to write the impossibility of such a thing right in the description of the video. And Dmitry Korzhevsky is not the only one.

Let's look here, I quote:

What would happen if the base magnets were spaced and oriented like in this drawing? Would it give it stability in the axial plane, and do away with the mirror requirement?

Or here, I quote:

On a Mendocino Motor why does one side float free while the other has a tip to a wall? I know the question might sound trivial but I have worked up the idea why not use the same magnets used to levitate as a counter force on both sides of the shaft? I attached a very rough jpg of what I mean. the green magnets at the end of the shafts is what im referring to. is there some theory or law preventing this?

That is, people all over the world want to get rid of mechanical axle support. I studied poorly at school and the impossibility of creating a completely magnetic suspension without crutches was also never obvious to me. On occasion, over a cup of tea, I asked my boss, a world-famous scientist (not a physicist, an applied mathematician), this question: “Why is it actually impossible?” And you know, it wasn’t obvious to him either!

On the above forums, no one really explained why this is impossible. At best, they quoted some kind of Earnshaw's theorem, which was not very digestible. So, it says the following: “Any equilibrium configuration of point charges is unstable if nothing acts on them except the Coulomb forces of attraction and repulsion.” Do you understand? I don't. Suppose I can accept the fact that we are talking about charged particles, and not about magnets. But what next?

When something is unclear to me, I draw a picture. For simplicity, it will be in two-dimensional space. Let's imagine four fixed unit charges at the corners of the square and a free charge at the center of the square. Like that:

Is the free charge not in a state of stable equilibrium? After all, wherever he moves, he approaches one of the fixed charges, increasing the repulsive force! Let's try to draw a map of the potential energy of a free charge. I didn’t do well at school and skipped physics, so we’ll draw our knowledge from Wikipedia. So, if we have only one fixed charge in space, then it creates an electrostatic potential throughout space.

Formula for the electrostatic potential (Coulomb potential) of a point charge in a vacuum:

In all speculative experiments, all my coefficients are equal to either zero or one. Therefore, the charge q is unit, the unclear k is also unit. That is, one fixed charge creates a potential measured by the formula 1/r, where r is the distance to the charge.

The potential energy of a free unit charge in the field of our fixed charge is also equal to 1/r. (Generally speaking, the energy is equal to k*q1*q2/r, but we choose the coefficients so that it is convenient to calculate). For several charges, all potentials are simply added together. Let's draw a map of the potential energy of our free charge, I do this using sage:

var('x,y') def unit_potential(a,b,x,y): return 1/(sqrt((xa)^2 + (yb)^2)) def system_potential(x,y): return unit_potential( 1,1,x,y)+unit_potential(-1,1,x,y)+unit_potential(1,-1,x,y)+unit_potential(-1,-1,x,y) contour_plot(system_potential(x ,y), (x, -2, 2), (y, -2, 2), cmap='hsv', contours=30, region=5-system_potential(x,y), figsize=12, colorbar=True )

Here is a map, I pinched out the points where the potential energy goes to infinity:

A local energy minimum is clearly visible in the center of the square. Wherever the particle moves from the center, the energy will increase, so from small disturbances it will clearly want to return back to the center, this is a point of stable equilibrium. Was Earnshaw lying? No, he didn't lie. The problem is that I didn't draw the picture well. And many are mistaken exactly the same as I am. Stop now, think, where did I go wrong?

In this case, the error is that in two-dimensional space a fixed charge creates a potential measured by the formula -ln r, where r is the distance to the charge, and not 1/r at all. Let's take my word for a while and allow me to change the Coulomb formula in an unclear way, then the correct code will look like this:

var('x,y') def unit_potential(a,b,x,y): return -ln(sqrt((xa)^2 + (yb)^2)) def system_potential(x,y): return unit_potential( 1,1,x,y)+unit_potential(-1,1,x,y)+unit_potential(1,-1,x,y)+unit_potential(-1,-1,x,y) contour_plot(system_potential(x ,y), (x, -2, 2), (y, -2, 2), cmap='hsv', contours=30, region=5-system_potential(x,y), figsize=12, colorbar=True )

Here is a picture of a potential energy map:

Please note that there are no local minima on the map. The center of the square is a saddle point, that is, a point of unstable equilibrium. As soon as the free charge moves even a micron from the center of the square, it will certainly roll off and fly out of the square, accelerating and accelerating.

