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. 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!
  2. 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.
  3. L

    Flux of constant magnetic field through lateral surface of cylinder

    If the question had been asking about the flux through the whole surface of the cylinder I would have said that the flux is 0, but since it is asking only about the lateral surfaces I am wondering how one could calculate such a flux not knowing how the cylinder is oriented in space. One could...
  4. C

    I Understanding the critic of this author for conventional S field

    I joined an article by Davis and Onoochin. I have troubles to understanding his critic of the conventional field momentum interpretation (sec. 3). More precisely, while I am able to check that eq. (23) is true, and that the field is of the order ##1/r^5##, I don't understand in which way this...
  5. Gbl911

    I Average Magnetic Field Between 2 Conducting Rods

    I am building small, simple version of a railgun using 2 copper bars and a couple of neodymium magnets to increase the magnetic field. I have also been trying to mathematically describe the magnetic field created by the conducting rods themselves. I am coming across some trouble when trying to...
  6. Y

    Programs Certificate and Degree in the same field?

    Hello friends, Is it a waste of time to get both a degree and certificate in the same field? If for example I received a graduate certificate and a masters degree in Mechanical Engineering, does having a flow blown masters degree sort of overwrite the certificate credential? While it's nice to...
  7. PainterGuy

    I Does the Higgs Field Consist of Higgs Bosons?

    Hi, I'd request you to keep it simple so it's accessible by a layman. Question 1: Is Higgs field made up of Higgs bosons? Higgs field has a positive value everywhere. Other fields such as electron field, hover around zero though virtual particles come into existence and decay almost...
  8. Mr_Allod

    Total Josephson current through junction with magnetic field

    Hello there, I am given a diagram of a Josephson Junction like so: With a magnetic field ##B = \mu_oH## in the z-direction. I'm reasonably sure ##d_x,d_y,d_z## are normal lengths, not infinitesimal lengths although that is up for debate. Using the above equations I rearrange the expression...
  9. BnayaMeir

    Surface Current and Electric Field

    Hi everyone! I'm pretty new in this forum, I found the topics here very relevant to my physics course. And here is my question: Given the following drawing, two infinite sheets (in y and z axis) of ideal conductive material. their thickness is infinitesimal (dx->0). The electric field is...
  10. A

    Understanding Electric Field of Positive & Negative Plates

    I am confused with the solution. It says ##\vec E = \frac{\sigma}{\epsilon_0}##. Shouldn't E = ##2*\vec E = 2*\frac{\sigma}{\epsilon_0}##? Electric field of the positive plate and electric field of the negative plate.
  11. ErikZorkin

    I Simulating physics: the current status of lattice field theories

    I recently watched this video by David Tong on computer simulation of quantum fields on lattices, fermionic fields in particular. He said it was impossible to simulate a fermionic field on a lattice so that the action be local, Hermitian and translation-invariant unless extra fermions get...
  12. A

    I Electric field is zero in the center of a spherical conductor

    Electric field is 0 in the center of a spherical conductor. At a point P (black dot), I do not understand how the electric field cancels and becomes 0. Electric field is in blue.
  13. A

    I Electric field, flux, and conductor questions

    1) Why is the electric field 0 at the bottom of Gaussian surface? Isn't the electric field on both sides of the surface, pointing down and outwards like a plane of charge? see image. 2) Why does a charge distribution with cylindrical symmetry have to be infinitely long? 3) My book says a...
  14. I

    Directing magnetic field lines by using diamagnetic material

    I want to improve the magnetic field strength at the surface of a magnet configuration by utilizing diamagnetic materials to guide the magnetic field lines. I have not the proper equipment to measure the effect myself but would this work? This is the initial configuration with four magnets side...
  15. A

    Sphere and electric field of infinite plate

    The solution says that the tension in the string in the negative x direction is balanced by the force of the plate on the ball (red). Why is the repulsive force of the ball on the plate (in blue) not included in this calculation?
  16. A

    I Electric field vector takes into account the field's radial direction?

    Does the electric field vector takes into account the field's radial direction? Usually when we calculate the electric field, we use ##\vec E = \frac{kq}{r^2}\vec j##, which is a straight line vector of a positive charge q's electric field. This electric field points from a positive charge q to...
  17. A

    Find the electric field everywhere resulting from two infinite planes

    What I don't understand is how come the electric field of the negative plane isn't pointing towards the positive plane (in blue) and cancelling out the electric field of the positive plane (in red). See image
  18. T

    A A question about charged particle in a magnetic field

    Suppose a molecule from our surrounding air (at ambient temperature) is being selected and is ionized. By some mechanical means, some velocity (say 100 m/s) is added to it and it has been put into a magnetic field perpendicular to its direction of motion. We all know how the molecule will behave...
  19. BigyanAdhikari

    I Acceleration at 1g in space -- Does it create a gravitational field?

