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. Elder1994

    Magnetic field due to the current flowing in a bent wire

    Hello, in this problem I'm supposed to calculate de magnetic field due to a bent wire at any point of the x-axis after the bending of the wires. It is obvious that the part of the wire that is parallel to the x-axis makes no contribution to the field so we can focus on the other part of the...
  2. E

    Quantum motion of a charged particle in a magnetic field

    Once I know the Hamiltonian, I know to take the determinant ##\left| \vec H-\lambda \vec I \right| = 0 ## and solve for ##\lambda## which are the eigenvalues/eigenenergies. My problem is, I'm unsure how to formulate the Hamiltonian. Is my potential ##U(r)## my scalar field ##\phi##? I've seen...
  3. Achintya

    What is the reality of the Electric Force & Field?

    Please help. Thank you.
  4. hugo_faurand

    I Find Einstein's Field Equation - Intuition with Strong Equivalence Principle

    Hello everyone ! I'm getting into General relativity. I wonder know how we find the Einstein's field equation. Maybe we can have an intuition with the strong equivalence principle. So if you can enlight me ☺️☺️ please Regards
  5. eedftt

    Mastering Physics Homework about Magnetic field

    Magnetic fields are sometimes measured by balancing magnetic forces against known mechanical forces. Your task is to measure the strength of a horizontal magnetic field using a 12-cm-long rigid metal rod that hangs from two nonmagnetic springs, one at each end, with spring constants 1.3 N/m ...
  6. G

    Line integral where a vector field is given in cylindrical coordinates

    What I've done so far: From the problem we know that the curve c is a half-circle with radius 1 with its center at (x,y) = (0, 1). We can rewrite x = r cos t and y = 1 + r sin t, where r = 1 and 0<t<pi. z stays the same, so z=z. We can then write l(t) = [x(t), y(t), z ] and solve for dl/dt...
  7. W

    Why a steel plate can shield magnetic field?

    If I put a very long steel plate above a coil with DC, the magnetic field above the plate will decrease because of the shielding of the steel plate. However, from the perspective of magnetci domain, some domains will be magnetized to turn to the direction of the magnetic field from the coil...
  8. F

    I Electromagnetic field according to relativity

    Hello, I am still trying to fully grasp the general idea of the EM field, which always travels at the speed of light regardless of the reference frame, and is represented by a tensor with 16 components in relativity theory. My understanding is that, depending on the observer's frame of...
  9. K

    Clarification on field intensity (electromagnetism)

    Is the intensity of a general electromagnetic wave always the norm of its Poyinting vector? Or are there other notions of intensity?
  10. B

    Electric Force and Field homework problem

    This is my homework. And we don't have online lessons. But my teacher gave this for homework. I couldn't solve this problem. Can you help me?
  11. John Greger

    A Solving Problems in "Conformal Field Theories" by Francesco

    I have been following the book called "Conformal Field Theories" by Francesco, also known as "the yellow pages". I do this for fun but, of course, sometimes it gets rather technical. Do there exist solutions to the problems in this book? I haven't found a solutions manual available. Many...
  12. Rongeet Banerjee

    Work, Energy and Power for a Particle moving in a Potential Field

    But yet again my text says that option 4 is correct.
  13. Seyit KAPLAN

    How is the thermal conductivity affected by magnetic field?

    Hello, I'm searching for how magnetic field affects the thermal conductivity of the metal (such as steel in solid form). If someone suggests any article about it will be very helpful.
  14. P

    Electric field in the Spherical Cavity

    a. For the question a the solution is If the uniform charge density is ρ then the charge of the sphere up to radius r is q = ρ * (4/3)*π * r3; Hence the electric field is E = (ρ *4π*r^3)/(3*εο*r^2); E = (ρ*r)/(3εο); b. I don't understand what is superposition? How to proceed? Please advise.
  15. Bilbo B

    Magnetic field affecting a circular loop

    If the magnetic field is constant then no change in flux will bring no induced emf nor any induced current. With the loop is in rest position the external magnetic field will exert a force but to calculate that force with the help of magnetic field isn't obvious. If this were a charged loop, the...
  16. D

    Thick square loop made of Al with the top half exposed to a B field

  17. F

    Proving Existence of b in F for Field with Char p and Reducible f

    Suppose ##f## is reducible over ##F##. Then there exists ##g, h \in F## such that ##g, h## are not units and ##f = gh##. If there exists ##b \in F## such that ##b^p = a##, then ##(x - b)^p = x^p - b^p = x^p - a##, using the fact that ##F## has characteristic ##p##. So, if such a ##b \in F##...
  18. G

    Electrostatic field of a sphere

    What is the electrostatic field of a non-conductive sphere (it's radius is R) which has a density charge distribution inside? ρ0 and R are parameters. I started solving this with Gauss's law: then: Solving the integral: This means the electrostatic field of the sphere in r is: Can you...
  19. S

    A Evanescent field of a waveguide

    If we look to the figure, in several articles they mentioned that the part of the mode field, the tail of the mode field diameter, travel in the cladding, this maybe I can understand that because of a little change between the refractive indices of the core and the cladding, then the transverse...
  20. L

    A Solve Conditions to Preserve Bondi Gauge Vector Field

    I'm going through the "Advanced Lectures on General Relativity" by G. Compère and got stuck with solving one set of conditions on the subject of asymptotic flatness. Let ##(M,g)## be ##4##-dimensional spacetime and ##(u,r,x^A)## be a chart such that the coordinate expression of ##g## is in Bondi...
  21. A

    I 'Off-shell' particle in an external field....

