Magnets Magnetic Physics of

tahanson43206 · 2024-03-05 13:11:52

SpaceNut, we had several topics with the word "magnet" in the title, but none were devoted to the subject itself.

The need for a topic dedicated to magnetism and magnets arises from an interesting proposal by Void, to attract suitable asteroids using magnetic force.

The Earth's magnetic field reaches far into space, and it influences the behavior if ions there. The Earth's field is strong enough to cause compass needs to move, if the pivot is free of resistance.

Thus, the idea of using a magnetic field at a distance of 100 meters (an arbitrarily chosen asteroid diameter) seems reasonable.

What is NOT clear from a conceptual vision, is what the actual strength of a magnetic field might be at 100 meters, since the strength of a magnetic field falls off at the cube root of the distance, compared to gravity, which falls off as the square root of the distance.

This topic is available for members to build up a solid collection of references and "Real Universe" observations.

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tahanson43206 · 2024-03-05 13:12:20

This post is reserved for an index to posts NewMars members may contribute over time.

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tahanson43206 · 2024-03-05 13:21:13

To open this new topic, I asked Google to look for citations about magnets, magnetism and related information....

In looking through the list below, I see that some references are confused about the rate at which magnetic force falls off.

It should be possible for anyone (NewMars member or otherwise) to confirm for themselves that magnetic force falls off as the cube root of distance, rather than the square root, as is the case with gravity.