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  20. DC2

    A Calculation of EM fields induced by an antenna in the near field

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  21. C

    A Relation of Electromagnetic Field & Field Tensor

    hello, 1. according to Robert Wald, General Relativity, equation (4.2.22) the magnetic field as measured by an observer with 4-velocity ## v^b ## is given by ## B_a = - \frac {1}{2} {ϵ_{ab}}^{cd} F_{cd} v^b ## where ## {ϵ_{ab}}^{cd}##, the author says, is the totally antisymmetric tensor (for...
  22. K

    I Gravitational Field with Dark Energy: Observable Effects?

    since the cosmological constant observed is that there is a small amount of energy in empty space, and in general relativity anytime there is energy there is curvature and therefore gravity, how to calculate gravitational field with dark energy and does it have any observable effects on matter...
  23. F

    Finding Total Number of Optical Field Modes for Visible Light

    ρ_kdk = k^2/π^2 dk is the density of field modes (what we are trying to solve for here), and as ρ_kdk = ρ_λdλ, and k=2π/λ, we can rearrange this to get ρ_λdλ = 8π/λ^4dλ This is where my confusion lies. I am not sure what to do next. I know this equation physically means the number of modes per...
  24. B

    I Incident electric field attenuation near a metallic plate

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  25. strangerep

    A Meaning of "symbol" in algebraic field theory?

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  26. Mr_Allod

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  27. Tertius

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  28. Tan Tixuan

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  29. Harikesh_33

    I Question regarding the use of Electric flux and Field Lines

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  30. J

    What is the electric field inside an infinite cylinder?

    3 cm is inside the cylinder. We can use a gaussian cylinder to enclose the inside of the cylinder up to 3 cm. Because the outer cylinder is infinite there is no flux out of the end caps with the inner cylinder. There is also no charge enclosed in the cylinder. So the electric field 3cm away from...
  31. J

    Electric Field Inside a Gaussian Surface with Point Charge q

    If I have a point charge q right outside of a gaussian surface, it makes sense that the flux is zero inside the surface because the electric field going in equals the electric field going out. However, how would the electric field be zero inside? Wouldn't it just take on the electric field of...
  32. G

    Calculating Electric Field at Point P(0,0'03,0'04)

    At point ##P(0,0'03,0'04)## the field caused by the sphere is added to the field caused by the plane. First, ##E_\sigma## $$E_\sigma=\dfrac{\sigma}{2\varepsilon_0}=\dfrac{0,2\cdot 10^{-6}}{2\varepsilon_0}=11299,44\, \textrm{V}/\textrm{m}$$ Then, ##E_0##: Because ##r<R##...
  33. D

    A Possible Ways to reduce Electron field Emission threshold

    Hello there, i wanted to ask if anyone knows a process or mechanism, that reduces the electric field that is requiered to tunnel an electron. When i use the work function of 4 eV (Aluminum) i get with Schottky-Nordheim approach a field of 870 kV/mm to tunnel an electron. Measurements tho just...
  34. G

    Flux of the electric field that crosses the faces of a cube

    a) $$\phi_T=\phi_F-\phi_I=10^4\cdot 4\cdot 4-10^4\cdot 4\cdot 4=0\, \textrm{Nm}^2/\textrm{C}$$ b) $$\phi_F=\underbrace{300\cdot 4}_{\vec{E}}\cdot \underbrace{4\cdot 4}_{\textrm{area}}=19200\, \textrm{Nm}^2/\textrm{C}$$ $$\phi_0 = 300\cdot 0\cdot 4\cdot 4=0\, \textrm{Nm}^2/\textrm{C}$$ Then...
  35. G

    Doubts about the electric field created by a ring

    I have the calculation of the electric field created by a ring of radius ##R## uniformly charged with a linear density of charge ##\lambda## at any point on the axis perpendicular to its surface (##z## axis), but I have some doubts about it. I'll leave you the calculation done first: In ##x##...
  36. J

