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. milkism

    Field transformations in the z-direction

    Question: Eq. 12.109: My solution: We’ll first use the configuration from figure 12.35 in the book Griffiths. Where the only difference is that v_0 is in the z-direction. The electric field in the y-direction will be the same. $$E_y = \frac{\sigma}{\epsilon _0}$$ Now we're going to derive the...
  2. T

    I Prefactor in Canonical Quantization of Scalar Field

    Hey all, I am encountering an issue reconciling the choice of prefactors in the canonical quantization of the scalar field between Srednicki and Peskin's books. In Peskin's book (see equation (2.47)), there is a prefactor of ##\frac{1}{\sqrt{2E_{p}}}## whereas in Srednicki's book (see equation...
  3. K

    A Two level system under electric field without RWA

    Hello! I have the following Hamiltonian: $$ \begin{pmatrix} 0 & -\Omega\sin(\omega t) \\ -\Omega\sin(\omega t) & \Delta \end{pmatrix} $$ where ##\Delta## is the energy splitting between the 2 levels, ##\Omega## is the Rabi frequency of the driving field and ##\omega## is the frequency of the...
  4. quarkyphysicsgirl

    Electric field generated by solenoid

    The formula we are given is E=(1/2r)(alpha)R^2(muo)Ioe^-(alpha)t. However, I am struggling to figure out what each of the symbols stands for in the formula...can someone help me out? Like super confused on what alpha is in this case.
  5. S

    B Magnetic Field & Particle Spin: Does It Matter?

    Does the magnetic field caused by moving particles depend on the particle spin value? Eg a stream of say protons spin 1/2 is creating a magnetic field. If the particles are (say) lithium nuclei spin 3/2 instead, does that create the same strength field ? (same conditions of course)
  6. H

    Electric field at the center of the equilateral triangle

    I've found the distance from each point to the center, which is equal to r=20x1.732/3 = 11.55 cm. I find out that E2 and E3 due to -4µEyC on x-direction canceled each other. The E2y = E3Y = EY = E2Ycos60 = E2/2 = [(KQ2)/r^2]/2 So the net E-field: E = E1 +E2y+E3Y =kQ1/r^2 + [(KQ2)/r^2]/2 +...
  7. Tertius

    I Can a scalar field model account for the cosmic redshift?

    A minimally coupled scalar field can model a cosmological fluid model where And where the equation of state can be the standard ## \omega = \frac {p} {\rho}## I can see how this does a fine job modeling matter, because as the scale factor increases, the density will go as ##\frac {1} {a^3}##...
  8. C

    A Macroscopic field - seeking a thorough exposition

    Of the classical books about EM, I found that Jackson's is the only one that touches with some rigour the subject of deriving the macroscopic field from the microscopic one. Unfortunately, I am quite disappointed by the derivation of Jackson. In the reference he gives, he says that a couple of...
  9. rmrribeiro

    Wrong B-Field Result: What Should I Have Done?

    I integrated B within the limits of a (from 0 to 0.007) teh result was 3.64E-10 T and it was wrong. the correcto one would be 5.8 E-4 T and it is a major diference (aprox 1 million times ) Waht shoud I have done? Regards
  10. neroE

    Electric Field due to a disk of radius R in the xy-plane

    Hello, This question, which I found in various electricitiy and magnetism books (e.g. Introduction to electrodynamics grif.). There are many variations of this question, I am mainly interested in the following setup of it: -Suppose there is a charged disk of radius R lying in the xy-plane, and...
  11. milkism

    Magnetic- and Electric- field lines due to a moving magnetic monopole

    Question: My answer: What it looks like for an electric charge: Am I correct? If you want I can hand out my Latex on how I got to it, it will refer to the book Griffiths a lot.
  12. D

    A (nonconservative) electric field is induced in any region in which...

    TL;DR Summary: A (nonconservative electric field is induced in any region in which) A. there is a changing magnetic flux B. there is a changing magnetic field C. the inductive time constant is large D. the electrical resistance is small E. there is electrical current there can be more than one...
  13. H

    I Gravitational Field Existence in Void: Philosophical Inquiry

    This is of a more philosophical inquiry. If two particles are in a void and moving apart, if they are sufficiently far apart, like say the distance between two galaxy cluster walls, does the gravitational field between them still fundamentally exist? I'm trying to understand if gravity will...
  14. huangdaiyu

    A Steady state confined flow field: Is it cyclic?

