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. P

    Induced EMF due to motion of a wire perpendicular to a magnetic field

    This question appeared in a university entrance exam.Basically, if magnetic flux passing through a surface of a loop changes over time ,only then e.m.f will be induced to that loop.But here only a straight line is used and there's no chance of forming any area.So by definition there's no chance...
  2. StenEdeback

    I Best book for Lagrangian of classical, scalar, relativistic field?

    Hi all experts! I would like to read about the Lagrangian of a classical (non-quantum), real, scalar, relativistic field and how it is derived. What is the best book for that purpose?Sten Edebäck
  3. P

    Calculating eletric potential using line integral of electric field

    So, I am able to calculate the electric potential in another way but I know that this way is supposed to work as well, but I don't get the correct result. I calculated the electric field at P in the previous exercise and its absolute value is $$ E = \frac {k Q} {D^2-0.25*l^2} $$ This is...
  4. StenEdeback

    Deduction of formula for Lagrangian density for a classical relativistic field

    Hi, I am reading Robert D Klauber's book "Student Friendly Quantum Field Theory" volume 1 "Basic...". On page 48, bottom line, there is a formula for the classical Lagrangian density for a free (no forces), real, scalar, relativistic field, see the attached file. I like to understand formulas...
  5. D

    What is high breakthrough field?

    I have read in the following article the expression "high breakthrough field": https://link.springer.com/article/10.1557/PROC-871-I9.6 I tried to find out in the internet what is the definition of that and what it refers to in the transistors but I couldn't find anything! Thank you in advance!
  6. X

    I Magnetic field strength of a stack of magnets

    I know that for a single cylindrical neodymium magnet, the formula $$ \displaystyle{\displaylines{B(z)=\frac{μ_0M}{2}(\frac{z}{\sqrt{z^{2}+R^{2}}}-\frac{z-L}{\sqrt{(z-L)^{2}-R^{2}}})}} $$ shows the relationship between the magnetic field strength and the distance between the magnet. I was...
  7. warhammer

    I Electric Field & Interplay between Coordinate Systems | DJ Griffiths

    Hi. I believe I have what may be both a silly and or a weird query. In many Griffiths Problems based on Electric Field I have seen that a coordinate system other than Cartesian is being used; then using Cartesian the symmetry of the problem is worked out to deduce that the field is in (say) z...
  8. B

    Velocity of a relativistic particle in a uniform magnetic field

    d(ɣmv)/dt = qvB (dɣ/dt)mv + ɣm(dv/dt) = qvB Substituting gamma in and using the chain rule, it ends up simplifying to the following: ɣ^3*m(dv/dt) = qvB Now, I am confused on how to solve for v.
  9. V

    I Electric field of a moving charge that's abruptly stopped

    Hello everyone, This is in reference to fig 5.19 (screen shot attached - please read the paragraph which says "Figure 5.19 shows the..."). I don't get why the field outside of the sphere of radius ct acts as though the particle would have continued its motion. Author's words : "The field...
  10. Salmone

    I How an induced electric dipole vibrates with EM field

    If we have an electromagnetic wave like the one in the picture and a molecule which is, in the image, the small black ball with electron cloud being the part with "minus sign" in it, does the molecule with its cloud start to oscillate, once the EM wave hits it, as an induced electric dipole...
  11. arjun_ar

    Calculate the magnetic field from the vector potential

    I am trying to derive radial and axial magnetic fields of a current carrying loop from its magnetic vector potential. So far, I have succeeded in deriving the radial field but axial field derivation gives me trouble. My derivation of radial field (eq 1) can be found here. Can anyone point out...
  12. G

    I What kind of tensor is the gradient of a vector Field?

    (1,1)or(2,0)or(0,2)?And does a dual vector field have gradient?
  13. JandeWandelaar

    I Uniform gravitational field possible in GR?