Generative AI is experimental. Learn more
Magnetic field strength decreases as the distance from the source increases. This relationship is known as the inverse cube law.
Experiment of The Month | Millersville University
For example, if the distance between a magnet and an object doubles, the magnetic field strength drops to 1/4 of what it was.
Magnetic fields are produced by magnetic dipoles, which can be permanent magnets or current-carrying loops of wire.
Here are some activities to learn about magnetic fields and distance:
Magnetic Field Strength vs. Distance
This science project from Education.com explores the relationship between magnetic field strength and distance.
Magnetic Fields and Distance
This activity from TeachEngineering includes learning objectives such as explaining how permanent magnets work and identifying the SI units of a magnetic field.
How do magnetic field lines show strength?
Formula for magnetic field strength
Ask a follow up...
Show more
Magnetic field strength decreases by distance cubed, and light intensity decreases by distance squared. Therefore, magnetic field strength decreases more rapidly with distance than does light intensity. Q5. Magnet A is twice as strong as magnet B.
Change in Magnetic Field Strength with Distance - TI Education
TI.com
https://education.ti.com › media
PDF
People also ask
How is magnetic field strength related to distance?
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Magnetic Fields and Distance - Activity
TeachEngineering
https://www.teachengineering.org › Activities
Show students the equations relating proportional magnetic field strength to distance. Explain that data are easier to analyze when in linear form, and then ...
Introduction/motivation · Procedure · Assessment
Magnetic Field Strength vs. Distance | Science project
Education.com
https://www.education.com › ... › Second Grade
How can we describe the relationship of magnetic field strength vs. distance? Kids will find out with this free science fair project idea.
Discussions and forums
Calculate magnetic field strength vs distance - Electronics & Electrical Engineering Forum
Site favicon
forum.allaboutcircuits.com
· Mar 16, 2021
What is the relationship between magnetic field strength and the distance from a pole called? - Quora
Site favicon
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· Feb 6, 2017 · 2 posts
What is the relationship distance between magnetic field lines and their strength? - Quora
Site favicon
www.quora.com
· Jan 8, 2018 · 1 post
Magnetic field strength dependent on distance from source
Physics Forums
https://www.physicsforums.com › ... › Electromagnetism
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2 answers
·
4 votes:
It's called the inverse square law. Field strength is the inverse of the square ...
Determine How Magnetic Field Varies With Distance
Instructables
https://www.instructables.com › Workshop › Science
This Instructabler describes how to make a scientific investigation to determine how magnetic field varies with distance. Two methods are presented ...
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Questions & answers
Quora
Question
What is the relationship between magnetic field strength and the distance from a pole called?
Answer · 4 votes
It's called the inverse square law. Field strength is the inverse of the square of the distance. Edit: This is likely the expected answer for 8th grade general science. But technically, a single magnetic pole doesn't exist in isolation. The magnetic field of a particle is unlike an electric field. The latter is a point-symmetric source; the former is more like a vector indicating the north direction while its negation indicates the south direction. North and south are not two distinct or separable points. The inverse square law is actually incorrect for magnetism. (Also the primary colors of paint are not red, blue and yellow, as anyone with a color printer should know.)
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Study.com
Question
What happens to magnetic field strength as distance increases?
Answer · 0 votes
As the distance between an object and a magnet increases, the strength of the magnetic field decreases. Think of this by looking at a simple flash light that is pointed on an object. Although you may be able to see the object if you are close by, the object can become difficult to see if the distance between the object and light is increased. The magnetic field's strength is said to follow a similar law close to Coulomb's law, where the distance or radius is inverse and squared. If the distance is halved, the force increased by four. However, if the distance is doubled, the force will decrease to a fourth of the value. That is why simple magnets need to be closer to one another in order to attract better.
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Brainly.co
Question
HELPP !! 50 POINTS How does distance affect the strength of the magnetic force?
Answer · 1 vote
Answer: Explanation: Depending on how far away the magnets are, will determine the strength of the magnetic force.
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Study.com
Question
What is the range of a magnetic field's strength?
Answer · 0 votes
Technically speaking there is no limit to the range of a magnetic field. Practically speaking however the field strength drops by the inverse square of the distance from the source. This means that if the distance between a magnet and the object in question doubles the magnetic field strength drops to 1/4 of what it was. It is a subjective decision by an observer or limited by the lower detection limit of measurement equipment as to when the field strength is negligible.
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Brainly.co
Question
What does the distance between magnetic field lines indicate
Answer · 5 votes
Answer: A magnetic field surrounds any electric charge in motion. ... The distance between the lines indicates relative strength of the magnetic field. The closer the lines are, the stronger the magnetic field is. Iron filings and a compass may be used to trace the shape, strength, and direction of magnetic field lines. Explanation:
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Stack Exchange
Question
Can a magnetic field be focused at a distance?
Answer · 2 votes
Probably not. Here's a simple argument. First, Gauss' law implies the magnetic field lines are closed. Thus the concentration of field lines would look something like the image below. However, then we could pick a closed rectangle-like loop as in the image, but with curved edges (in the image, the edges are straight). The edges would be chosen so that the first, HI, is always along a field line in the dense region, IJ is always perpendicular to magnetic field, JK, is always in opposite direction to the field in the non-dense region and KH is again perpendicular to magnetic field. However applying Ampere's law implies that the average magnetic field along HI should be the same or weaker than that of JK (the field is curved "outside", as implied by localization, which makes JK in practice shorter than HI), which is a contradiction. Note that for electromagnetic waves my derivation is not valid, as Ampere's circuital law actually depends on the time-derivative of electric flux through H…
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Stack Exchange
Question
Why is the magnetic field proportional to distance?
Answer · 5 votes
Because there are no magnetic charges the simplest system that can generate a magnetic field is a dipole. The field from a dipole (magnetic or electric) falls off as 1/r3 rather than 1/r2 so the field strength falls off more rapidly with distance than the electric field from a point charge. Increasing the current or the number of turns will certainly increase the strength of the magnetic field from a solenoid, but you're still stuck with the 1/r3 dependance on distance. You might be interested in this article that describes one way of channeling a magnetic field over (relatively) large distances.
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Quizlet
Question
Does the magnetic field strength decrease more rapidly with distance closer to the wire or farther from the wire? Explain.
Answer · 0 votes
Much like gravity, the magnetic force obeys the inverse square law with distance. Which means, it responds exponentially to the distance in between. $$ F \propto \dfrac{1}{r^{2}}$$ Where $F$ is the magnetic force and $r$ is the distance in between. The graph also indicates how the magnetic field strength plummets as the distance from the wire increases. For example, at distance $0\text{ cm}$, the magnetic strength is $20\, \mu\text{T}$, and at a distance of $10\text{ cm}$, the magnetic field strength is $2\, \mu\text{T}$. Hence, the magnetic field strength decreases more rapidly with distance farther from the wire. The magnetic field strength decreases with distance farther from the wire.
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Generation Genius
Question
How can you increase the strength of the magnetic force on an object like a paper clip?
Answer · 0 votes
Magnetic forces get stronger as the magnet moves closer to the object it is attracting or
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Answers
Question
What happens to the strength of the magnetic field as you move away from a magnet?
Answer · 0 votes
When you increase the distance the force is decreased, and when you decrease the distance you increase the force.
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Feedback
Magnetic field strength and distance: collect your own data
YouTube · Dr Richard Walding
4K+ views · 2 years ago
30:21
... students to make it in the lab. The experiment is filmed in a way to allow viewers to collect their own data. It shows the reading on an ...
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Magnetic Fields Varying as an Inverse Cube
Millersville University
https://www.millersville.edu › physics › experiments
For electric dipoles, the field strength decreases more rapidly with distance; as R -3 . Magnetic monopoles have never been observed. Instead, the basic ...
Action-at-a-distance and magnetic fields | IOPSpark
IOPSpark
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The space around a magnet where the magnetic force acts is called the magnetic field (just as the space around a mass where the gravitational force acts is ...
How Does Magnetic Field Vary With Distance? : 5 Steps
Instructables
https://www.instructables.com › Workshop › Science
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Magnetic forces (video) - Khan Academy
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The strength of a magnetic force depends on the strength of the magnets and the distance between them. Stronger magnets can even levitate trains! Understanding ...
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The strength of a magnetic field, B , some distance d away from a straight wire carrying a current, I , can be found using the equation B = ? I 2 ? d , ...
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In this video I give some ideas about how to get this experiment to work. It is very tricky. The magnetic field about a wire diminishes ...