    For a fluid that is confined to a finite region with no sources and sinks, are the only options for the flow field a) static, and b) cyclic? The example I have in mind is Rayleigh convection in a shallow dish heated from below, where convection cells are formed beyond a certain temperature...
  15. A

    A Schrodinger equation in quantum field theory

    What is the Schrodinger equation in QFT? is it the nonrelativistic approximation of a Klein-Gordon scalar field? or Is there more? I have read that the Schrodinger equation describes a QFT in 0 dimensions. I accept every answer
  16. S

    I Noether currents for a complex scalar field and a Fermion field

    For a complex scalar field, the lagrangian density and the associated conserved current are given by: $$ \mathcal{L} = \partial^\mu \psi^\dagger \partial_\mu \psi -m^2 \psi^\dagger \psi $$ $$J^{\mu} = i \left[ (\partial^\mu \psi^\dagger ) \psi - (\partial^\mu \psi ) \psi^\dagger \right] $$...
  17. Grizzly_1

    B Confusing diagram of a rotating coil in a magnetic field

    Hello all, I am currently studying for a physics a-level qualification in the UK, I use the AQA specification and I am having trouble understanding this image representing a scenario I found in my textbook. The first image in the three part diagram shows this rotating coil and to me, it makes...
  18. Idontknowhatimdoing

    Electric Field Inside Cylindrical Capacitor

    we know that flux is equal to the area integral of electric field dotted with dA and we can set this equal to charge enclosed divided by epsilon naught. Thus, in this case, the integral simplifies to E * A = (q_enclosed)/(ε_naught) when we choose a cylindrical gaussian surface with radius of r...
  19. A

    B Magnetic field coil ferrous material for loudspeaker

    Hi everyone. there are materials called soft magnetic and they are halfway between a permanent magnet and a ferromagnetic material. I would like to try to make a field coil loudspeaker where the maximum amount of flux-density is very important but also the amount of current it takes to create...
  20. J

    Thin heartshaped wire - Calculate the magnetic field at the origin

    Hi, So I know I am to use Biot Savarts law dB= (my_0/4pi)* (I dl x (r-r')/|r-r'|^3 where r=0 because its in origo and r'=r'_c(r'_hat). This makes (r-r')= -r'_c(r'_hat) and |r-r'|^3= r_c^3. From previous questions, I have defined dl' as the infinitesimal displacement of r'(phi) when phi' is...
  21. notgoodatphysics

    Electric field of a cylinder given the electric field of a ring

  22. J

    Struggling With Part C of Electric Field Calculation

    I've figured out parts A and B but I'm struggling with Part C. I used the equation V = kQ1/r1 + kQ2/r2 where Q1 = -4.4e-12C ; k = 8.98755e9 r1 = 0.026 m Q2 = 27.4e-12 and r2 = .051-.026 My answer (8.329 V) is wrong but I have no idea why. Please help if you can.
  23. V

    Why is electric field at the center of a charged disk not zero?

    The electric field strength at the center of a uniformly charged disk should be zero according to symmetry of concentric rings about the center, where each ring is contributing to the electric field at the center of the disk. For a thin ring of uniform charge distribution the formula is ##E =...
  24. O

    Motional EMF in the presence of a time-varying magnetic field

    My question arises from the following problem: We have a uniform magnetic field into the page, decreasing at a constant rate dB/dt< 0, causing the bar to move to the right. Find the velocity of the bar as a function of the time, and in terms of the known parameters: the resistor R and the...
  25. Pushoam

    Electric field at a point close to the centre of a conducting plate

    Since the electric field due to a conducting plate is twice the electric field due to a plastic plate having same charge density, the electric field at the point P will be twice in case of conducting plate and hence it is 20 volt per metre. Is that correct?
  26. A

    Relationship between magnetic field lines and magnetic field

    As stated in the problem, I want to demonstrate mathematically that field line density is directly related to the magnitude of B. How would I be able to do this, other than simply using the flux equation and showing that for a higher flux in the same area, the magnetic field must be rise...
  27. kuruman

    Finding a Parametric Solution for Particle Trajectory in Magnetic Field

    This is a solution to a problem inspired by another thread. It is posted here to separate it from the multiple choice question which was the subject of that thread. A parametric solution for the trajectory can be found quite easily if the motion is modeled as a particle with charge ##q##...
  28. 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##...
  29. 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...
  30. 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.
  31. C

    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!
  32. C

    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...
  33. 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)...
  34. C

    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!
  35. C

    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!
  36. C

    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...
  37. C

    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!
  38. 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...
  39. C

    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...
  40. C

    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!
  41. 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...
  42. 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...
  43. 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...
  44. 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...
  45. 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...
  46. 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...
  47. 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?
  48. 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...
  49. 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...
  50. 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...
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