Field Definition and 1000 Threads

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics.
The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and p-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements.
The relation of two fields is expressed by the notion of a field extension. Galois theory, initiated by Évariste Galois in the 1830s, is devoted to understanding the symmetries of field extensions. Among other results, this theory shows that angle trisection and squaring the circle cannot be done with a compass and straightedge. Moreover, it shows that quintic equations are, in general, algebraically unsolvable.
Fields serve as foundational notions in several mathematical domains. This includes different branches of mathematical analysis, which are based on fields with additional structure. Basic theorems in analysis hinge on the structural properties of the field of real numbers. Most importantly for algebraic purposes, any field may be used as the scalars for a vector space, which is the standard general context for linear algebra. Number fields, the siblings of the field of rational numbers, are studied in depth in number theory. Function fields can help describe properties of geometric objects.

View More On Wikipedia.org
Recent contents Top users
  1. B

    A Energy levels shifts in a time-varying electric field

    Hello! I have 2 levels of the same parity with energies ##E_1 < E_2##, and another level of opposite parity a distance ##E## from the ##E_2##. I also have that ##E_2 - E_1 << E##. I have a laser on resonance (I am trying to scan along the resonance and find it) with the transition from ##E_2##...
  2. M

    A Maxwell theory invariant under dual field strength tensor application

    Hello everybody! I know in classical field theory adding in the Lagrangian density a term of the form Fαβ (*F)αβ (where by * we denote the dual of the field strength tensor) does not change the EOM, since this corresponds to adding a total derivative term to the action. However when computing...
  3. N

    Proving SHM for charged spring mass system in electric field

    For part (f) Solution is I don't understand why the bit highlighted in yellow is true. Would anybody be kind enough to help.
  4. M

    Electric field at surface of lead-208 nucleus

    For this problem, The solution is, However why is ## r = (208)^{1/3}(1.20 \times 10^{-15} m)## Many thanks!
  5. M

    Integrating electric field of rod

    For this problem, However, I am trying to solve this problem using an alternative method compared with the solutions. My method is: ##\vec E = k_e \int \frac {dq} {r^2} \, dx ## ##\hat r## ##\vec E = k_e \int \frac {\lambda} {x^2 + d^2} \, dx## ## \hat r## If I let ## \hat r = \frac {-x\hat i...
  6. M

    I Electric field in a rotating frame

    Hello! I have a radially pointing electric field i.e. at a given radius, R, the electric field has the same magnitude and points radially around that circle of radius R. I have a particle moving around that circle of radius R, with uniform velocity (ignore for now how it gets to move like that)...
  7. M

    Magnetic Field Due to a Curved Wire Segment

    For this problem, The solution is, However, why did they not use limits of integration for the integral in red? When I solved this, I used as limits of integration. I see that is not necessary since you get the same answer either way, but is there a deeper reason? Many thanks!
  8. M

    Magnetic Field Surrounding a Straight Conductor

    For this problem, Part of the solution is, However, would someone please tell me where they got the sine function circled in red from? Many thanks!
  9. M

    Electric field due to n charges

    For this problem, The solution is, However, should they be a vertical component of the electric field for the expression circled in red? I do understand that assuming that when the nth charge is added it is placed equal distant for the other charges so that a component of the electric field...
  10. M

    Electric field due to two charges

    The problem and solution is, However, I would like to know why it cannot be solved like this: Thanks for any help!
  11. noowutah

    Find the electric field of a long line charge at a radial distance

    TL;DR Summary: Find the electric field of a long line charge at a radial distance where the potential is 24V higher than at a radial distance r_1=3m where E=4V/m. Answer: 29.5V/m. Never mind: I retract this question. The integral apparently is supposed to diverge! I apologize for not reading...
  12. M

    Symmetry behind charged spring-mass system in Electric field

    For this problem, If we assume that x = 0 is where the spring connects to the wall, then the rest position of the mass-spring-electric field position is x = EQ/k and the max position is x = 2EQ/k. Is there a reason for the symmetry between the rest position and max position? (The symmetry...
  13. M

    Alternative method of finding Electric flux from non-uniform field

    For this problem, The solution is, However, why can the differential area not be: I tried integrating and got, Can someone please tell me what I have done wrong? Thank you!
  14. Kostik

    A Static Gravitational Field: Why Must ##g_{m0} = 0##?

    In Dirac's "General Theory of Relativity", he begins Chap 16, with "Let us consider a static gravitational field and refer it to a static coordinate system. The ##g_{\mu\nu}## are then constant in time, ##g_{\mu\nu,0}=0##. Further, we must have ##g_{m0} = 0, (m=1,2,3)##." It's obvious that...
  15. L

    B Quantum field theory and wave particle duality

    I recently watched this lecture "Quantum Fields: The Real Building Blocks of the Universe" by David Tong where the professor provides a succinct explanation of QFT in about 6 minutes around the midway mark. The main point being that there are fields for particles and fields for forces and the...
  16. J

