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.

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  1. Adesh

    Will the induced electric field be circumferential?

    Let’s say we have a right handed Cartesian system and magnetic field goes in positive z direction, and let’s assume that the magnitude of magnetic field varies with time. Now, if I draw a circle with radius ##r## in the ##x-y## plane and let the magnetic field pass through it and vary with...
  2. H

    Electric field from a charge q1

    Hi, I have a charge q1 = -10 * 10^9. The the coordinatesare (3,4)m. I found the electric field vector that is (-2160i -2880j) n/c. My questions is if I add a charge q2 to the the coordinates(0,0) is the electric field stay the same?
  3. T

    I Electromagnetic Radiation in Strong Gravity Field

    Does the electric and magnetic fields of electromagnetic radiation remain perpendicular in the presence of an intense gravity field? If not, what is the physical ramifications of this?
  4. Elder1994

    Verify Stokes' Theorem for this vector field on a surface

    I do not understand how can I parameterize the surface and area and line differentials.
  5. B

    Revolving disc in a magnetic field

    I know that there is a known equation to calculate the magnetic field strength of a rotating disc which I have made use of in here Do you agree that revs per minute of the disc turns out to be 9337 rpm? Thanks for any help! Much appreciated
  6. B

    Calculate the charge flowing in the coil if the field is removed

    Hello So here is my question Not so sure how to approach this question This is what I have worked out so far which is the magnetic field strength of the solenoid, not sure if this comes in helpful though Thanks for any help!
  7. D

    I Sufficient condition for a vector field to be conservative

    Homework Statement:: F is not conservative because D is not simply connected Relevant Equations:: Theory Having a set which is not simply connected is a sufficient conditiond for a vector field to be not conservative?
  8. S

    Why is the Answer C? Electric Field in Conductors

    The answer according to the key is C. I thought the answer would be E since the electric field inside a conductor is always zero. Can someone explain why the answer is C?
  9. B

    Where Will Earth's Magnetic Field Cancel a Wire's B-Field?

    Hey everyone So this is question shown below I'm not so sure where to begin with this, but I thought I'd work out the net magnetic field first How would I work out the magnetic field strength that is acting on the vertical current-carrying wire. Since I do not know what d is in this case...
  10. W

    I Propagator of a Scalar Field via Path Integrals

    I don't understand a step in the derivation of the propagator of a scalar field as presented in page 291 of Peskin and Schroeder. How do we go from: $$-\frac{\delta}{\delta J(x_1)} \frac{\delta}{\delta J(x_2)} \text{exp}[-\frac{1}{2} \int d^4 x \; d^4 y \; J(x) D_F (x-y) J(y)]|_{J=0}$$ To...
  11. D

    Work of a vector field along a curve

    let ##f : R^3 → R## the function ##f(x,y,z)=(\frac {x^3} {3} +y^2 z)## let ##\gamma## :[0,## \pi ##] ##\rightarrow## ##R^3## the curve ##\gamma (t)##(cos t, t cos t, t + sin t) oriented in the direction of increasing t. The work along ##\gamma## of the vector field F=##\nabla f## is: what i...
  12. D

    Outward flux of a vector field

    My idea is to evaluate it using gauss theorem/divergence theorem. so the divergence would be ## divF = (\cos (2x)2+2y+2-2z ( y+\cos (2x)+3) ) ## is it correct? In this way i'ma able to compute a triple integral on the volume given by the domain ## D = \left\{ (x, y, z) ∈ R^3 : x^2 + y^2 +...
  13. S

    Antisymmetry of the electromagnetic field tensor

    I am trying to answer exercise 5 but I am not sure I understand what the hint is implying, differentiate with respect to ##p_\alpha## and ##p_\beta##, I have done this but nothing is clicking. Also, what is the relevance of the hint "the constraint ##p^\alpha p_\alpha = m^2c^2## can be ignored...
  14. S