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How the strength and direction of the magnetic field are ...
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Magnet strength and action at a distance | IOPSpark
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Strength of a Magnet at Various Distances - Physics
Art of Smart
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CK-12
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https://en.wikipedia.org › wiki › Magnetic_field
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1 answer
·
0 votes:
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tahanson43206 · 2024-03-05 13:23:35

This is an invitation for anyone (NewMars member or otherwise) to help to answer the question:

Can a magnet pull a suitable asteroid towards the magnet with sufficient strength to reach the outer edge of the asteroid, if the diameter is given as 100 meters?

If so, how strong must the magnet be to achieve the desired effect?

What practical considerations apply in this situation?

A corollary question is: If such a magnet is moved into the center of a rubble pile asteroid, can it pull the entire asteroid toward itself, and thus provide a way to pull on the entire asteroid, if the asteroid presents a risk to Earth or Mars?

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SpaceNut · 2024-03-05 20:08:39

The article you are starting with fore fusion is a super conductor and these fall into ceramic or glass which is insulative and the power going through the conductor makes a field. The fields could be DC is similar to a permanent magnet where an AC is fluctuating and is quite different too in that flipping field has the frequency of the cycle.

tahanson43206 · 2024-03-05 20:59:42

For SpaceNut re #5

Thanks for noting the fusion magnet is using superconducting material to increase the magnetic field strength.

I'm not sure you've had time to read Void's idea, so i'll try to quickly summarize what I understand about it.

From what I can gather, Void is thinking about a powerful magnet capable of pulling an entire asteroid. To my knowledge (admittedly limited) no one has previously considered such an idea. Once again, Void is far ahead of the rest of us.

Can you answer the question posted in #4? For simplicity, assume you are working with a standard metal washer that is made of magnetic material (ie, soft iron). The washer has a mass of one gram, and it is 40 meters away from the powerful magnet. How strong would the magnet need to be to exert a force of one gram on the magnet?

It is entirely possible you have the background to handle this question. The answer would be given in Teslas, so you can round to the nearest Tesla.

Bing/CoPilot found an asteroid that would meet Void's criteria... it is 2007 UN12. That asteroid is 40 meters in diameter, so it is a good candidate for Void's idea.

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tahanson43206 · 2024-03-05 21:02:18

The following is from a conversation with CoPilot/Bing on Windows 10 using Edge ...

Copilot
You’re welcome. I understand that you are looking for a free flying asteroid that has magnetic materials and is less than 100 meters in diameter. Based on my web search, I found one possible candidate for your experiment. It is an asteroid that belongs to the M-type, which means it has a metallic composition. It is also less than 100 meters in diameter, as you requested. ?
The name of this asteroid is 2007 UN12. It was discovered in 2007 by the Catalina Sky Survey. It has an orbital period of about 1.3 years and a minimum orbit intersection distance with Earth of about 0.02 AU. It has an estimated diameter of about 40 meters and a high albedo of about 0.4. It is also classified as a potentially hazardous asteroid, which means it has a chance of colliding with Earth in the future. You can read more about it here: 2007 UN12 - Wikipedia.
I hope this helps you with your experiment. Please note that I cannot guarantee the accuracy or completeness of this information, as it is based on web sources that may be outdated or unreliable. You may want to do further research on this or other asteroids before proceeding with your experiment. ?

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tahanson43206 · 2024-03-06 10:31:25

The following table was prepared by ChatGPT4 to show how magnetic field strength falls off with distance:

Magnetic Field Strength at Distances up to 40 Meters
We've calculated the magnetic field strength at each meter from the magnet, starting with an initial strength of 1 Tesla at the magnet's pole. Here's how the field strength decreases over distance, following the inverse cube law:
1m - 1.000 Tesla
2m - 0.125 Tesla
3m - 0.037 Tesla
4m - 0.0156 Tesla
5m - 0.008 Tesla
6m - 0.00463 Tesla
7m - 0.00292 Tesla
8m - 0.00195 Tesla
9m - 0.00137 Tesla
10m - 0.001 Tesla
11m - 0.000751 Tesla
12m - 0.000579 Tesla
13m - 0.000455 Tesla
14m - 0.000364 Tesla
15m - 0.000296 Tesla
16m - 0.000244 Tesla
17m - 0.000204 Tesla
18m - 0.000171 Tesla
19m - 0.000146 Tesla
20m - 0.000125 Tesla
21m - 0.000108 Tesla
22m - 0.000094 Tesla
23m - 0.000082 Tesla
24m - 0.000072 Tesla
25m - 0.000064 Tesla
26m - 0.000057 Tesla
27m - 0.000051 Tesla
28m - 0.000046 Tesla
29m - 0.000041 Tesla
30m - 0.000037 Tesla
31m - 0.000034 Tesla
32m - 0.000031 Tesla
33m - 0.000028 Tesla
34m - 0.000025 Tesla
35m - 0.000023 Tesla
36m - 0.000021 Tesla
37m - 0.000020 Tesla
38m - 0.000018 Tesla
39m - 0.000017 Tesla
40m - 0.000016 Tesla
Conclusion at 40 Meters:
At a distance of 40 meters, the magnetic field strength diminishes to 1.5625 x 10^-5 Tesla. Given this extremely weak magnetic field, the force exerted on a 1-gram washer made of highly magnetically susceptible material (like Electrolytic Iron) would be too weak to be measured. This illustrates the challenges in utilizing magnetic fields over large distances for practical applications, especially when attempting to exert forces on small objects.

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tahanson43206 · 2024-03-06 15:08:44

As a follow up to the work done earlier today, to explore Void's idea of using magnetic force to assist with asteroid mining, I asked ChatGPT4 to try to explain what a Tesla of magnetic force might be. Here is a first try at an answer:

Understanding the Tesla: A Summary by ChatGPT
In our exploration of magnetic fields, particularly focusing on what constitutes a Tesla (T) and its practical implications, we've compared several scenarios where magnetic fields of this magnitude play a crucial role. Here's a summary to provide a clearer understanding of the significance of a Tesla:
MRI Machines: These medical imaging devices typically operate with magnetic fields ranging from 1.5 to 3 Teslas. Some specialized systems can reach up to 7 Teslas or more, showcasing the substantial effect Tesla-level fields have at the atomic and molecular scales for diagnostic purposes.
Laboratory Magnets: High-field lab magnets can achieve fields upwards of 10 to 45 Teslas in short pulses, used in material science, physics, and chemistry research to investigate material properties under extreme magnetic conditions.
Particle Accelerators: Superconducting magnets in facilities like the Large Hadron Collider (LHC) generally operate around 7-8 Teslas. These magnets are essential for manipulating and focusing high-energy particle beams for groundbreaking scientific research.
Neodymium Magnets: Among the strongest permanent magnets commercially available, neodymium magnets can exceed 1 Tesla at their surface. Their use spans a wide range of applications, from motors in electric vehicles to personal electronics.
Contextualizing the Tesla in Asteroid Mining:
While a 1 Tesla magnetic field is potent, especially in controlled environments or at close range, its influence diminishes quickly with distance due to the inverse cube law. This presents challenges in applications like asteroid mining, where distance and material properties significantly impact the feasibility and effectiveness of magnetic manipulation.
This summary, prepared by ChatGPT, aims to demystify the concept of a Tesla and illustrate its importance across various technological and scientific applications. Understanding these principles helps bridge the gap between theoretical physics and practical, real-world applications, illuminating the challenges and possibilities inherent in working with magnetic forces.
Feel free to ask for further clarification or exploration on this topic or any other areas of interest.

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Calliban · 2024-03-06 16:33:19

A strong field isn't strictly necessary, as the gravity field you are working against is extremely weak. But the ability to switch the field on and off is valuable, as you must otherwise mechanically remove material from the magnet. Using electromagnets, it is possible to pump ferromagnetic material.

tahanson43206 · 2024-03-07 13:08:12

Noting post #10 by Calliban ...

The post below is follow up regarding strengths of various magnets ...
https://www.coolmagnetman.com/magflux.htm