    Silly question but could someone explain why a real, 'observable' particle is said to be 'off-shell' in an external field? @A. Neumaier 's excellent FAQ notes that the mass shell constraints ceases to have meaning in this case. I'm just not fully clear on why (probably obvious) given that energy...
  22. G

    Electric field around a sphere with an internal charge distribution

    I'm just going to skip some of the step since I only need help with understanding the last part. After rearranging the equation stated at "Relevant equation" (and skipping some steps) we will get: E * 4*pi*e0*R^2 = integral pv * 4*pi*R^2 dR E = 1/(4*pi*e0*R^2) * 4*pi * integral pv*R^2 dR E =...
  23. Bell

    What's the direction of electric field in Laguerre-Gaussian beams?

    As for Laguerre-Gaussian beams, the direction of wave vector is helical, and how about the direction of electric field? I found that there was little literature mentioned this.
  24. jisbon

    Determine the points where the net magnetic field is zero

    Unsure about this, but here is my attempt: B from the first wire: ##\dfrac {\mu _{0}I}{2\pi r} ## B from the second wire: ##\dfrac {\mu _{0}I}{2\pi r} ## Let the point be (x,y) Can I state that: ##\dfrac {\mu _{0}I}{2\pi y}+\dfrac {\mu _{i}\left( I/3\right) }{2\pi x}=0## Hence the magnetic...
  25. Jackoyo

    Solving Electric Field & Potential: Jack Needs Help!

    Hi everyone, I have abit of trouble with this question. Please help! Given charges +q, +2q, −5q and +2q are placed at the four corners ABCD of a square of side a, taken in cylic order from the bottom left corner. Find the electric field E and the potential V at the centre and verify that they...
  26. B

    Torque on a rectangular coil in a uniform magnetic field

    So this was a section taken out from a question which I am trying to do shown below I have drawn a sketch to help me visualise of what is going on I have used Fleming's left hand rule to help me determine what direction the force is facing on each side of the coil. For the last part in...
  27. Yassin98

    The Equivalence Principle -- Is this a way to distinguish between a gravitational field and an accelerated rocket?

    If we are in a cabine in a gravitational field and inside, we have a racket and a ball. We put strings in each side of the racket and we connect the racket to the ceiling of the cabine. This strings only allows us to keep the weight of the racket. Then, we drop a ball to the racket. We do this...
  28. Z

    Electric field direction on a grounded conducting sphere

    I am required to find the direction of the electric field on the surface of a grounded conducting sphere in the proximity of a point charge ##+q##. The distance between the center of the sphere and the point charge is ##d## and using the method of images we find that the charge of the sphere is...
  29. Leo Liu

    Understanding the Electric Field of a Charged Sphere

    This page claims that "[t]he electric field outside the sphere is given by: ##{E} = {{kQ} \over {r^2}}##, just like a point charge". I would like to know the reason we should treat the sphere as a point charge, even if the charges are uniformly distributed throughout the surface of the...
  30. jisbon

    How is the Electric Field in Energy Bands Calculated?

    This is my attempt at this question, and I'm probably wrong, will need some help/guidance from the experts here :/ i) (ii) Since energy band given by ##6.67x^2##, can I assume that electric field is simply the energy difference from 0-3m divided by 3m? In this case, would the answer simply be...
  31. bhaskarporey

    About changes in a magnetic field

    Here i tried this way (see picture) Please tell me am i right or wrong. Also they says find the change in magnetic field with time using Faraday's law in a rectangular loop. How can i solve that??
  32. A

    How can we deduce the kink effect in the electric field?

    Hi. In videos online the kink is explained as a delay in the electric field when charges accelerate. Does this mean we can deduce the existence of kinks from coloumb law. Does the simple form of plane electromagnetic waves which is well treated in most books really exist. What is the...
  33. T

    Vector field equality Curl Proof of Moving Magnet & Conductor Problem

    The moving magnet and conductor problem is an intriguing early 20th century electromagnetics scenario famously cited by Einstein in his seminal 1905 special relativity paper. In the magnet's frame, there's the vector field (v × B), the velocity of the ring conductor crossed with the B-field of...
  34. T

    Is There an Electric Field Within the Cavity of a Polarized Hollow Conductor?