    A Mode Field Diameter and Penetration depth of the Evanescent field

    Source: fobasics.blogspot.com Source: scirp.org As it is shown in the first pic above that the mode field diameter is defined as the mode field decreases to 1/e (in intensity 1/e^2), if I take the mode field and subtract the core's diameter then I divide it by 2, should I get the penetration...
  15. Spinnor

    I Photon Double Slit Experiment: Visualizing the Guiding Field

    Does the following picture which I think shows the guiding field for electron in the double slit experiment have a corresponding image when the experiment is done with photons? Thanks for any help.
  16. Spinnor

    I Bohm guiding field, electrons in a semiconductor "box"

    Consider the light sensor in a modern camera. Light can give energy to electrons and populate the numerous "boxes" of our light sensor with extra electrons. Those boxes will temporally store the electrons till they are counted. I would like to understand this process with my "Bohmian" glasses...
  17. D

    Compute the flux of a vector field through the boundary of a solid

    is it correct if i use Gauss divergence theorem, computing the divergence of the vector filed, that is : div F =2z then parametrising with cylindrical coordinates ##x=rcos\alpha## ##y=rsin\alpha## z=t 1≤r≤2 0≤##\theta##≤2π 0≤t≤4 ##\int_{0}^{2\pi} \int_{0}^{2} \int_{0}^{4} 2tr \, dt \, dr...
  18. D

    Flux of a vector field through a surface

    Given ##F (x, y, z) = (0, z, y)## and the surface ## \Sigma = (x,y,z)∈R^3 : x=2 y^2 z^2, 0≤y≤2, 0≤z≤1## i have parametrised as follows ##\begin{cases} x=2u^2v^2\\ y=u\\ z=v\\ \end{cases}## now I find the normal vector in the following way ##\begin{vmatrix} i & j & k \\ \frac {\partial x}...
  19. A

    Magnetic field lines and magnetic flux density

    I'm trying to understand the relationship between the "number" of field lines passing through a region and the magnetic force in this region.I understand that the drawings are of course conceptual: we cannot draw "all" the field lines (although can be visualized with iron fillings).Also the...
  20. F

    Electric field Difference between Electrostatics and Electrodynamics

    Hello everyone, I have been pondering on the behavior of the E field in conductors. In electrostatics (where the charges are not moving): a) Electric fields are time- independent but position-dependent b) Electric fields are always zero inside a charged or uncharged conductor. At the...
  21. PeroK

    I What do we do with the massive scalar quantum field in QFT?

    I'm learning some QFT from QFT for the Gifted Amateur. Chapter 11 develops the massive scalar quantum field but they don't seem subsequently to do anything with it. I've looked ahead at the next few chapters, which move on to other stuff, which leaves me wondering what we we actually do with...
  22. S

    I Electric Field Directly Ahead of or Behind a Moving Charge

    Since it is stated that ##E'_x = E_x##, I am going to set a special case where ##z' = z = 0##, ##E_x## in (5.10) reduces to, ##E_x = \frac{1}{4 \pi \epsilon_0}\frac{Q}{x^2}## However, ##E'_x## in (5.13) reduces to, ##E'_x = \frac{1}{4 \pi \epsilon_0}\frac{Q}{\gamma^2 x'^2}## There is an...
  23. Athenian

    [SR] - Test Particle inside the Sun's Gravitational Field - Part 4

    So, here's an attempted solution: With ##r_{min}##, $$r_{min} = \frac{1}{B + \frac{\beta}{\alpha^2}}$$ With ##r_{max}##, I get: $$r_{max} = \frac{1}{B - \frac{\beta}{\alpha^2}}$$ or $$r_{max} = \frac{1}{\frac{\beta}{\alpha^2}}$$ Other than this, I and the team have absolutely no idea on how...
  24. jisbon

    How Do Symmetrical Charges Affect Net Electric Field in a Circle?

