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. L

    Direction of electric field vector on the surface of charged conductor

    Consider a negatively charged spherical conductor. On the surface of it, what is the direction of its electric field? Well, the definition of the direction of an electric field is the direction a positive test charge would go if placed at that point. But... it wouldn't move anywhere! So is the...
  2. H

    A Vector field vs. four scalar fields ("QFT and the SM", Schwartz)

    This is the statement in question: But if they were scalar fields, they would not transform at all. How could they contribute differently if they didn't change?
  3. H

    I What does "transforms covariantly" mean here?

    The Lagrangian, $$\mathcal L(x)= \frac 1 2 \partial^{\mu} \phi (x) \partial_{\mu} \phi (x) - \frac 1 2 m^2 \phi (x)^2$$ for a scalar field ##\phi (x)## is said to be Lorentz invariant and to transform covariantly under translation. What does it mean that it transforms covariantly under translation?
  4. A

    Magnetic Field Due To Wire Of Semi-Circular Cross Section

    I proceeded as follows Current in sector ##d\theta=## is: $$dI=\int_{x=0}^{x=R}{\frac{I}{\pi R^2/2}\times\frac{d\theta}{2\pi}\times 2\pi x dx}$$ Field due to sector ##d\theta## is therefore $$dB=\int^{x=R}_{x=0}{\frac{\mu_○}{2\pi x}\times\frac{I}{\pi R^2/2}\times\frac{d\theta}{2\pi}\times 2\pi x...
  5. K

    Calculation of the magnetic field of four quarter circles

    The diagram is something like this, and I want to calculate the magnetic field at the center using the Biot-Savart law. In this case, do the magnetic fields formed by the quarter circles pairwise alternate with each other?
  6. tanmays1

    Does levitation distance increase as Fg increase???

    I'm trying to conduct an experiment where I calculate the magnetic field strength of a magnet, by comparing the levitation distances between two magnets. My experiment involves using different masses to anchor down magnetic repulsion between 2 magnets. Fg = Fm. The formula I am using for this...
  7. R

    B Help with science experiment magnets/force/acceleration/speed

    Sorry, I guess I should have remembered all of this from my school days, but right now I have forgotten so much that I need some help. I am developing some simple experiments for school children (age ca. 12). This one involving magnets. I am not asking for detailed calculations, that is way...
  8. H

    Quantum 'Quantum Field Theory for the Gifted Amateur' by Tom Lancaster and Stephen J. Blundell as a self-study textbook?

    I'd like to hear your professional opinion on and experience with using Quantum Field Theory for the Gifted Amateur by Tom Lancaster and Stephen J. Blundell as a self-study textbook. Thank you.
  9. Slimy0233

    Calculating the flux of a vector field

    I am not sure why latex is not rendering, but here is the question. The answer is ##\frac{a^2}{8}## and for the love of my life, I don't know how. Can you please help me with this?
  10. paulimerci

    What is the magnitude of the field at point R?

    I've no idea how to solve this problem. The sign of the charge is not mentioned, so I'm assuming the charge is "+". The charge exerts an outward electric field. Since two lengths of the right-angle triangle are given, I use the Pythagorean to find the hypotenuse, which is the distance between q...
  11. paulimerci

    Find the magnitude of the electric field at point P

    There are two identical spheres with the same charge that are the vertices of an equilateral triangle. ##+3 \mu C## will exert an outward electric field, which is drawn in the FBD below (see the attached pic), Since the horizontal force components (1x and 2x) are equal and opposite at point P...
  12. T

    A The field equations of elasticity

    First, my ignorance... I know there are classes of equations: Laplace, Poisson, Wave, Diffusion, etc. (I suppose Laplace is a subset of Poisson, but that is not the issue). Into what category of mathematical equations would you place the field equations of elasticity (stress/strain/displacement)?
  13. carlsondesign

    I What is the official name for a Field Series in mathematics/physics?