Experiments with magnets and our surroundings
How strong are magnets?
Typical Values
Here is a list of how strong some magnetic fields can be:
Smallest value in a magnetically shielded room 10^-14 Tesla    10^-10 Gauss
Interstellar space 10^-10 Tesla    10^-6 Gauss
Earth's magnetic field 0.00005 Tesla    0.5 Gauss
Small bar magnet 0.01 Tesla    100 Gauss
Within a sunspot 0.15 Tesla    1500 Gauss
Small NIB magnet 0.2 Tesla    2000 Gauss
=>>> I found a 1 Tesla 10,000 Gauss bar magnet for sale on Amazon <<<=
Big electromagnet 1.5 Tesla    15,000 Gauss
Strong lab magnet 10 Tesla    100,000 Gauss
Surface of neutron star 100,000,000 Tesla    10^12 Gauss
Magstar 100,000,000,000 Tesla    10^15 Gauss
What is a Tesla? It is a unit of magnetic flux density. It is also equivalent to these other units:
1 Weber per square meter
10,000 Gauss (10 kilogauss)
10,000 magnetic field lines per square centimeter
65,000 magnetic field lines per square inch.
Now, 1Gauss is about 6.5 magnetic field lines per square inch.
If you place the tip of your index finger to the tip of your thumb, enclosing approximately 1 square inch, four magnetic field lines would pass through that hole due to the earth's magnetic field!
Measuring the strength of magnets
1. Hall-Effect Devices

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tahanson43206 · 2024-03-07 13:24:30

For Calliban re #10

There are many gravity fields in play in a rubble pile asteroid ...

First of all, there is the Sun, which I expect is superior to all others.

Second, all the major planets are in play, to varying degrees based upon mass and distance.

Third, every individual mass within the rubble pile has it's own gravity.

The challenge is to find a magnetic field that is strong enough to reach through the entire asteroid with strength greater than the inertia/momentum of the particles. We have seen (thanks to Void's original inspiration) that magnetic force falls off as the Inverse Cube of distance, while the force of gravity falls off as the Inverse Square of distance. Thus, even a 1 Tesla magnet has negligible (immeasurable) strength just a few meters from the pole. It is for that reason I am doubtful of the ability of a magnetic field humans could produce to move an entire asteroid.

In other words, I am pretty sure that even a 1 Testla magnet would have no effect on asteroid material a short distance away from the magnet.

An example for our readers might be the experiment often performed with bar magnets and iron filings. A bar magnet dipped into a pile of iron filings will attract a number of filings, but the number attracted is limited by the strength of the magnet. The Inverse Cube Law is at work. After some distance, the magnetic force is insufficient to overcome the pull of Earth's gravity on the iron filings.

I believe that the same effect would be apparent on one of Void's iron rich asteroids. An experimenter might dip an electromagnet into the material and apply a current. Magnetic susceptible material would jump to the magnet, which could then be withdrawn so that the collection could be delivered to a bag or other container for shipment to a processing center. The reach of such a magnet would be limited by the Inverse Cube Law, but it might be sufficient to facilitate efficient harvesting of a suitable asteroid.

Such a magnet could NOT move the entire asteroid.

We appear to have come full circle. Void reminded me today that his original idea had to do with asteroid mining, and we seem to have found a way to accomplish that.

(th)

SpaceNut · 2024-03-07 18:40:18

Artificial Magnetosphere - Electromagnetic Induction

tahanson43206 · 2024-03-09 10:04:57

For SpaceNut re #13

Thank you for the interesting link. It appears to be a link to an entire topic devoted to the challenge of creating a magnetosphere.

This post is an in-depth study of the Earth's magnetic field, with the goal of obtaining a reasonable estimate for it's strength.

Estimating the Earth's Core Magnetic Field Strength
This report, prepared by ChatGPT, presents a theoretical calculation to estimate the Earth's magnetic field strength at its core, using the observed magnetic field strength at the Earth's surface and applying the Inverse Cube Law.
Background
The Earth's magnetic field is a complex and dynamic phenomenon, essential for navigation, animal migration, and protecting the planet from cosmic radiation. While direct measurements of the magnetic field strength at the Earth's core are not possible due to the inaccessible depths involved, we can use mathematical laws and surface observations to make educated estimates.
Observations and Calculations
Observed Surface Magnetic Field Strength: The strongest magnetic field strength recorded at the Earth's surface is approximately 67 nanoteslas (nT).
Radius of the Earth: 6371 kilometers (km), which is equivalent to 6.371 x 10^6 meters.
Methodology
Using the Inverse Cube Law, we sought to estimate the magnetic field strength at the Earth's core based on the known surface strength. The Inverse Cube Law allows for the calculation of the field strength at a different point if the distance and initial field strength are known.
Calculation Steps
1. Apply the Inverse Cube Law: B_core = B_surface x (r_Earth)^3, where B_surface is the magnetic field strength at the surface, and r_Earth is the radius of the Earth.
2. Calculate the Estimated Core Magnetic Field Strength: With B_surface = 67 nT and r_Earth = 6.371 x 10^6 meters, the estimated core magnetic field strength was calculated to be approximately 1.73 x 10^13 Tesla.
3. Validate the Calculation: Using the estimated core magnetic field strength, we then applied the Inverse Cube Law in reverse to estimate the surface magnetic field strength from the core, which resulted in a calculated value of 67 nT, matching the observed surface value.
Conclusion
This theoretical exercise demonstrated that applying the Inverse Cube Law to the observed surface magnetic field strength yields an estimated core strength that, when recalculated for the surface, matches the original observation. This approach offers a fascinating insight into how mathematical models can be used to infer conditions in inaccessible parts of the Earth's system, despite the simplified nature of the calculation and the complex reality of the Earth's magnetic field generation process.
Report Prepared by ChatGPT

