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. 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...
  2. 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...
  3. 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...
  4. 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...
  5. jisbon

    Electric field in Energy Bands

    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...
  6. 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??
  7. 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...
  8. 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...
  9. T

    Electric field within the cavity of a hollow conductor polarized by external charges?

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  10. 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...
  11. 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...
  12. 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)...
  13. Pispi Choudhury

    Courses Math & Physics Courses for Quantum & Statistical Field Theory

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  14. 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...
  15. 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.
  16. 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...
  17. 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!
  18. 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...
  19. 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...
  20. 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.
  21. 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!
  22. 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...
  23. 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...
  24. 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...
  25. 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...
  26. 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...
  27. H

    Electric field from a charge q1

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  28. T

    I Electromagnetic Radiation in Strong Gravity Field

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  29. Elder1994

    Verify Stokes' Theorem for this vector field on a surface

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  30. B

    Revolving disc in a magnetic field

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  31. 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!
  32. 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?
  33. 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?
  34. B

    How far from this current-carrying wire will the Earth's magnetic field cancel the B-field from the wire?

    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...
  35. 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...
  36. 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...
  37. 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 +...
  38. 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...
  39. 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...
  40. 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.
  41. 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...
  42. 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...
  43. 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}...
  44. 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...
  45. 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...
  46. 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...
  47. 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...
  48. 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...
  49. 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
  50. 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?
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