    Calculating Area & Direction of Magnetic Field

    Hi, the problem statement is above. I have some questions about how to calculate the area and the direction of the magnetic field of this problem. As the magnetic flux, my professor have defined it as Phi= integral(B dS)=(Area)e_x B= (Area_triangle + (L^2/2) *(β + α(t)))*B e_z. How can one know...
  17. J

    Derive an expression for the radial charge distribution of an E field

    I know we're supposed to attempt a solution but I'm honestly super confused here. I think the second an third terms of the del equation can be cancelled out because there is only an E field in the r hat direction, so no e field in the theta and phi directions. That leaves us with ##\nabla \cdot...
  18. Flannel

    Force Field Experiments: Realistic Physics Questions

    Hello everyone! Events in my current webnovel have reached the limit of confidence in my physics reasoning, so I'm here to ask for confirmation of my estimates of what would happen from experimentation with force fields. While the setting is fantasy/magic based rather than superscience, I still...
  19. sal1854

    I Direction of motion of particles with total spin under magnetic field

    According to Chapter 8 of Griffiths' book Introduction to Electrodynamics, the magnetization force that acts on a magnetic dipole is $$F_M=\nabla (m \cdot B)$$, where ##m## is the magnetic moment and ##B## is the magnetic field. For a paramagnetic or diamagnetic particle...
  20. T

    I Origin of the Earth's magnetic field

    How can the motion of liquid iron consisting of neutral iron atoms in the earth's core create magnetic fields? What am I missing?
  21. Ahmad Sadek

    B Can a rocket flame be controlled by a magnetic field?

    An idea came to my mind after I saw the plasma reactor and how the plasma floats through a magnetic cage that prevents the nuclear reactor from melting, to make a magnetic cage that prevents the rocket engines from melting while they are working. Is this possible, or does it take a critical...
  22. G

    I Magnetic Field vs Spacetime: Effects on Inertia Disk

    By following article a magnetic field can produce a least a minimum distortion in spacetime. If we have a inertia disk spinning 50% inside of a strong closed magnetic field may we suppose that we will create an unbalanced in the angular disc moment producing a propulsion without mass variation...
  23. Addez123

    Can't find potential of vector field

    1. To find the solution simply integrate the e_r section by dr. $$\nabla g = A$$ $$g = \int 3r^2sin v dr = r^3sinv + f(v)$$ Then integrate the e_v section similarly: $$g = \int r^3cosv dv = r^3sinv + f(r)$$ From these we can see that ##g = r^3sinv + C## But the answer is apparently that there...
  24. M

    No Limits of Integration for Electric Field Integral?

    For this problem, The solution is, However, why have they not included limits of integration? I think this is because all the small charge elements dq across the ring add up to Q. However, how would you solve this problem with limits of integration? Many thanks!
  25. M

    Finding Area of Ring Segment to Find Electric Field of Disk

    Hi! For this problem, Why is the area of each ring segment dA equal to (2π)(r)(dr)? However, according to google the area of a ring segment (Annulus) is, Many thanks!
  26. arivero

    I Is Connes' Model of the Higgs Field a Wormhole?

    As this week the great topic is the Einstein-Rosen bridge and duals of it, I wonder what happened with the interpretation of Connes NCG model as two sheets of spacetime whose separation is measured by the higgs field. How is it different of a ER bridge?
  27. M

    Electric Field of a Uniform Ring of Charge

    Hi! For this problem, The solution is, However, why did they not include constants of integration in their working shown in red? Many thanks!
  28. alan123hk

    B Decompose the E field into conservative and non-conservative parts

    Recently, I've seen several discussion threads here about splitting electric fields. I want to express my opinion. Of course, this is just a calculation method, not a basic physical concept, but it is also useful in some cases, at least not wrong. The following is an example of the out of a...
  29. livio

    Estimates of voltage drop with distance in weak electric field

    If I resolve the equation in 0, imposing a voltage value of 5 mV, it gives a non real solution, therefore I cannot resolve it for R=1 because I do not know which voltage value to impose. I am sure this is simpler than I am putting it :) thanks for any advice!
  30. C

    Why Does Positive Charge Exert Elec. Field Beyond Neg. Charge?

    I wonder how it is possible that a positive charge can exert el. field beyond negative charge? Shouldn't they "connect" and therefore positive charge should stop to have el. field beyond neg. charge? I mean, I am obviously wrong about that, but can someone please explain why/how el. field from...
  31. ermia

    Finding Constants: Potential and Field Analysis

    I have wrote all feilds and potentials and I want to find the constants. My first question is " when we say in the a<x<2a the potential is V(x)" then the potential in the a is V(a) or V(0) ( cause it is 0 in our new area) ? Second one is " when I want to write the gausses law for the point x=a I...
  32. M

    A The Effect of Time-Varying Electric Fields on a 2-Level Ion System

    Hello! If I have an ion which can be treated as a 2 level system, in a time varying electric field (the variation of the field doesn't need to be on or close to resonance, but for simplicity we can assume it is an oscillatory field) can I simply separate the problem into a center of mass motion...
  33. redirmigician

    I Acceleration in an electron field

    I read something about accelerators using nanotubes. I am a little concerned about the design mentioned in the "High Density with Perpendicular Carbon Nanotubes" part of this paper(https://doi.org/10.3390/photonics8060216). Can wakefield acceleration be done in an electron field? Or maybe I...
  34. A. Neumaier