    Mentors’ note: this thread is forked from https://www.physicsforums.com/threads/free-fall-in-curved-spacetime.1016510/ But what if the gravity field is homogeneous? Like that of an infinite massive plane? The objects in the ship will stay where they are. An infinite massive plane is quite...
  14. L

    A Vector analysis question. Laplacian of scalar and vector field

    If we define Laplacian of scalar field in some curvilinear coordinates ## \Delta U## could we then just say what ##\Delta## is in that orthogonal coordinates and then act with the same operator on the vector field ## \Delta \vec{A}##?
  15. Tesla In Person

    Electric field strength at a point due to 3 charges

    I got E. 13q as the answer. That is what i did: The electric field due to +q at origin 0 should equal the electric fields of charges -3q and the new charge placed at 2x. So applying the equation above like this; k*(q) / (2^2) = -3q*k + (k*C)/ 4 solving for C the new charge added, gives 13q. I...
  16. Salmone

    I Two molecules with different polarizability in an EM field

    If I have two separated and non-interacting molecules with different constants polarizabilities ##\alpha_1## and ##\alpha_2## and I send an EM field of frequency ##\omega## first on the molecule no.##1## and then on the molecule no.##2## so that the two molecules will have a dipole moment...
  17. JandeWandelaar

    A What is the cause of the Mexican hat potential of the Higgs field?

    The Higgs mechanism is an ingenious mechanism inspired by condensed state physics. The famous Mexican hat potential ensures a VEV value of about two times the mass of the Higgs particle (which, as an aside, is of comparable order as the W and Z vector bosons, the difference though is that its a...
  18. Delta2

    Potential energy of a sphere in the field of itself

    My attempt was to consider spherical shells of radius ##r## (##r\leq R##))and thickness ##dr## and then the potential energy of this shell would be in the field only of the "residual" sphere of radius ##r## (a result also known as shell theorem) $$U_{dr}=G\frac{\rho\frac{4}{3}\pi r^3 \rho 4\pi...
  19. WMDhamnekar

    Computing##\displaystyle\int_C f\cdot dr ## for the given vector field

  20. Tesla In Person

    Flux of Electric field through sphere

    My attempt: We have 3 charges inside 2 +ve and 1 -ve so i just added them up. 4 + 5 +(-7) = 2q Then there is a -5q charge outside the sphere. I did 2q + (-5q)= -3q . The electric field flux formula is Flux= q/ E0 . So i got -3q/E0 which is obviously wrong : ) . After quick googling , I...
  21. Tesla In Person

    Electric Field Inside a Conducting Sphere: Is it Always Zero?

    Is the electric field inside a sphere always 0? Even if we have charges on the surface?
  22. Eclair_de_XII

    B Can the continuity of functions be defined in the field of rational numbers?

    I argue not. Let ##f:\mathbb{Q}\rightarrow\mathbb{R}## be defined s.t. ##f(r)=r^2##. Consider an increasing sequence of points, to be denoted as ##r_n##, that converges to ##\sqrt2##. It should be clear that ##\sqrt2\equiv\sup\{r_n\}_{n\in\mathbb{N}}##. Continuity defined in terms of sequences...
  23. N

    Electric Potential Field Calculation

    I've already tried to calculate the potential with respect to the 3 segments and then apply superposition (V1+V2+V3). However, I was not very successful. My error I think is in the calculation of the radii, mainly of the line segment that is on the z axis. Can anybody help me? I need some light...
  24. HelloCthulhu

    Mathematically expressing field driven water autoionization

    I recently read a paper on using an electric field to drive water autoionizaton. I'm trying to figure out how to use the Laplace equation on pg 9; 4th paragraph; to solve for voltage. I'm also interested in how this equation would change if I replaced the hemispherical tip with a parallel plate...
  25. T

    B Notation for a "scalar absolute field"?

