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. 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}...
  2. 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...
  3. 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...
  4. 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...
  5. 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...
  6. 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...
  7. jisbon

    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
  8. 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?
  9. 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...
  10. 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...
  11. 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...
  12. 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...
  13. 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...
  14. 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...
  15. 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...
  16. 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...
  17. 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...
  18. H

    Electric field between two capacitor plates (proof)

    See attached image
  19. 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...
  20. 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
  21. 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) =...
  22. 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
  23. P

    Magnetic field in 3 dimensions

    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 =...
  24. 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...
  25. M

    I Field Renormalization vs. Interaction Picture

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  26. sunrah

    Job Skills Astro MRes after PhD in another field?

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  27. 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...
  28. 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...
  29. 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...
  30. 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?
  31. 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...
  32. 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.
  33. 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...
  34. Z

    Find the electric field on the surface of a sphere using Coulomb's law

    Note that the solution is 5625 V/m in z direction which is found easier using Gauss' law, but I want to find the same result using Coulombs law for confirmation. Lets give the radius 0.04 the variable a = 0.04m. ##\rho## is the charge distribution distributed evenly on the surface of the...
  35. icesalmon

    Engineering Determination of a magnetic field for a project

  36. Diracobama2181

    Potential and E field for a non homogeneous charge Density

    Based on the conditions, I found that $$V(x)=\frac{a^2}{\pi^2} ρ_0sin(πx/a)$$ would be a solution to Laplace's equation for $$|x|\leq a$$ and $$V(x)=cx+d$$, where c and d are constants. From the boundary conditions, $$\frac{dV(a)}{dx}=\frac{a}{\pi} ρ_0cos(πa/a)=ac$$, $$c=\frac{a\rho}{\pi}$$ and...
  37. astrocytosis

    Magnetic field of a rotating cylinder with permanent polarization

    I am struggling to get my work to match the posted solutions to this problem. I understand part (a) but can’t get the integral to work out for (b). I know I have to use Biot-Savart and add up the components from the the surface and volume currents. The cylinder is very long, so I need to make a...
  38. Saptarshi Sarkar

    Electric field inside a polarized dielectric sphere

    My attempt: I know from Gauss' law in dielectric ##\nabla .D = ρ_f## where ##D = ε_0E + P##, so as ##ρ_f = 0## (as there is no free charge in the sphere) => ##\nabla .D = 0## => ##ε_0\nabla .E = \nabla .P## from this I get ##E = \frac {-kr^2 \hat r} {ε_0}## But, I know that for a uniformly...
  39. D

    Metal bar moving in a magnetic field

    When the bar moves ,magnetic force is applied to all the charged particles.At the equilibrium the electric and magnetic force have the same value and the positive and negative charges are accumulated in the two sides of ab. Eq=qVB, E=VB Here is when cannot continue,I thought it like a capacitor...
  40. G

    I A one dimensional example of divergence: Mystery

    I am trying to understand “divergence” by considering a one-dimensional example of the vector y defined by: . the parabola: y = -1 + x^2 The direction of the vector y will either be to the right ( R) when y is positive or to the Left (L). The gradient = dy/dx = Divergence = Div y = 2 x x...
  41. fisher garry

    Magnetic field inside a solenoid

    I have a problem with the derivation above I don't get how Can someone derive this and illustrate this visually for example by using Figure 2 or using another drawing?
  42. C

    Electrical potential of a thin wire in an E field

    Assume that an infinite metallic plate A lies in the xy-plane, and another infinite metallic plate B is parallel to A and at height z = h. The potential of plate A is 0, and the potential of plate B is constant and equal to V. So, there is a uniform electrostatic field E between plates A and B...
  43. S

    The electric field of a piecewise uniform 1D charge distribution

    This is not really homework, but I'm having trouble understanding it intuitively. I came across this when learning about the space charge layer of a diode. The solution I know simply uses the 1D form of Gauss's law: ##\vec{\nabla} \cdot \vec{E}## = ##\dfrac{\rho}{\epsilon_0}## becomes...
  44. Saptarshi Sarkar

    Calculating total charge when the electric field is given

    I first tried to use the Gauss' law equation E.A = q/ε0 to find the total charge enclosed. The answer came out to be q(enclosed) = 4πqε0e^(-4r). So for r approaching infinity, q(enclosed) approached 0. Next, I tried the equation ∇·E = ρ/ε0, calculated rho to be -4qε0e^(-4r)/r^2 and total...
  45. T

    Spin Orbit Coupling + Magnetic Field

    I am pretty confused where to even start with this question, which is not a good thing less than a week before the final :(. One thing in particular that I don't get is that I thought we were using the Clebsch-Gordon coefficients for ##\vert jm \rangle ## states, not for ##\vert J, J_z \rangle...
  46. AndrewC

    Magnetic field intensity, flux density and magnetization of coax cable

    Inner conductor radius = 1cm outer conductor radius = 10cm region between conductors has conductivity = 0 & 𝜇r = 100 𝜇r = 1 for inner and outer conductor Io = 1A(-az) 𝑱(𝑟) = (10^4)(𝑒^-(r/a)^2)(az) Problem has cylindrical symmetry, use cylindrical coordinate system. Find the total current...
  47. askcr9

    A slab? Infinite area? Electric field? Help please

    The first time I saw this question I had no idea how to do it (as you can see in the figure, I lost a lot of points :s) because I was confused on how to even approach it with area of the slab from all sides being infinity. Right? That's problematic, no? Today, I just tried the problem again for...
  48. fisher garry

    E field calculation for q sphere

    I have some questions about this answer. Why do they use absolute value when writing in the limits in the integral underlined with orange? And how do they get from this value where I have underlined with orange to the answer for E outside the sphere. Can someone do the rewriting? And last why is...
  49. Moara

    Wire rotating inside a magnetic field

    For a infinitesimal wire of lengh dx, the induced potential difference in an uniform B field perpendicular to it's motion is : dE=B.Vp.dx, where Vp is the component of the velocity perpendicular to the wire. Looking to the big wire I tried to take an arbitrary point express dE in function of...
  50. Moara

    Electron moving inside a region of homogeneous electric field

    a) since the eletric field is perpendicular to the inicial velocity, the x component is constant, hence Vf.cos45=Vo. This gives Vf=0,6√2.C b) Ei=γi.Eo , γi=5/4 , Ef=γf.Eo , γf=5/(2√7) Finally, Ei+e.E.d=Ef. Apparently this is incorrect, why??
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