I have no way of knowing if further checking by NewMars members will verify or falsify the field strength estimate above, but I'm offering this as a starting point for further work. It should be possible to validate the number by adding more actual observations to the computation. For example, it appears that readings may have been taken from orbit, and since those would be actual readings, they can be used as observational data along side the actual physical readings taken all over the planet at the surface.

The greatest value measured anywhere on the surface appears to be 67 nT at one of the magnetic poles.

(th)

tahanson43206 · 2024-03-09 20:22:05

In an effort to better understand the Earth's magnetic field, I consulted with ChatGPT4 and with Gemini and the consensus seems to be that an ESA site is the best place to look for readings of magnetic field strength in orbit around the Earth.

https://earth.esa.int/eogateway/missions/swarm/data

A swarm of three satellites has been in operation for 10 years, collecting vast amounts of data, and apparently all of that data has be archived in easy-to-access form. I was only looking for two pieces of data so am overwhelmed by ten years of readings from the entire orbital space around the planet.

That said, if there is anyone in the forum readership who is interested, the web site at the link above invites you (or anyone) to visit.

(th)

SpaceNut · 2024-03-10 09:48:17

Earth has natural magnetic materials of a variety of elements that rely on domain alignment which can be changed after heating the material, and as it cools alongside of a material of different orientation means we can change the alignments. This is what happens with the modern CD that we enjoy storing data on.

Why are rare earth elements so rare?

Of course earth also contains convection cause fields and so much more that is part of the internal structure of the earth its self.
Most are aware of all of the forms derived from a power source whether its DC,AC of IF/RF in nature of the formed field.

Recently I found that we have another form in https://en.wikipedia.org/wiki/Antimagnetic_watch
Introducing the ‘antimagnet’

SpaceNut · 2024-03-23 11:33:36

Bismuth is so strongly repelled from magnets, it levitates. How?

tahanson43206 · 2024-11-07 19:53:15

The research described in this article makes progress in understanding how ferromagnetism works.
What some may find interesting in the story is how a 19th century puzzle provides a model for what the researchers found...