    The notation I think best describes it is ## F = \lVert\int^{space}_s|\vec{V}|ds\rVert ## So you have a vector field V in a 3d space. For each point you integrate over all of space (similar to a gravitational or electromagnetic field) *but* vectors in opposite directions do not cancel, they...
  26. LCSphysicist

    EMF induced in a wire loop rotating in a magnetic field

    To solve this problem, we need to evaluate the following integral: $$\epsilon = \int_{P}^{C} (\vec v \times \vec B) \vec dl$$ The main problem is, in fact, how do we arrive at it! I can't see why a Electric field arises at the configuration here. The magnetic field of the rotating sphere is...
  27. K

    I Postulating a minimum gravitational field strength

    this paper postulating a minimum gravitational field strength postulating a minimum gravitational field strength (minimum curvature) and a minimum acceleration but otherwise leaving Gr could reproduce MOND [Submitted on 25 May 2022] MONG: An extension to galaxy...
  28. Delta2

    I Is a Unified Field Theory the Key to Understanding the Universe?

    Is there any approach in any books out there, where we consider that in universe exists only one field, let it be called the Unified Field (UF), in which all of the known fields (gravitational, EM field, quark field, gluon field, lepton field, Higgs Field, e.t.c.) are just components (pretty...
  29. S

    I Magnetar Mystery: How Do Neutrons Generate a Magnetic Field?

    If a magnetar is a neutron star, how do the neutrons composing the star generate a magnetic field? A neutron has zero charge, so it generates no magnetic field.
  30. Adgorn

    Relativistic particle in uniform magnetic field (solution check)

    My solution was as follows: $$\frac {d\overrightarrow p} {dt}=q \frac {\overrightarrow v} {c}\times \overrightarrow B_0$$ The movement is in the ##[yz]## plane so ##|\overrightarrow v\times \overrightarrow B_0|=vB_0##, therefore: $$\biggr |\frac {dp} {dt}\biggr |= \frac {qvB_0} {c}.$$ On the...
  31. T

    I EM wave propagation: respective phase of E and M field

    Hi alltogether, I have been confused about a certain topic of EM wave propagation: it´s clear to me that E and M field are perpendicular to each other (I know Maxwell´s equations well). But: sometimes you can find on the internet that both fields are in phase...
  32. gremory

    A S-Matrix in Quantum Field Theory

    Hello, i need help with the S-matrix. From what i understand, with the S-matrix i would be able to compute the scattering amplitude of some processes, is that correct? If so, how would i be able to do that if i have some field ##\phi(x,t)## in hands? Is that possible?
  33. P

    I Exploring Newton's Third Law in an Imaginary Magnetic Field

    Hi, here's a theoretical problem that I am trying to find a satisfactory answer for. Imagine a coil that is temporarily switched on an off and generates a magnetic field that permeates through space. Now imagine a charged particle passing through this field, at time that the coil is already...
  34. W

    I Energy Density in E Field: Does it Contribute to Inertial Mass?

    Hi. I'm not sure where to put this question, it concerns particles, mass-energy equivalence and various things. Classical electromagnetism seems to be as sensible a place as any. There is energy stored in an E field. Energy density (at position r, time t) = \frac{1}{2}...
  35. shivajikobardan

    MHB Why do we need "total length" field in ipv4 datagram?

    TBH I don't really understand the question that I am asking myself. I was inspired asking this from my textbook "Do you know". Can you make me understand why would not we require total length field? Here's a similar question-...
  36. G

    I Solving the EM field equations to produce the desired vector field

    So, we have A, the magnetic vector potential, and its divergence is the Lorenz gauge condition. I want to solve for the two vector fields of F and G, and I'm wondering how I should begin##\nabla \cdot \mathbf{F}=-\nabla \cdot\frac{\partial}{\partial t}\mathbf{A} =-\frac{\partial}{\partial...
  37. HelloCthulhu

    How does an electric field create velocity in a water bridge?