    Suppose we have a hollow metallic conductor, just a thin metallic shell forming a large hollow cavity. It is then polarized by electric charges placed nearby externally. The equilibrium electric field must be parallel to the surface normals of the shell, there must be no tangential component...
  35. JD_PM

    A Commutation relations between HO operators | QFT; free scalar field

    I am getting started in applying the quantization of the harmonic oscillator to the free scalar field. After studying section 2.2. of Tong Lecture notes (I attach the PDF, which comes from 2.Canonical quantization here https://www.damtp.cam.ac.uk/user/tong/qft.html), I went through my notes...
  36. Dusan Stan

    How can magnetic fields be shielded from external sources?

    It seems I don't understand how magnetic shielding supposed to work. I tried shielding a wire, using some ferrites, but it doesn't work. I assumed the magnetic field will concentrate in the magnetic material, bypassing the meter magnetic loop, so less will be measured by the meter. I thought the...
  37. P

    Effective potential in a central field

    Hi, I am confused by a point which should be relatively simple. When we consider classical motion of a particle in a central field U(r), we write the total energy E = T + U, where T is the kinetic energy. The kinetic energy contains initially r, r' and φ' (where ' denotes the time derivative)...
  38. Pispi Choudhury

    Courses Math & Physics Courses for Quantum & Statistical Field Theory

    Summary:: What are the relevant mathematics/ mathematical physics courses for studying quantum field theory and statistical field theory? I'm a physics undergraduate currently in my junior(third) year, thanks.
  39. G

    Engineering Calculating the total electric field from two charges

    a) Should be pretty straight forward, from the equation E = kQ/R , we see that scaling is simply 1/R. b) Here is gets a bit trickier. We know that q acts as a source (E-field points outwards) and -q acts as a sink (E-field points inwards). If the distance is far away do we consider the Q1 and...
  40. R

    Is Nuclear Engineering a Stress-Free Field?

    I'm in nuclear engineering and things seem fairly laid back. Everyone comes of as a bit of a party animal and faculty don't seem to be in any particular rush to get things done.
  41. B

    Find the mass of this ion moving in a magnetic field

    I had started off by equating the magnetic force to the centripetal force, to find the mass of a single ion. Then calculated the mass of what 1 mole of that ion would weigh to identify what the ion is Hence the answer would be nitrogen (option C) as 1 mole of nitrogen weighs 14g Would this...
  42. B

    Deflection distance for an electron beam in an electric field

    Have tried doing this question but I'm a bit confused on where I'm going wrong This is what I have done but get a value that doesn't match to any of the options given above? Any help would be really appreciated, Thanks!
  43. F

    Determining the path of a particle in a field

    This is not a specific homework question, but more of a general query. If provided with a simple vector field indicating forces (for example, an electrical field), can you use integration to determine the path of a particle placed in that field, if also provided with some initial conditions...
  44. E

    Potential energy of a dipole in an external field

    I'm considering the arrangement shown below. Let the positive charge be ##q##, and the negative be ##-q##. To derive the potential energy of this configuration, one usually adds the potential energies of both of the charges in the external field, taking the zero volts equipotential of the...
  45. H

    Magnetic field vector using F = qV * B

    Hi, I'm trying to find the magnetic field B using F = qV * B. I have F = (3i + j + 2k) N V = (-i +3j) * 10^6 m/s q = -2 *10^6 C Bx = 0 I don't know how to resolve a 3 dimensional vector equation. B = F/qV makes not sense for me.
  46. K

    Determine the magnetic field inside and outside the cable using Ampere's law

    My attempt: I realized after i had tried to solve the problem that the current must be constant in the cables. But no information about where the cables has radius a and b is given so how would I go about to find an expression for the current? Thanks in advance!
  47. JD_PM

    Other Classical and Field books on symmetries

    Hi. I am interested in finding books dealing with symmetries. Specifically books that make me understand assertions like, and I quote Orodruin's #10 and #16 here https://www.physicsforums.com/threads/find-a-transformation-that-leaves-the-given-lagrangian-invariant.984601/, 'a rotation in the...
  48. F

    B field between the plates of a charging capacitor (Ampere's law)

    A standard example consider a capacitor whose parallel plates have a circular shape, of radius R, so that the system has a cylindrical symmetry. The magnetic field at a given distance r from the common axis of the plates is calculated via Ampere's law: \oint_\gamma {\mathbf B} \cdot d{\mathbf...
  49. HAgdn

    Magnetic field Cancellation in AC wires (How?)

    Why do the magnetic fields in-between the wires does not seem to cancel? Even those outside each wire? (the fields do are not in opposite direction). Yet most of the people I have talked to until now says that such magnetic fields do cancel? I am confused...
  50. 1

    Help with the trajectory of a particle in a magnetic field

    Hello, sorry for this stupid question but I am getting confused with equations and I need some help. My problem is the following: I have a dipole with known magnetic field (B=0.234T), it is 110.6 mm long and 89 mm wide (mechanical dimensions). I want to calculate deflection of particles...
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