https://getpocket.com/explore/item/a-ch … wtab-en-us

Pocket worthyStories to fuel your mind
A Child’s Puzzle Has Helped Unlock the Secrets of Magnetism
People have known about magnets since ancient times, but the physics of ferromagnetism remains a mystery. Now a familiar puzzle is getting physicists closer to the answer.
Quanta Magazine
Marcus Woo
Quanta Magazine
More from Quanta Magazine
The ‘Weirdest’ Matter, Made of Partial Particles, Defies Description
Puzzling Quantum Scenario Appears Not to Conserve Energy
Computer Scientists Prove That Heat Destroys Quantum Entanglement
15puzzle_2880x1620.gif
The 15-puzzle asks players to slide numbered tiles around a grid. When the numbers are replaced by the spins of electrons, the puzzle can be used to explain how permanent magnets work. Credit: Dave Whyte for Quanta Magazine.
For a few months in 1880, entire swaths of the United States succumbed to an addiction the likes of which had never been seen. “It has become literally an epidemic all over the country,” wroteThe Weekly News-Democrat in Emporia, Kansas, on March 12, 1880. “Whole cities are distracted, and men are losing sleep and going crazy over it.” The epidemic spread to Europe and as far as Australia and New Zealand.
The disease was a new obsession: a frustratingly simple mechanical game called the 15-puzzle. Still familiar today, it consists of a four-by-four grid in which you slide 15 numbered tiles around, trying to put the numbers in sequence.
The game seems quaint by today’s standards, but in 1880, it was all the rage. “No child is too puerile to be beneath its entertaining powers, and no man is too vigorous or in too high station to escape its fascination,” the News-Democrat wrote . The frustration, perhaps, stemmed from the mathematically proven fact that only half of the puzzle configurations are solvable (likely unbeknownst to the addicted).
Now, nearly 140 years later, the 15-puzzle is of interest again, this time not as a distraction, but as a way to understand a seemingly unrelated and much more complex puzzle: how magnets work.
Permanent magnets such as the ones on your refrigerator are magnetic because of a phenomenon called ferromagnetism. In a ferromagnet, the spins of electrons align, collectively generating a magnetic field. More specifically, metals such as iron, cobalt and nickel demonstrate itinerant ferromagnetism, which refers to the fact that their electrons can move around freely within the material. Each electron also has an intrinsic magnetic moment, but to understand exactly how and why all those magnetic moments align in a magnet demands calculating the quantum interactions among all the electrons, which is prohibitively complex.
“Itinerant ferromagnetism is actually one of the hardest problems in theoretical condensed matter physics,” said Yi Li, a physicist at Johns Hopkins University.
But Li and two graduate students, Eric Bobrow and Keaton Stubis, may be just a bit closer to solving the problem. Using the mathematics of the 15-puzzle, they expanded a well-known theorem that describes an idealized case of itinerant ferromagnetism. In their 2018 analysis, published in the journal Physical Review B, they extend the theorem to explain a broader and more realistic system, potentially leading to a more rigorous model of how magnets work.
“This is a beautiful paper,” said Daniel Arovas, a physicist at the University of California, San Diego. “Especially because rigorous results for the case of itinerant ferromagnets are rather few and far between, I really like this work.”
Hole Hop
At the most basic level, electrons in a metal have to abide by two big constraints. First, they’re all negatively charged, so they all repel one another. In addition, electrons must obey the so-called Pauli exclusion principle, which states that no two particles can occupy the same quantum state. This means that electrons with the same property of “spin” — which is proportional to the electron’s magnetic moment — cannot occupy the same quantum state around an atom in the metal. Two electrons with opposite spins, however, can.
It turns out the easiest way for an ensemble of freely moving electrons to satisfy both their mutual repulsion and the constraints of the Pauli exclusion principle is for them to stay apart and for their spins to align — and thus become ferromagnetic.
But this is just a simplified sketch. What’s eluded physicists is a detailed model of how such an organized pattern of aligned spins emerges from the countless quantum interactions between individual electrons. For example, Li explained, an electron’s wave function — the complex mathematical description of its quantum properties — can be entangled with another electron’s wave function. To fully understand how the behavior of individual particles leads to the collective phenomenon of ferromagnetism, you would need to keep track of the wave function of every electron in a system as it continually reshapes every other electron’s wave function through their mutual interactions. In practice, this widespread entanglement makes the full, rigorous equations needed to describe ferromagnetism impossible to write down.
Instead, physicists like Li are trying to glean insight by studying simpler idealized models that capture the underlying physics of ferromagnetism. In particular, her recent work expands on a milestone discovery made more than 50 years ago.
In the mid-1960s, two physicists heralding from opposite sides of the globe independently derived a proof that explained why electrons should align and create a ferromagnetic state. David Thouless, a physicist then at Cambridge University who would go on to win the Nobel Prize in 2016, and Yosuke Nagaoka, a physicist visiting the University of California, San Diego, from Nagoya University at the time, published their proofs in 1965 and 1966, respectively. Their result, called the Nagaoka-Thouless theorem (also Nagaoka’s theorem), relies on an idealized system of electrons on an atomic lattice. So while it didn’t explain real-world magnets, it was nevertheless important because it showed, for the first time, in principle why electron spins should align. And because their analyses were mathematical proofs, they were exact, unburdened by the approximations typical in physics.
To understand the theorem, imagine a two-dimensional square lattice. Each vertex can accommodate two electrons of opposite spins, but the theorem assumes that it would require an infinite amount of energy for two electrons to occupy a single site. This ensures only one electron resides in each slot. In this configuration, each electron can be spinning either up or down. They don’t have to be aligned, so the system is not necessarily ferromagnetic.