    In this case, I know there won't be any net efield in the x direction because it cancels out with each other. The problem is dealing with the y axis. Am I supposed to presume an angle for each of them or what should I do instead? Thanks
  25. VictorMedvil

    A Superconductivity: Current and Magnetic Field Limitations

    Why when a certain current limit is breached is superconductivity destroyed in a material, what atomically causes this effect when J > Jc? Secondary question what causes H0's value to be higher or lower atomically and chemically for a given material?
  26. P

    Charged proton enters an electric field

    I tried to do Net force with electric field = E x q minus the gravitational force= mg. However, this gives me a negative net force suggesting the proton is moving downwards. I'm not sure this is correct as the initial velocity was horizontal. Was there no gravitational force before? Am I missing...
  27. B

    Field Evaporation of Contact Charged Spheres: Exploring Possibilities

    If you were to positively contact charge a small ~1 mm diameter sphere using a Van de Graaff generator, and were to charge it sufficiently high enough that field evaporation began to occur, what would happen? Would the rate of evaporation increase exponentially as the field strength would...
  28. jisbon

    Energy band gap when there is an electric field

    So I have just been taught this topic but this question seems to be one of a kind and I can't seem to figure it out. What I've learnt: When there is a positive electric field applied to the right, for example, the electrons that are free moving in a crystal (aka conducting band) will oppose...
  29. Athenian

    [SR] - Test Particle inside the Sun's Gravitational Field - Part 3

    Below, I have already solved - I assume - correctly for question 1. Question 2, I am nearing to what I believe is the solution. Question 3, I simply have no idea where I should begin considering that it is interconnected with question 2. With that said, below is the lengthy and somewhat tedious...
  30. Adesh

    Calculating the magnetic field of an infinite solenoid

    Here is the image ## \tan \theta _1 = \frac{a}{z} ## ## \tan \theta _2 = \frac{a}{l+z}## where l is the length of the solenoid and z is the distance from the forward center to the point P. My doubt is how ##\theta_1## going to become 0 and ##\theta_2## ##\pi## as the length of solenoid...
  31. Athenian

    [SR] - Test Particle inside the Sun's Gravitational Field - Part 2

    To begin with, I posted this thread ahead of time simply because I thought it may provide me some insight on how to solve for another problem that I have previously posted here: https://www.physicsforums.com/threads/special-relativity-test-particle-inside-suns-gravitational-field.983171/unread...
  32. P

    Magnetic field Calculation of a Square Wire Loop (with a changed segment)

    I tried to solve the above i have one confusion here. I have marked the areas as shown B2 = B4 = 0; B1 , B5 Out of Page ; B3, B6 Into the Page. B1 and B5 Calculation Now main doubt is regarding the B field of the finite wire let us say 1. I took the derivation of the infinite wire as below from...
  33. Decimal

    Spectral density of radiative electric field

    So I have to find an expression for ##\vec{A}(\omega)##, $$\vec{A}(\omega) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} \vec{A}(t)e^{i\omega t} dt.$$ This point is where my confusion comes up. In the answer sheet they integrate over the retarded time ##t_r##, so the integral is...
  34. saadhusayn

    A Calculating the ghost field in the Becker and Becker paper

    This is the paper that I refer to. I'm trying to figure out the ghost action (Equation 2.16) in the background field gauge. I am attempting to use Srednicki's (chapter 78) expression for the ghost field in the background gauge. However, I am missing out on a √g coefficient in front of the term...
  35. H

    Electric field between two capacitor plates (proof)

    See attached image
  36. Athenian

    [Special Relativity] Test Particle Inside the Sun's Gravitational Field

    Below is an attempted solution based off of another user's work on StackExchange: Source: [https://physics.stackexchange.com/questions/525169/special-relativity-test-particle-inside-the-suns-gravitational-field/525212#525212] To begin with, I will be using the following equation mentioned in...
  37. D

    Electric field involving 4 point charges in a rectangle

    I am stuck on the following question (Image attached of my work) appears to make sense until i try to take a limit as c--->0 because the result should be 0. Am i missing something, if so can't you point me in the right direction. Thank you
  38. P

    How do I find the angle and direction of a magnetic field?