When I received an obvious contradiction with Earnshaw's theorem, I realized that I had made a mistake somewhere and began to look for an error. It is best to look for an error consistently from the very beginning. I sighed heavily and went to read what Maxwell’s equations are. At school, I didn’t do very poorly, my grades were excellent. But I obviously didn’t gain knowledge in all subjects. For example, I only had nightmares about Maxwell’s equations after school, but at university and beyond I simply didn’t have to deal with them.

But it turned out that everything is extremely simple, especially if we are only interested in electrostatics! There are four Maxwell equations according to the following laws:

1. Gauss's law, it will be useful to us. For now, let’s leave all sorts of divergences aside, “on fingers” this is simply the law of conservation: energy does not come from nowhere and does not go anywhere. 2. Gauss's law for a magnetic field - the same eggs, side view. And I’m not interested in the magnetic field yet, because... the conversation comes from charged particles, let's skip it. 3. Faraday’s law: if we move magnets, they generate an electric field, this is interesting, we’ll look in more detail later. 4. Ampere's law: if we move an electric field, we generate a magnetic one. Fuck it, it's not interesting.

So, these four laws connect two vector fields E and B, the electric field and the magnetic field. These vector fields are functions that have four arguments (x,y,z,t), and each four arguments are associated with one three-dimensional vector. The magnetic is not very interesting to us in this case; let’s consider the field E(x,y,z,t). Moreover, we do not forget that we are interested in electrostatics, so E is constant in time. It is very convenient to consider this vector field as a certain river, where we tell each and every point of the river where the water flows and at what speed.

Faraday's law says that in the case of a time-constant field E (we are talking about electrostatics) there are no vortices.

How is electrostatic potential related to electric field? It’s very simple: if the field E is irrotational (our case), then it is possible to create such a landscape u that by covering it with a meter layer of water (at all heights!) and “releasing” this water, the speed and direction of the water flow will give rise to the field E. If in clever words , then we can find a scalar function u such that its gradient is equal to the field E.

Gauss's law says the following: take a small region of space. If we did not specifically place a charge in it, then the amount of “water” that flows into this area is equal to the amount that flows out. If you want to show off, you can say that the divergence of the E field is zero.

Let me remind you that the field E is the derivative of the scalar function u. If its divergence is zero, then this means that the Laplacian of the function u is equal to zero. Laplacian is a fancy word for the "curvature" of a function. In the case of a function of one variable, the Laplacian is simply the second derivative. The second derivative is zero only for a constant or linear function (logically, the curvature is zero). In the case of a function of two variables, the Laplacian is the sum of two partial derivatives. If it is zero, then curvature in one direction must be canceled out by curvature in the other direction. That is, chips are allowed:

But a function with a zero Laplacian has no local minima (or maxima). That is, chips are allowed, but hills are not:

Imagine that we dip a wire ring (well bent) into soapy water. Then the soap film forms a surface with zero Laplacian:

This will be the so-called minimal surface. The soap film tries to reduce its area. It is logical that if there was a certain local maximum on it, then by smoothing it out, we would get a film of a smaller area. That's why they don't exist. So, the electrostatic potential is a kind of minimal surface; it does not have local maxima (in places where we did not specifically place a charge).

The function 1/r has a zero Laplacian in three-dimensional space, but not in two-dimensional space! If we want to draw two-dimensional examples, then we need to solve the Dirichlet problem, I already talked about it in one of my previous articles. For 2D this would be the -ln r function.

Update: good comment by chersanya, clarifying the essence of magic.

So, returning to our example with one free charged particle. The electrostatic field potential has no local minima, and, as a consequence, the potential energy of one particle does not have local minima. Therefore, one particle cannot be in a state of stable equilibrium in a constant field. Congratulations, we have just proved Earnshaw's theorem. But what about more complex systems? How can this theorem be applied to them?