Now take away one electron. What remains is a vacancy called a hole. An adjacent electron can slide over into the hole, leaving behind another vacancy. Another electron can scoot over into the new opening and leave behind another new hole. In this way, the hole effectively hops from one site to another, shuttling around the lattice. Thouless and Nagaoka found that in this scenario, with the addition of just a single hole, the electrons would spontaneously align. This was, they proved, the lowest energy state, one that’s ferromagnetic.
For the system to be in the lowest energy state, Arovas explained, the hole must be free to roam without disturbing the configuration of electron spins — a process that would require extra energy. Yet as the hole moves, the electrons also move around. For the electrons to move without altering the configuration of spins, the electrons must be aligned.
“Nagaoka’s theorem is one of the few examples with which you can mathematically prove instances of ferromagnetism,” said Masaki Oshikawa, a physicist at the University of Tokyo. “But from a physics point of view, it’s very artificial.”
For example, it costs a lot of energy for two electrons to overcome their mutual repulsion and settle in the same site — but not infinite energy, as the theorem demands. The Nagaoka-Thouless picture also only applies to simple lattices: two-dimensional lattices of squares or triangles, or a three-dimensional cubic lattice. In nature, however, ferromagnetism arises in many metals with all kinds of structures.
If the Nagaoka-Thouless theorem really explains ferromagnetism, then it should apply to all lattices. People assumed this was likely the case, Li said. “But no one really gave a clear proof.” That is, until now.
Spin Tiles
In 1989, Hal Tasaki, a physicist at Gakushuin University in Japan, extended the theorem somewhat, finding that it would apply as long as a lattice has a mathematical property called connectivity. Take the simple case of a square lattice with one moving hole. If, after moving the hole around, you can create every configuration of spins while preserving the number of spin-up and spin-down electrons, then the connectivity condition is satisfied.
But other than the square and triangle lattices and the three-dimensional cubic, it wasn’t clear whether the connectivity condition would be satisfied in other cases — and thus whether the theorem applies more generally.
To tackle this question, Li started by focusing on the six-sided honeycomb lattice. As her students, Bobrow and Stubis, worked on the problem, they realized it resembled that 19th-century obsession: the 15-puzzle. Just swap the labels on the tiles from numbers to up or down spins, and the puzzle becomes equivalent to a Nagaoka ferromagnet, with a hole that moves through a lattice of electrons.
The puzzle is solved when you can reorder the tiles to make any sequence you want, which is precisely the meaning of the connectivity condition. So whether the connectivity condition is satisfied for a given lattice becomes a question of whether an equivalent puzzle with that lattice structure is solvable.
It turns out that back in 1974, a mathematician named Richard Wilson, now at the California Institute of Technology, had figured it out, generalizing and solving the 15-puzzle for all lattices. As part of his proof, he showed that for almost all nonseparable lattices (which are those whose vertices remain linked even after removing one vertex), you could slide the tiles around and get any configuration you want, so long as you make an even number of moves. The only exceptions are single polygons larger than a triangle, and something called the θ 0 (“theta zero”) graph, in which a vertex in the center of a hexagon is connected to two opposite vertices.
The researchers could then directly apply the results of Wilson’s proof to the Nagaoka-Thouless theorem. For a system of electrons and a single hole, they proved that the connectivity condition is satisfied for nearly all lattices, including common structures like the two-dimensional honeycomb and the three-dimensional diamond lattices. The two exceptions — polygons larger than a triangle and the θ 0 graph — are not structures you would find in a realistic ferromagnetic anyway.
Hole Explosion
Using the 15-puzzle is a fresh and potentially fruitful approach, said Sriram Shastry, a physicist at the University of California, Santa Cruz. “I like the fact that they brought in new language, a new set of connections with graph theory,” he said. “The connection I think is rich — it can be a rich source of insights in the future.” But while the study takes a significant step forward, problems remain.
One complication is that the Nagaoka-Thouless theorem doesn’t always work when the moving hole has to take an odd number of steps as it loops around a lattice, Shastry said. Perhaps the most glaring problem, however, is that the theorem requires the presence of exactly one hole — no more, no less. In metals, though, holes are abundant, often filling up half the lattice.
But physicists have tried generalizing the theorem to multiple-hole systems. Using numerical calculations, physicists have shown that Nagaoka ferromagnetism seems to work for a square lattice of finite size that’s up to 30 percent filled with holes. In the current paper, the researchers applied exact analytic techniques to the two-dimensional honeycomb lattice and the three-dimensional diamond lattice. Nagaoka ferromagnetism seems to exist as long as the number of holes is less than the number of lattice sites raised to the 1/2 power for the honeycomb, or the 2/5 power for the diamond.
These exact solutions could lead to a more complete model of itinerant ferromagnetism. “This is just one small step forward to set up some rigorous mathematical starting point for future study,” Li said.
Marcus Woo is a science journalist based in the San Francisco Bay Area. His work has appeared in WIRED, New Scientist, National Geographic, Smithsonian, NPR, the BBC, and other publications.

(th)

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#8 2024-03-06 10:31:25

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#9 2024-03-06 15:08:44

Re: Magnets Magnetic Physics of

Understanding the Tesla: A Summary by ChatGPT

#10 2024-03-06 16:33:19

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#11 2024-03-07 13:08:12

Re: Magnets Magnetic Physics of

#12 2024-03-07 13:24:30

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#13 2024-03-07 18:40:18

Re: Magnets Magnetic Physics of

#14 2024-03-09 10:04:57

Re: Magnets Magnetic Physics of

Estimating the Earth's Core Magnetic Field Strength

Background

Observations and Calculations

Methodology

Calculation Steps

Conclusion

Report Prepared by ChatGPT

#15 2024-03-09 20:22:05

Re: Magnets Magnetic Physics of

#16 2024-03-10 09:48:17

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