What is Field: Definition and 1000 Discussions

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.

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  1. guyvsdcsniper

    Help finding the Electric field at the center of charged arc

    I am having trouble understand where area circled in red. I get that lamda is Q/L. The charge is +Q. Length is pi/R/2. I am having trouble understanding why the length is pi/R/2? Is it because the circumference of a circle is 2*pi*R and since we have broken this problem down to just...
  2. Ugnius

    Thin Charged Isolated Rod -- Find the electric field at this point

    Hi , I've been trying to manage a solution in my head and i think I'm on the right path , i just need some approval and maybe some tips. So it's obvious I can't solve this without integration because law's only apply to point charges , and i can't shrink this object to a point as i could do with...
  3. Sagittarius A-Star

    COVID Electric field of moderate strength inactivates Covid-19 virus

    A science team from the university of Kassel (Germany) proved with a physical model, that a moderate electric field inactivates the Convid-19 virus. Source: https://www.nature.com/articles/s41467-021-25478-7 via...
  4. karush

    MHB -b.1.1.4 directional field as t \to \infny

    from the spoiler its #4 hope I got instructions right $y'=-1-2y$ $\begin{array}{ll} rewrite &y'+2y=-1\\ exp &u(t)=\exp\ds\int 2 \, dx=e^{2t}\\ product &(e^{2t}y)'=-e^{2t}\\ &e^{2t}y=\ds\int -e^{2t} \, dt=\dfrac{-e^(2 t)}{2}+c\\ hence &y(t)=\dfrac{-e^{2t}}{2} +...
  5. Tertius

    A Difference Between Scalar Field Solutions in Curved Spacetime

    A general free field Lagrangian in curved spacetime (- + + +), is given by: L = -1/2 ∇cΦ ∇cΦ - V(Φ) when the derivative index is lowered, we obtain: L = -1/2 gdc∇dΦ ∇cΦ - V(Φ) then we can choose to replace V(Φ) with something like 1/2 b2 Φ2 so: L = -1/2 gdc∇dΦ ∇cΦ - 1/2 b2 Φ2 ** I will...
  6. V

    Dipole placed in a uniform electric field

    My understanding is that the uniform electric field ##\vec E## cannot be the net electric field since the dipole creates its own electric field as shown in first diagram below, which must superimpose with the uniform electric field. So, yes, the uniform electric field ##\vec E## around the...
  7. bob012345

    I Divergence of the Electric field of a charged circular ring

    In a previous thread* the field in a charged ring was discussed and it was shown to be not zero except at the center. In *post #45 a video is referenced that says the field diverges as one gets close to the ring and it was argued that at very close distances the field looks like an infinite line...
  8. Noki Lee

    I Investigating the 1d Equation: Charges & Field Disparity

    Can we apply the 1d equation (dE/dx = labmda/epsilon0)dEdx=λϵ0 to the first and the second figures? But, in the 2nd case, if we integrate the charge density, some field exists between the two charge densities. Intuitively, it should be like the last figure. What's wrong with this?
  9. D

    I Cartan's Understanding of Einstein Field Equation

    About a week ago I was reading about Cartan's geometric interpretation of the Einstein Field Equation Gij + Λgij = κTij According to Cartan, this equation expresses the idea (sum of moments of rotation for the faces of a little 3-cube) = 8π * (amount of energy-momentum within that 3-cube) As...
  10. L

    B Project on pulsars/electromagnetic quantum field

    Hello, I am a high school student doing a project on pulsars and I would love to get into the deeper workings of a quantum electromagnetic field, but I am a bit lost. Since pulsars are neutron stars, who send out a beam of electromagnetic waves, I was quite curious about it's workings. So...
  11. O

    AC power loss in lossy dielectric (capacitor) with DC-bias field

    Hi there, if a dielectric (capacitor) is described with a constant permittivit eps (or C) and loss-tangent DF, how much energy ist lost when charging the capacitor by 1V? For example: C=1, DF=0.1. When charging from 0 to 1V, the lost energy (in J) is ...? When charging from 1V to 2V, the lost...
  12. K

    Electric field in a rotating rod in a magnetic field

    The first part of the problem seems easy enough, the free electrons in the wire would move in a circle owing to an electric field that would be induced in the rod which would provide the centripetal force for the same (Please correct me if I am wrong). So we have $$eE=mω^2x$$, where e is the...
  13. L

    Central force field derivation

    The total energy of the particle is ##u^2 / 2 - k/R##. When ##u^2 \gg 2k/R##, we take the total energy to be ##u^2/2## only. By the conservation of energy, we have: $$ \frac{u^2}{2} = \frac{w^2}{2} - \frac{k}{p} $$ Take the angular momentum expression ##l = bu##, we can replace ##u## with...
  14. wnvl2

    I Rindler Transformation & 't Hooft's Introduction to General Relativity

    I am reading 't Hooft introduction to general relativity. https://webspace.science.uu.nl/~hooft10 ... l_2010.pdf In this text 't Hoof derives the Rindler transformation. A little bit further he writes My question is, how does he come to that formula $$\rho^{-2}g(\zeta)$$
  15. rudransh verma

    B Why Are Electric Fields Perpendicular on a Charged Conductor's Surface?