Here is another example proposed by my boss, which was supposed to refute Earnshaw's theorem. Let's fix two charges and create a moving body consisting of a weightless inextensible stick with charges at both ends:

Intuitively, if we slightly move the stick to the left (right), then one of the ends will approach the fixed charges, and they will push it away, returning the stick to its original position. Where's the catch? Let's draw the electrostatic potential of two fixed charges:

var('x,y') def unit_potential(a,b,x,y): return -ln(sqrt((xa)^2 + (yb)^2)) def system_potential(x,y): return unit_potential( 0,1,x,y)+unit_potential(0,-1,x,y) contour_plot(system_potential(x,y), (x, -2, 2), (y, -2, 2), cmap=' hsv', contours=30, figsize=12, colorbar=True)

How to draw the potential energy of our stick charged at the ends? The stick has three degrees of freedom (two for translation and one for rotation), so the graph will be four-dimensional. Let's try to ignore the rotation and only allow the stick to move parallel. Let's fix a point on a stick, for example, its center, and draw a map of the potential energy of the stick for the position of its center. Then the total potential energy of the stick is the sum of the potential energies of the charges at the end:

var('x,y') def unit_potential(a,b,x,y): return -ln(sqrt((xa)^2 + (yb)^2)) def system_potential(x,y): return unit_potential( 0,1,x,y)+unit_potential(0,-1,x,y) def energy(x,y): return system_potential(x+2,y)+system_potential(x-2,y) contour_plot(energy( x,y), (x, -3, 3), (y, -2, 2), cmap='hsv', contours=30, figsize=12, colorbar=True)

So the energy of the stick has four peaks (each of the two ends of the stick hits each of the two charges). As expected, the stick will not want to move horizontally. She will run away vertically!

This is logical, because where did we get energy from? We added up the potential energies of each charge. We know that the potential energy of each charge is a function with zero Laplacian. Their sum will also have a zero Laplacian. That is, the potential energy of any

(not just our stick!) a charged body cannot have minima in a constant electric field!

The mental image of magnetic and electric fields in people who have not worked closely with physics is deceptive. The brain deceives us by drawing pictures of energy minimums. Unfortunately, this is not the case, and it seems difficult to actually create a Mendocino engine without a support.

What loopholes could there be? Earnshaw's theorem (if we make an effort and generally apply it to magnets) is applicable only to systems of stationary permanent magnets

.

1. We can try to create a dynamic magnetic field 2. Diamagnetism and all sorts of superconductors are also not included in the scope of Earnshaw’s theorem 3. Moving bodies in general and rotating bodies in particular are also not considered, the most famous example is the Levitron

So, all is not lost. Yes, using any of these things will completely kill the conciseness of the Mendocino engine, but the magic of things freely floating in the air will cover everything!

It was Earnshaw's theorem that showed the impossibility of the existence of solid matter, thus rejecting the existing model of the structure of the atom. As a result, a planetary model of the atom was built.

Mendocino motor. Types and device. Operation and Application

Perpetual motion technology has been interesting at all times. That is why many scientists, including ordinary people, are trying to solve the issue of its creation. It is believed that the creation of a perpetual motion machine will produce a world revolution and make its creator a famous and rich man. But it is necessary to take into account that science currently rejects the possibility of its development, because it would have to violate physical laws. Engines of this kind constantly appear on the network, but so far this problem has not been solved.

One of these engines is the Mendocino engine. This invention is often called a solar perpetual motion machine. It has no wires, hoses or other cables through which power is supplied. And if you don’t know how it works, then this engine can be called fantastic. It can rotate just like that and at the same time be in a levitating state. But it's not that simple.

Kinds

The Mendocin engine appeared in 1994 thanks to the efforts of the American Larry Spring. The engine got its name from the area of ​​Mendocino, which is located on the California coast. For a long period of time, this unit was located in the Lari store. After some time, it became very popular among local residents. This was explained simply - the rotor was spinning without stopping, while being practically in limbo.

In the unique design of the Spring engine, the axle rested on the glass thanks to pointed heels. However, modern designs have changed somewhat. Today the axis is literally levitating. On one side, the axis rests only on the air space. Only on the other side does the rotor axis rest against the wall to ensure a balanced position. This design allows the device to operate indefinitely, but subject to one condition - the presence of solar energy.

Device

The Mendocino motor, like most electric motors, includes a rotor and stator in its structure. However, at its core, the unit is not a standard engine. In this case, the stator is a stand that has a permanent magnet, as well as a magnetic support. The rotor is made in the form of a dielectric frame with a set of solar cells.

The batteries are activated when photons from the sun fall on them. Thanks to this, the batteries begin to create electric current. This current is directed to coils that are wound around the rotor. When electric current passes through the coils that surround the rotor, a magnetic field appears. Due to the interaction of this field with the stator field, that is, arising from a permanent magnet, the rotor begins to rotate.

The small device requires only a few watts of power, allowing the rotor to spin quite quickly. However, for industrial units a few watts of power will not be enough; solar cells will be required an order of magnitude larger.