    I've been working on developing infinitesimal recursion (what I call continuous hierarchy), but I ended up arriving at "field series" instead. My searches didn't seem to come up with anything reasonable (battlefield the video game series), so I'm wondering what the official name for a field...
  14. harryharns

    C/C++ Perform RK4 between 2 Clusters in Magnetic Field

    I have a simulated data of charged particles in a magnetic field. I have selected clusters, each cluster contains a set of points(x,z) and I want to perform RK4 between the first and second clusters and fill the positions in a histogram. I have selected the clusters with the initial...
  15. ergospherical

    I Average magnetic moment of atom in magnetic field ##B##

    from the partition function - am trying to show that ##\langle \mu \rangle = \beta^{-1} (\partial \log Z / \partial B)## where ##Z## is the canonical partition function for one atom, i.e. ##Z = \sum_{m=-j}^{j} \mathrm{exp}(\mu_0 \beta B m)##, and ##\mu = \mu_0 m##. The average...
  16. C

    Work energy KE theorem for a book being lifted up in a gravitional field

    For this, From the work kinetic energy theorem, if we assume that the book and the earth is the system, and that the finial and inital speed of the system is zero, then is the work KE theorem there is no net work done on the system. However, clearly there is work done on the system is shown by...
  17. D

    B Magnetic field and generator power output

    Hi, I am confused about whether decreasing the magnetic field used for a generator could increase the generator's power output. I used four equations: 1. Torque = Force x radius 2. Torque = NIAB (N = number of turns, I = current, A = area of armature, B = magnetic field). 3. emf =...
  18. StenEdeback

    Classical Looking for book about relativistic classical field theory

    Hi, I am trying to learn relativistic classical field theory as a preparation for studying quantum field theory. I am currently reading chapter 13 i Herbert Goldstein's Classical Mechanics edition 3, but I think that this book is a bit too brief and does not fully derive and explain the...
  19. W

    B Exploring the Electric Field of a Moving Charge

    How does an electric field of a moving charge, for example a moving electron, inside a wire looks like? Does it looks like this with distorted circular radial lines?
  20. I

    I Exploring Field Strength & Singularities

    Why doesn't this work if the field is strong? Or does it work as long as there are no singularities? Mentor's Note: Original thread title was, "Calculating rest mass by integrating T_{00} over a 3 volume for static metric"
  21. M

    Electric field amplitude question regarding intensity calculations

    My question is specifically with calculating the intensity. The book solution is I=P/(4*pi*r^2) but would this not give me a weaker electrical amplitude in the final calculation after plugging it in to I=(1/2)*√(ε0/μ0)*(E02) ?
  22. CaliforniaRoll88

    Multi-part question involving a vector field

    (a) ##\vec G=24xy\hat a_x+12(x^2+2)\hat a_y+18z^2\hat a_z## @ ##P(1,2-1)## ##\vec G=24(1)(2)\hat a_x+12(1^2+2)\hat a_y+18(-1)^2\hat a_z## ##\vec G=48\hat a_x+36\hat a_y+18\hat a_z## (b) I am not sure how to get this part started. Could someone point me in the right direction?
  23. ahmadphy

    I The effect of a field on a particle depends on the particle velocity?

    Assume there is a force (vector field) on the space .....does the effect of this field on the particle(the change of momentum) at some position depend on the speed at that position? And is it related to the time interval dt the particle experiences this force ? Can i say dt=dx/v? And is that...
  24. milkism

    A diamagnetic sphere in a uniform magnetic field

    Problem: Solution part a) where formula 6.14 is just M x n. We need to do part b without seperation of variables, I'm quite stuck. Will B just be the magnetic field inside a solenoid? How can I find the other fields.
  25. W

    B Seeing both B field lines and E field lines at the same time

    After watching this clip Electric Field Lines Lab I wonder if it is possible to see both electric field lines and magnetic field lines at the same time by swapping the two nails in the video with two bar magnets, as the conductors as we understand bar magnets are metals and metals are good...
  26. M

    Electron in a time variable magnetic field

    TL;DR Summary: Find acceleration of electron in dB/dt >0 Hello. Here is a problem that i'm not so sure about: Inside a solenoid there is a time-dipendent magnetic field B, so we have dB/dt = b (constant). We want to know the acceleration of an electron: a) placed in the center of the solenoid...
  27. S

    I Gravitational field modulation of a moving mass

    If a mass is moving it has a wavelength - my question is, is the gravitational field of the mass then modulated accordingly ?
  28. R

    A Bound states of an electron trapped in a dipole field

    The problem of bound states of an electron trapped in a dipole field is being studied by Alhaidari and company. (See, for example, https://arxiv.org/ftp/arxiv/papers/0707/0707.3510.pdf). It is not clear to me why the point dipole approximation is used everywhere in such calculations. Can't an...
  29. R

    A Aligning effect in uniform field

    I would like to discuss the nature of the following effect. At whatever angle and with whatever initial speed the particle fly into a uniform potential field, over time the directions of the instantaneous velocity and field strength converge. The kinematics and dynamics here are trivial, but I...
  30. L