    If there is a spherical conductor like this with excess charges on its surface then this is the field it sets up. Each e- would have there own radial field of lines. And all the e-s would exert a force on each charge and the net field on each charge and thus net force would be zero. All the e-s...
  16. K

    Studying How to keep track of the latest results in your field?

    Hello! What is the best way to focus on the most relevant, recently published papers in your field (in my case physics). Usually I just go on arXiv (hopefully most of them will be there), choose my field of interest and start scrolling. But that takes a lot of time. Searching for key words helps...
  17. W

    What Causes the Non-Circular Distribution of Magnetic Fields in Steel Ducts?

    Hi all, first time on this forum. I know this may sound like a stupid question, but how does the magnetic field distribute? I am working on FEMM and i am analysing magnetic losses on steel ducts. I was checking the flux density and the magnetic field distributions and i was surprised when i...
  18. Poetria

    A gradient field (analysing a picture)

    I think it is increasing as you move from one level curve to the other with bigger value. Am I right?
  19. rudransh verma

    I What is the cause of electrical disturbances?

    The electromagnetic field can carry waves, some of them are light, others are used in radio broadcast. These oscillatory waves are distinguished by their frequency. The usual pickup that we get from electric currents in the circuits in the walls of the building has a frequency of about 100...
  20. E

    A Some basic questions about effective field theory (EFT)

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  21. V

    Electric field at a point within a charged circular ring

    I have broken the ring into a top arc and a bottom arc. First, let's assume an imaginary charge of +1 C is placed at point P. We will determine the force on this unit charge from top and bottom arcs. The charges in the top arc will result in electric fields that will all cancel each other...
  22. F

    B Tangential part of the potential electric field

    Good afternoon to everybody. I have may be a stupid question according to the tangential part of the electric field near the surface of the conductor. Why is it zero? The normal part is zero on the distance of Debye cause of screening. But is this situation the same for horizontal direction...
  23. P

    I Is the Higgs field in its present form phlogiston-like?

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  24. E

    Relationship between magnitude of current and magnetic field

  25. rudransh verma

    Flux of a non-uniform E field through the face of a cube

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  26. MichPod

    I Calculating Force & Acceleration in Electrostatic Field (B=0)

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  27. snoopies622

    I What are the implications of Einstein's field equations having two free indices?

    I'm trying to understand Killing fields and I've hit a bump right away. Suppose I'm in a plane and using the Cartesian coordinate system, and I consider the swirly vector field \vec{v}=(-y,x) . It is easy to show that \nabla _{y} v^x + \nabla_{x} v^y = -1+1=0 . But if I convert this to polar...
  28. V

    Electric Field on the surface of charged conducting spherical shell

    When I look at the relevant equations, then there is no mention of field for a point on the surface of the shell, so it gets confusing. On the other hand, I feel the radial E will get stronger as we approach the surface of shell and magnitude of E will approach infinity.
  29. Einstein44

    I Calculating Magnetic field strength of a magnet

    I have been trying to calculate the magnetic flux thought a single loop of wire occurring from a magnet (meaning it has a nonuniform field), so I have the following equation: Φ=∮BdAcosθ Now my problem is that I do not know how to calculate the magnetic field strength (B)of that magnet (which...
  30. M

    I Formula For Calculating Field Angle (Beam Angle) of Light?

    There are actually several questions. The formula for calculating the "beam angle" of light (emitted from lights bulbs, flashlights etc) is - α = 2 arcCos ( 1 - Lm ÷ ( 2 π Cd )) Where, α = beam angle in degrees Lm = luminous flux (Lumens) Cd = luminous intensity (Candela) The above formula...
  31. V

    Engineering What is the magnetic field of a half toroid?