    Again struck up with the direction of the magnetic field, i suppose now the field not simple along the x axis. How to find the angle and the direction of the field. My attempt is B1 = (μ*i)/(2*π*r) = (4*π*4)/(2*π*4) = 200nT where r = d2 = 4; is the field due to i1. B2 = (μ*i)/(2*π*r) =...
  39. S

    Correct statement regarding a conductor in a magnetic field

    By using Fleming's Left Hand Rule, I got the force acting on proton is directed upwards so my answer is (d) but the answer key is (a). So the force acting on proton is actually downwards? Thanks
  40. P

    What Is the Magnetic Field at Point P on the Y-Axis?

    The problem is as above, My attempt is as below but there is lot of effort in terms of imagining and not very confident, Required the magnetic field on the y-axis let us say point P. The magnetic field due to the x-axis wire is out of the paper at P with the values as R=2.0m, i =30A. B1 =...
  41. P

    Magnetic field due to two loops

    My attempt is the magnetic field due to loop1 and loop2 should get added The magnetic field due to loop1 is B1 =(μ0 * Φ * i)/(4*π*r) = (4*π*(2*π)*0.004) /(4 *π*0.015) = 1670nT. I assumed this value should be less than 100nT. What is the reason? The other question is "Loop 2 is to be rotated...
  42. M

    I Field Renormalization vs. Interaction Picture

    When introducing renormalization of fields, we define the "free Lagrangian" to be the kinetic and mass terms, using the renormalized fields. The remaining kinetic term is treated as an "interaction" counterterm. If we write down the Hamiltonian, the split between "free" and "interaction" terms...
  43. sunrah

    Job Skills Astro MRes after PhD in another field?

    How would you respond to postdoc applicants if they hold a PhD in another field (physical oceanography) and then complete an MRes in astrophysics? Just interested because I don't qualify for funding for any of the astronomy and cosmology PhD positions I'v found in the UK (even though I am...
  44. S

    Gradient Force of an optical near field

    Source: Principles of Nano-Optics, for Lukas Novotny and Bert Hecht.The equations above represent the electric field in the second medium when a light hit a surface and the condition of TIR (total internal reflection) is satisfied. Actually this is what called Evanescent field. The point is if I...
  45. A

    A Visualizing water molecular dynamics in an electromagnetic field

    I am interested in showing a visualization of water molecules in a time-varying electric/magnetic field as part of my PhD work. I would like something like this visualization: , but with an external time-varying field applied. At first, I thought of simply animating water molecules...
  46. G

    Gauss-Theorem on a solid dielectric sphere

    The load system formed by the point load and the load distribution generates two regions in space corresponding to r<1m and r>1m, i.e. inside and outside the sphere. Given the symmetry of the distribution, by means of the Gaussian theorem we can find the modulus of the field at a distance r from...
  47. Tiptoeingelephants

    I Is a Particle Simply the Manifested Kinetic Energy of Its Quantum Field?

    Trying to better understand quantum field theory, I've read that particles are created when it becomes an exitation of its quantum field. Would it then be right to think of a particle as the manifested kinetic energy of its field?
  48. S

    Other What field would be the most employable in 2020

    Hi everyone! I have during my time here on PF have posted a number of threads related to employment demand for STEM majors, including the following back in 2017: https://www.physicsforums.com/threads/which-stem-field-could-be-the-most-employable-in-2017.898554/ I wanted to take the...
  49. S

    Engineering Magnetic field near a rectangular bus bar

    An old field theory notebook has given me a formula for a long straight conductor that H = I/2πd which suggests 2.3873T at 0.2mm. Is it a reasonable approximation to use this as a basis for selecting the sensor? Any help much appreciated.
  50. sakh1012

    A Dirac Field quantization and anti-commutator relation

    Can anyone explain while calculating $$\left \{ \Psi, \Psi^\dagger \right \} $$, set of equation 5.4 in david tong notes lead us to $$Σ_s Σ_r [b_p^s u^s(p)e^{ipx} b_q^r†u^r†(q)e^{-iqy}+ b_q^r †u^r†(q)e^{-iqy} b_p^s u^s(p)e^{ipx}].$$ My question is how the above mentioned terms can be written as...
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