    I've been researching water bridges and electrowetting to learn the effects of electric fields on water molecules but something continues to confuse me: if polar molecules can only rotate in an electric field, how is the water moving? Anyone familiar with this phenomenon? Any help is greatly...
  38. J

    The acceleration of protons using a changing magnetic field

    If we increase the magnetic field, the radius of the particle's circular path will decrease which increases the tangential acceleration. How do I find the tangential acceleration. Do I use derivatives?
  39. Ahmed1029

    I Average electrostatic field over a spherical volume

    this formula in the picture is the average electrostatic field over a spherical volume of radius R. It is the same expression of the electrostatic field, at the (position) of the point charge, of a volume of charge of uniform density whole entire charge is equal to (negative)q. My question is...
  40. J

    A curved wire rotating in and out of a magnetic field

    If I'm correct then the maximum change in magnetic flux occurs when the semi circle crosses the point at which it's plane is parallel with the magnetic field and minimal when it crosses the point at which the magnetic flux is maximum ( perpendicular with the field). I'm having trouble writing a...
  41. A

    Calculating a large toroid's magnetic field

    ##B_0 = \frac{\mu_0(N)(I)}{2\pi r}## ##\frac{N}{2\pi r} = 200## ##B_{net} = B_0 + B_m = (1+X)\times B_0## Plugging in the numbers: ##B_0 = 4\pi\times 10^{-7}(200)(1.5) = 3.8\times 10^{-4}## ##B_{net} = (1+X)\times B_0 = (1+ 3\times 10^3)\times 3.8\times 10^{-4}## = 1.13 T But the answers says...
  42. wnvl2

    Salt bridge and electical field

    Is it correct that a salt bridge in a Galvanic cell makes that there is no electrical field between the solutions of the two hallf cells? Does that mean that the electrical potential (I do not write electrochemical potential) between both solutions is zero in an ideal world? Is it also possible...
  43. J

    EMF generated in a blood cell by an oscillating magnetic field

    At first I tried plugging everything in with 60Hz in the numerator but that did not work. I was told to think about sinusoidal waves and derivates but I'm not sure how that works. Any ideas? Thanks a lot
  44. A

    Magnetic field generated by an infinitely long current-carrying wire

    Can someone explain how there can be a radial magnetic field? I thought the magnetic field was always tangent to the circle using the right hand rule where you wrap your fingers around the current and point your thumb in the direction of the current.
  45. J

    Find magnetic field at center of rotating sphere

    if a sphere rotates, it's like multiple currents going around in a circle. I can find the magnetic field of each of those currents at the center point of the circle and add them together. We can integrate with respect to y and R. y ranges from 0 to 5 cm away from the center of the loop and the...
  46. A

    Understanding Torque in a Magnetic Field with Loop

    I am confused about this, do the black arrows represent the direction of magnetic force? The torque ##\tau = -IABsin\theta##, where I = current A is area of loop and B is magnetic field strength and I am a little confused how ##\theta## here is 45 degrees when the angle between the normal for...
  47. Field physics

    Welcome! Investing, Athletics & Physics: Exploring New Fields

    Hi I am new to the forum. I am a investor, athlete, and a "field physicist." I got a bachelors in physics and Hopefully soon can get a phD. I like to do basic experiments that demonstrates physics.
  48. Sunny Singh

    I Understanding Landau Levels for 3D Fermionic Gas in Magnetic Field

    I am a beginning graduate student and I've been assigned a paper which uses landau levels for 3d fermionic gas in uniform background magnetic field. I am having trouble finding a proper source which deals with solution of dirac equation in such a case. With the two papers that i have found which...
  49. EmmanKR

    Finding the Frequency Domain and Time Domain magnetic field

    I was wondering if anyone could walk me though a better explanation on how to get the given results for these two questions. The solutions posted by my professor aren't really clear to me so if anyone is able to better explain how to get the solution it would be much appreciated!
  50. link223

    Calculating the Electric field for a ring

    What i don't understand is why we are able to replace the ring with 'two oppositely charged superposed disks'? Just trying to understand.. So we have a uniform charge which means that this'll just be a simplification of the problem than, correct? Thanks in advance.
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