    Hello, I would like to know what is the magnetic field of a half toroid. Would we use Ampere's law? So, considering that the integral is equal to BA, we would have BA = μo * i, then B = (μo * i) / r. But, using Biot-Savart's law, by symmetry, it seems that the vector sum of all the...
  32. A

    I References for Hamiltonian field theory and Dirac Brackets

    I'm looking for complete and detailed references on constrained Hamiltonian systems and Dirac brackets. While my main interest is electrodynamics, I would prefer a complete exposition of the theory from the ground up. So far, my knowledge about the topic comes from books in QFT, like Weinberg...
  33. DaniV

    The 1-loop anomalous dimension of massless quark field

    I tried as first step to find Z_q the renormalization parameter, to do so I did the same procedure to find the renormalization parameter of the gauge field of the gluon A^a_\mu when a is representation index a \in {1,2,...,N^2-1} such that A^{a{(R)}}_{\mu}=\frac{1}{\sqrt{Z_A}}A^{a}_{\mu}...
  34. jackiepollock

    How to picture a radial field around a 3d object?

    Hello! First off, for a), I am not too sure how to picture a radial field around a 3d object. I know that this spherical metal dome is basically a enlarged version of an atom, but since with problems on radial field around an atom, I don't have to consider its diameter, I'm not sure how the...
  35. jackiepollock

    I Why are field lines parallel in a uniform field?

    For a uniform field like this, I imagine the two plates that creates it are made of multiple atoms with charges, which are points sources that create radial fields. We know that radial fields don't have parallel fields lines, so how are parallel fields lines form when the field is made of...
  36. N

    B Why do moving charged particles feel a force in a magnetic field?

    When I read things about magnetism on internet, I don't understand at all about one thing: If a moving particle receive a force if it's in a magnetic field, so it should accelerate, so what happen if we change the referential so that the particle now don't move? The particle shouldn't receive...
  37. J

    A Connection between QFT renormalization and field regularization?

    While in QFT we remove infinite energy problem with renormalization procedure, asking e.g. "what is mean energy density in given distance from charged particle", electric filed alone would say $$\rho \propto |E|^2 \propto 1/r^4 $$ But such energy density would integrate to infinity due to...
  38. V

    Magnitude of electric field E on a concentric spherical shell

    The only explanation that I have seen in textbooks is that since the outer spherical shell is symmetrical relative to internal charged spherical shell so field every where on the outer shell is same in magnitude at every point on it. I can understand that electric field needs to be...
  39. Shreya

    Force on a dipole in a non uniform electric field

    I have figured out how the force is towards left in the first case. I think it is due to the larger force on - charge. Please help me out with the second and third case.
  40. Shreya

    Flux density and Divergence of Electric field

    I think Flux density is flux/Volume. Or is it flux/ Area Please be kind to help
  41. jjson775

    Electron motion after collision in a magnetic field

    Isn‘t this a straightforward problem of conservation of momentum? See attached.
  42. K

    B Rock Found in Field Turns Out to Be 4.6GY Old Meteorite

    A small fragment of rock found lying in a field in Gloucestershire in the UK may not have looked like much to the casual passerby ... I'm curious how this was found, if not by a "casual passerby". Did there happen to be a meteorite expert wandering in the Gloucestershire field?
  43. S

    I Does a field affect the originating entity?

    Eg a gravitational field exists around a mass (A), and no doubt influences any other local mass (B) - but.. - does that field also exert an influence on the original mass (A) ? ie a sort of radial force outwards ?
  44. Leo Liu

    Vector space and 3D flow field

    Could someone explain the green highlight to me, please?
  45. Paige_Turner

    B The Impact of Tidal Force on a Star's Magnetic Field and Supernova

    And if a star is severely stretched by tidal force, how does that affect its magnetic field and the process of it collapsing at the start of a supernova?
  46. P

    Magnetic field - General current in a cube

    I could solve a similar (rather, a specific case of the above) where the current entered through a corner and left from the corner opposite to it along the body diagonal of the cube. For this specific case, I was able to easily exploit symmetry to deduce the answer (0). However, I cannot think...
  47. M

    Expression for the magnitude of an electric field

    k(q_1-q_2)/(R^2+x^2)^3/2
  48. merlyn

    B Maxwell-Boltzmann distribution under the influence of a gravitational field

    Could anyone suggest a simple video showing the Maxwell-Boltzmann distribution under the influence of a gravitational field? I trying to show a flat earther idiot how pressure gradients arise in a simple manner. Thank you all. DF
  49. docnet

    B Entangled protons in a uniform magnetic field?

    In NMR, protons that are near each other exchange energy with each other by photon emission and absorption. Proton A is spin down, and proton B is spin up. proton A emits a photon and becomes spin up, and proton B absorbs this photon and becomes spin down. have the two protons been entangled?
  50. B

    Show that the given electric field is a plane wave

    A wavefront is defined as a surface in space where the argument of the cosine has a constant value. So I set the argument of the cosine to an arbitrary constant s. ## k(\hat{u} \cdot r - c t) + \phi = s ## The positional information is is in r, so I rearrange the equation to be ## \hat{u}...
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