    A Ensembles in quantum field theory

    Then please explain how the transition in conceptual language from a single quantum field (extending all over spacetime, or at least over the lab during a day) to an ensemble of particles can be justified from the QFT formalism.
  35. cwill53

    Average Electric Field over a Spherical Surface

    The picture above shows the integral that needs to be evaluated, and the associated picture ## \cos\alpha ## can be obtained via the law of cosines. I'm simply confused as to where the ##\cos\alpha ## comes from in the first place. I just don't see why ##\cos\alpha ## is necessary in this...
  36. Green dwarf

    I Time Dilation in Gravitational Field: Potential vs Field Strength

    I'm wondering is whether it is the gravitational potential (in J/kg) at a point in space that determines the rate of passage of time, or whether it is the gravitational field strength (in m/s2). To clarify, suppose you had a very heavy hollow spherical shell. The gravitational potential would...
  37. N

    Find the Electric field at point p

    Since q3=q4 and they are opposite to each others they cancel out But as soon as I try to find the electric field of one of the charges, I need the radius which is not given. By isolating the electric field for radius E=(kq)/r^2 I now have two unknowns
  38. Povel

    A Exploring the Electric Field of a Moving Charged Spherical Shell

    The electric field inside a charged spherical shell moving inertially is, per Gauss's law, zero. If the spherical shell is accelerated, the field inside is not zero anymore, but it gains a non-null component along the direction of the acceleration, as mentioned, for example, in this paper. The...
  39. Y

    Why Does the Electric Field Component Ex Require a Negative Sign?

    hello i would like to understand to something. here is the drew now for my question: i was able to find Ey and here is my correct answer: when i try to find Ex i didnt understand something, i found the correct answer but i need to put minus before and i want to know why? here is my solution...
  40. H

    I Why is the electric field in y direction in a TE guided wave?

    Hi, I have a fairly simple question, but the answer is probably not as simple. I'm not sure to understand why in a guided wave (TE), the electric field is in the y direction. I know ##E_z = 0##, but why ##E_x = 0, B_y = 0##?
  41. S

    I How do I visualize the magnetic field?

    Hi! So my question is this, I have done measurements with an magnetic field meter around a transformer from 0.5 meter away (then measure some points around) and then I moved out 0.5 meters and so on until I reached a nearby building. So my issue now is I want to visualize this to my customer...
  42. V

    Why does a diamagnetic rod align perpendicular to a magnetic field?

    I know that each material is made up of tiny magnets due to electrons orbiting the nucleus and also from electron spinning about its own axis. In ferromagnetic or paramagnetic rod these tiny magnets align with the applied field causing the net field in the rod to increase. But for diamagnetic...
  43. T

    I How is an interference pattern changed by an external field?

    For example, through the diffracting of electrons and their wave-particle dualistic nature, its possible to produce an interference pattern. And for the resulting electron beam, it will still be affected by the Lorentz force and be deflected by it. At the same time, the position(s) of the...
  44. J

    A The Probability Distribution of a Bosonic Field when Emitted

    If a bosonic field is probabalistic, and if it can be emitted (suddenly coming into existence), what determines its probability distribution when it is emitted from a fermion? In other words, one thinks (or at least I think) of a fermion field as already being in existence and already having...
  45. N

    I Questions about a Conductor in an Electric Field

    Dear PF, I have a question regarding a conductor in electric filed. I have formulated my question in attached PDF file ... would please be so kind and advise me please... Thanks you in advance ...
  46. C

    I Where do discontinuities in the electromagnetic field occur?

    Obviously at point charges, but where along boundaries? Would they theoretically occur in superconductors since they can carry infinite current (J -> infinity)?
  47. G

    I Geodesic in Weak Field Limit: Introducing Einstein's Relativity

    I'm reading《Introducing Einstein's Relativity_ A Deeper Understanding Ed 2》on page 180,it says: since we are interested in the Newtonian limit,we restrict our attention to the spatial part of the geodesic equation,i.e.when a=##\alpha####\quad ##,and we obtain,by using...
  48. T

    B Thinking about a calendar epoch based on the field of physics

    I was thinking about an idea of a calendar epoch based on the field of physics, and its thinking about which event should be used as the epoch so I have this idea and just wondering what you guys might think about it I initially landed upon two ideas, one pretty ancient and the other kinda...
  49. gentzen

    I Are there asymptotic QFT/QED states in a constant magnetic field?

    It is "easy" to produce experimental setups that could and should for all practical purposes be described as having a constant background magnetic field everywhere, especially in the "asymptotic region" where the detectors are located. You can do this both in vacuum, and inside a solid sample...
  50. Barbequeman

    Calculating Velocity & Field Strength of Gas: Help Needed!

    For the first calculation of the velocity of the gas I use the first equation and this converted in meter would be look like this (first value as an example) v=299792458 m/s * (6.76813x10^-7-6.768x10^-7)/6.768x10^-7 =5836.03m/s or 0.0019c this was the velocity of the gas for the first spectral...
Back
Top