    Divergence of the Electric field of a point charge

    Hi, unfortunately, I am not sure if I have calculated the task correctly The electric field of a point charge looks like this ##\vec{E}(\vec{r})=\frac{Q}{4 \pi \epsilon_0}\frac{\vec{r}}{|\vec{r}|^3}## I have now simply divided the electric field into its components i.e. #E_x , E-y, E_z#...
  31. N

    I Electric Field Shielding by Conducting Sheets

    Dear Experts, When a thin conducting sheet with no charge on is placed at a certain distance from a point charge, does it shield the electric field caused due to the point charge from reaching the other side of the sheet. As an extension of that idea, when a conducting sheet or slab is placed...
  32. milkism

    Solving a Problem with Interchanging Field Tensors

    Exercise: Solution: The result is correct, but I'm unsure about equation from 29 to 30 where right-hand side became just the covariant dual field tensor. I assumed that I could interchange the covariant dual- and normal covariant field tensor, but don't think it's possible since matrices...
  33. O

    I need help with a Baldor dc motor, 15kw 240 volt ,150 volt field

    I need help with a Baldor dc motor, 15kw 240 volt armature and 150 volt field, I don't need the 15kw and so want to feed it with 150 volts to both field and armature, does anyone know the effect this will have on the motor, can I cause damage, what would be the approximate power output, I know...
  34. simonjech

    I Necessity of the field viewpoint

    I am learning QFT rn, so the question that I naturaly ask myself often is why do we have to use field operators in relativistic quantumn theory instead of operators with finite number of degrees of freedom which are used in non-relativistic quantumn mechanics? One of the reasons that I came...
  35. S

    Magnetic field force between two perpendicular wires

    There are a couple of problems with the same setup. On plugging in the values and solving for the integral, I am not getting the expected values of the force. Is there something wrong in the solution attached?
  36. K

    A Two level system in a thermal noise electric field

    Hello! I have a 2 level system with a dipole moment d. I want to simulate numerically the evolution of the system under an external sinusoidal electric field (far off resonant). This is straightforward using SE. However I also have on top of that another electric field, created by a coupling of...
  37. S

    I Propagator of massless Weyl field

    I have this Lagrangian for a free massless left Weyl spinor, so it’s just the kinetic term, that can be written embedding the field into a larger Dirac spinor and then taking the left projector in this way: $$i \bar{\psi} \cancel{\partial} P_L \psi$$ Srednicki says that the momentum space...
  38. AHSAN MUJTABA

    A Scalar Field Dynamics in Inflation

    I am facing a problem while wanting ##\phi## dynamics in a cubic potential; ##g\phi^{3}##. The equation of motion I get for my case is(this follows from the usual Euler-Lagrange equations for ##\phi## in cosmology--Briefly discussed in Carol's Spacetime Geometry, inflation chapter)...
  39. F

    Determining Radius from Magnetic Field of a Single-Wire Loop

    So I thought I knew how to do this problem but I've run into some issues that make the algebra feel impossible and I am beginning to feel like I'm taking the wrong approach, I ended up rewriting it in a doc because I was concerned maybe my handwriting was the cause of my error so the work is...
  40. O

    Understanding the energy of a dipole in a uniform electric field

    Griffith's E&M problem 4.7 asks to calculate the energy of a dipole in a uniform electric field and I ended up getting a different answer than the one given. I thought that calculating the energy/work done to construct the dipole is the same as dragging two point charges where one is d apart...
  41. Brian Tsai

    I A math confusion in deriving the curl of magnetic field from Biot-Savart

    I am recently reading "Introduction to Electrodynamics, Forth Edition, David J. Griffiths " and have a problem with the derive of the curl of a magnetic field from Biot-Savart law. The images of pages (p.232~p233) are in the following: The second term in 5.55(page 233) is 0. I had known...
  42. Drakkith

    I Units of q in Electric Field Equation

    Quick and possibly stupid question, but in the equation for calculating the electric field: ##{\mathbf E} = \frac{1}{4πe_0}\frac{q}{r^2} \hat {\mathbf r}## What unit is ##q## in? Coulombs? Although now that I think more on it I suppose it also depends on the units you're using to calculate the...
  43. jed1408

    I Can angular motion exist outside of a gravity field?

    I can't find the answer anywhere here's my question. can a centrifuge exist outside a field of gravity. More specifically, in a theoretical void of nothing without stars in view or any point of reference for comparison how could motion like spinning or acceleration exist?
  44. 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...
  45. 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...
  46. 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...
  47. 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.
  48. 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)
  49. 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 +...
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