    Calculating the Electric field inside an infinite planar slab using Gauss' Law

    Draw a Gaussian pill box that starts from 0 (half way between the slab) and extends towards 2 cm.$$A \times \int_{0}^{0.02} \rho dz$$ I'm not sure if I should multiply the integral by A (area) or V (volume) And if area would I multiply by 0.02^2? I'm confused here. Thanks for your help.
  37. S

    Exploring Electric Field Boundaries at a Charge Density Boundary

    Hey, I have a really short question about electrostatics. The boundary conditions are : \mathbf{E}^{\perp }_{above} - \mathbf{E}^{\perp}_{below} = -\frac{\sigma}{\varepsilon_{0}}\mathbf{\hat{n}} , \mathbf{E}^{\parallel }_{above} = \mathbf{E}^{\parallel}_{below}. My question is what is...
  38. A

    B Higgs field and nuclear reactions

    What happens to the higgs field when say a fusion reaction occurs. Like if mass is converted into energy and the higgs field gives a particle mass what happens to higgs field. I doubt this, but is the higgs field the mechanism that converts mass into gamma rays. Go easy on me I only have a high...
  39. SanaiBongchul

    Green's reciprocity theorem about current density and magnetic field

    I have to prove three equations above. For first two equations, I've been thought and made reasonable answer by using a definition of the electricfield. However, for third, I can't use a definition of a magnetic field due to the cross product Like J_2 X J_1 X (r_2 - r_1). I think three of 'em...
  40. F

    What's the best shape function/element in finite element methods for phase field simulation?

    What's the best shape function/element in finite element methods for phase field simulation?
  41. J

    Find the electric field of a charged arc a distance R away

    define charge at an infinitesimal length of arc $$dQ = \lambda R d \theta$$We only care about the x component of the electric field because the y components cancel due to symmetry $$dE_x = \frac{k_e dQ}{R^2} cos \theta$$ Integrate to add up the infinitesimal parts. A quarter circle means 90...
  42. J

    Find net velocity of charged particle in electric field (symbols only)

    We know the net force on the charged particle in the uniform electric field pointing up is mg - qE. To get acceleration, divide the net force by mass to get g - qE/m Plug into kinematic equation and get velocity by itself and substitute$$\sqrt{h(2g - \frac{q \sigma}{\epsilon_o m})}$$
  43. J

    Find the electric field at a point away from two charged rods

    λ1 = 3 microC/m λ2= -4 microC/m __________ . __________ l----L1---l-a1-l-a2-l-----L2---l (Not to scale) L1 = length of rod 1 (1m) a1 = length of end of rod 1 to point (0.7m) L2 = length of rod 2 (1m) a2 = length of end of rod 2 to point (0.3m) k = e field constant...
  44. Salmone

    I How to obtain Hamiltonian in a magnetic field from EM field?

    To calculate the Hamiltonian of a charged particle immersed in an electromagnetic field, one calculates the Lagrangian with Euler's equation obtaining ##L=\frac{1}{2}mv^2-e\phi+e\vec{v}\cdot\vec{A}## where ##\phi## is the scalar potential and ##\vec{A}## the vector potential, and then we go to...
  45. E

    I Commutation relations for an interacting scalar field

    Hi there, In his book "Quantum field theory and the standard model", Schwartz assumes that the canonical commutation relations for a free scalar field also apply to interacting fields (page 79, section 7.1). As a justification he states: I do not understand this explanation. Can you please...
  46. S

    Potential difference of a ring rolling in magnetic field

    I don't understand why there is potential difference between point A and O. Is there any change in magnetic flux experienced by the ring? I think the magnetic field passing through the ring's cross sectional area is constant Thanks
  47. F

    Find the flux of a vector field

    Question: Equation: Attempt: Can someone verify my answer?
  48. MatinSAR

    How Is the Electric Field Calculated for a Point Outside a Charged Ring?

    Hi ... How can I find the electric field due to a thin circular ring of radius a and charge q for points outside the plane of the ring? The distance from the center of the ring to the point of the electric field is large compared to the radius of the ring. I have answered it but I don't know if...
  49. bluesteels

    Exam Prep: Electric Field - Is It Zero?

    I'm having an exam soon so i want to make sure. Is the electric field here zero?? cause if i draw gauss surface covering both of them they should cancel out or am i wrong.
  50. R

    Displacement Electric Field Outside Dielectric Material

    I know that inside region 1, the D-field is zero as it is a conducting sphere, the E-field must be zero. It makes sense that in region 2 (inside the dielectric) there is a D-field. My question is, is there a D-field outside the dielectric material (r>R)? Obviously there will be an E-field, but...
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