How to make a magnet with your own hands

The operation of all levitators is based on a magnetic base. If you wish, you can make a magnet at home. For example, to turn a regular screwdriver into a magnetic one. You will need: 5 or 12 volt battery, copper wire, electrical tape, screwdriver.

  1. We take a screwdriver and wind 280 to 350 turns on it very tightly to each other.
  2. We wrap electrical tape over the wire, also carefully.
  3. We connect one end of the wire to the plus of the battery, the other to the minus and evaluate the magnetic effect.

Demonstration of the magnetic levitation heat engine mechanism

This is the simplest heat engine, the operation of which is based on magnetic levitation. You will need several ring neodymium magnets and thin sheet graphite disks. All this is an example of the simplest diamagnetic levitation without energy consumption. On our website you will find videos by Igor Beletsky with the development of this topic.

If you need a neodymium magnet, you can purchase it in a Chinese online store.

To create a heat engine, you just need to supply energy to it. It is interesting and spectacular to do this in a non-contact way. For example, with the help of the Sun. But you will need one more detail in the form of a curtain. Uneven heating creates a potential difference necessary for the operation of any motor. And if you use a concentrator, the rotation speed will increase noticeably. At the heating point, the diamagnetic properties of the magnet decrease, the disk seems to fall in this direction and begins to twist. If you influence a graphite disk with a laser pointer, it can be made to move in any direction.

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Where to buy a levitating motor

The channel “Dmitry Korzhevsky” brought to your attention an overview of an interesting levitating engine, which earlier in another video he called eternal, although with the caveat in the description that it is eternal not in the sense of the absence of energy sources, but simply the resource of its uninterrupted operation is at least 100 years. This video is below. Using it, you can assemble it with your own hands. If you have no desire to waste time and effort on it, then you can buy it in this Chinese store. Available with free shipping, various design modifications.

“So why is he spinning around so merrily? Why does it rotate? - you ask. And the whole secret is that the edges of the rotor are made of photocells, with 2 coils between them. Each photocell produces half a volt in direct sunlight. True, in the previous video, the sun is not shining. The rotor, however, rotates quite quickly. How so?

It's all about the magnetic suspension. These are levitating bearings with complete absence of friction. True, there is a fulcrum in this case, without it there must be at least one. The force acting on it is less than one gram, and the frictional force is negligible, so it does not require bright daylight sunlight to rotate. It is enough that it is simply not dark. In the evening, ceiling lighting is enough for him if it is dark outside. In this case, there may be units of millivolts on photocells, but it rotates.

At the moment, a 100 W incandescent lamp is shining on it at a distance of about 1 meter. If the engine is kept on the windowsill, it will rotate even at dusk. With the onset of darkness, naturally, it stops. And at dawn, when the first ray of the rising sun gilds the treetops, the engine wakes up. Because if the sun's rays hit the rotor, it spins so that the wind whistles if you turn your ear.

Friends, the engine is not such a useless thing. There is an impeller. Place it on the axle and take a look. Rotates. Not only that, but it still blows. We take a piece of cotton wool, it moves in space. A piece of foil. She's right there. In, blown away by the wind.

Thus, the engine performs some useful work, which the author did not even count on during the manufacturing process. Toy fan. On a hot sunny day it can even cool your face in bright sunshine.

Details about the design of the levitating engine

The engine is four-stroke. In the first 2 cycles, power is supplied to the first winding and to the second in turn. Then, when turning, a polarity reversal occurs. Again to the first, to the second. The polarity reversal occurs automatically due to the fact that the light falls on one side, and on the other side, respectively, the shadow. In this way, switching occurs without any mutating elements. The only difficulty in its manufacture may arise with balancing. It should be so careful that neither side outweighs at rest. More about the levitating motor from 5 minutes

An addition from the author of the videos for those who decided to assemble this levitating, spectacular engine.

Latest video

Assembly details

www.youtube.com/watch?v=5mERXljsdHY

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Do-it-yourself engine with a floating rotor

The Mendocino motor is the simplest brushless motor. Such an engine needs very little voltage, it runs on light, and even a simple light bulb is enough for it to move. There is no practical use for such a device, but contemplating the operation of a levitating engine relaxes you and makes you take your mind off everything.

Creating a Mendocin motor requires certain knowledge and skills, you will also need solar panels and neodymium magnets.

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