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. U

    Current induced from a changing magnetic field

    This question is motivated by Problem 7.12 in Griffiths Electrodynamics book. I have not included it in the homework section, because I have already solved it correctly. However, I question whether my solution which agrees with the solution's manual is correct. Relevant Equations: $$\Phi =...
  2. Q

    A Bosonic Field operator in this exponential

    If i had a bosonic field ##\phi(x)## and I took the exponential in the following way to get the operator $$W=e^{\imath f \phi(x)}$$ where ##f## is a parameter what effect would this have when acting on the vacuum ##|0\rangle##? Is it analogous to the space translation operator? Will it transform...
  3. hiddenname

    I Electricity & Planetary Magnetic Field Reversal: Implications & Studies

    What is/are the potential or theoretical effect on electricity due to reversal of planetary magnetic field? Would circuitry continue to function or would knowledge and understanding need re-evaluating as a result of a reversal of our magnetic field on planet Earth? Note: this is not a homework...
  4. V

    Time derivative jump of the electric/magnetic field

    So I just wanted to see if anyone could offer some suggestions. So in my mind this seems impossible, in the case of electric field a jump in time derivative of that field would indicated in my mind that electric charge was either introduced or removed from the system instantaneously which would...
  5. Bobjoesmith

    I Physical Meaning of a Quantum Field

    Sorry in advance if this question doesn't make sense. Anyway, I am reading a paper about quantum field theory and the Whitman Axioms (http://users.ox.ac.uk/~mert2060/GeomQuant/Wightman-Axioms.pdf), and it describes a field (Ψ) as Ψ:𝑀→𝑉⊗End(𝐷) where 𝑀 is a spacetime manifold, 𝑉 is a vector...
  6. looseleaf

    A Derivative of a Lorentz-Transformed Field

    Please help me understand this line from P&S, or point me towards some resources: Why is there another Lorentz transformation acting on the derivative on the RHS? Thanks
  7. D

    How to show that the electric field inside a spherical shell is zero?

    Hi! I need help with this problem. I tried to do it the way you can see in the picture. I then has this: ##dE_z=dE\cdot \cos\theta## thus ##dE_z=\frac{\sigma dA}{4\pi\epsilon_0}\cos\theta=\frac{\sigma 2\pi L^2\sin\theta d\theta}{4\pi\epsilon_0 L^2}\cos\theta##. Then I integrated and ended up...
  8. D

    How to calculate the electric field at a point on the axis of two rings

    Hi! I need help with this problem. I tried to solve it like this: First I calculated the electric field of each ring: Thus the electric field at a point that is at a distance z from the ring is ##E=\frac{Qz}{4\pi\epsilon_0(z^2+r^2)^{3/2}}##, Thuss for the upper ring, the electric field would be...
  9. P

    Measuring Magnetic Field Strength of a Cylindrical Magnet

    Hello, Today I am wondering if anyone can help me quantify the strength of the magnetic field created by a permanent cylindrical magnet. I have been able to find equations online for the strength of the field within the z axis, (ie. the longitudinal length) but I would like to know the strength...
  10. L

    Engineering Magnetic Field Question (saturating an iron plate)

    I know that the field inside sample is a combination of the demagnetizing field and whatever applied field you may have. So these two fields together influence how big a field you need in order to magnetize the sample.
  11. D

    Trick for conserved currents in classical field theory

    First I found the equations of motion for both fields: $$\partial_\mu \partial^\mu \psi = -\frac{\partial V(\psi^* \psi)}{\psi^*}$$ The eq. of motion with the other field is simply found by ##\psi \rightarrow \psi^*## and ##\psi^* \rightarrow \psi## due to the symmetry between the two fields...
  12. Snarlie

    Force & Torque on Electric Dipole in Magnetic Field

    Would this also be correct? Or is there something wrong with it? I appreciate all your responses! Snarlie
  13. D

    I Lorentz transformation of derivative and vector field

    I'm currently watching lecture videos on QFT by David Tong. He is going over lorentz invariance and classical field theory. In his lecture notes he has, $$(\partial_\mu\phi)(x) \rightarrow (\Lambda^{-1})^\nu_\mu(\partial_\nu \phi)(y)$$, where ##y = \Lambda^{-1}x##. He mentions he uses active...
  14. F

    Why is the magnitude of the electric field in a sphere the same?

    I was looking at a sphere that has a positive point charge at the center of a sphere with radius R. Now, I understand that the electric field is pointing outwards (in the direction of dA), so $$d\phi = EdA$$ However, I am told that since the magnitude electrical field is the same because the...
  15. gleem

    Examining the Impact of a Leading Researcher's Death on the Life Sciences Field

    What happens to research when a leading researcher passes away? A new study looked at this in the Life Sciences . The study observed that when a leading expert in a field died there was a significant increase in publications by new researchers (with new ideas?). it was also determined that...
  16. K

    Electric field using Gauss's Law, but in open cylinder given only r?

    I have no idea how to approach the problem using Gauss's Law. I found the electric field using superposition, and it was incorrect. I am assuming you treat the wire as a continuous electric field, and then also treat the pipe as a continuous electric field. I solved for this using...
  17. Pencilvester

    I Pullback of Vector Field in Relativity: Restrictions?

    Since coordinate transformations should be one-to-one and therefore invertible, wouldn’t there be no restriction on pushforwarding or pullbacking whatever fields we feel like (within the context of coordinate transformations)?
  18. G

    I Quantum field theory: an informative approach

    I'm looking for a book that describes the quantum field theory without going deeply in the theory with formulas or complex description of the mathematics under the theory. I know that this theory is really complex and it needs a deep knowledge of quantum physics in order to be understood. But...
  19. jisbon

    Electric field of an infinite charged plate

    So I figured to get e-field at point (4,4,0), I need to find the resultant e-field from the negatively charged particle and the plate ##E_{resultant}=E_{particle}+E_{plate}## ##E_{particle}=\frac{kq}{d^2}=\frac{(9*10^9)(-2*10^-6)}{4^2}=-1125N/C## Now for the plate is where I'm confused. If this...
  20. D

    Amplitude of an oscillating electric field

    Homework Statement: The amplitude of the oscillating electric field at your cell phone is 4.0 μV/m when you are 10 km east of the broadcast antenna. What is the electric field amplitude when you are 20 km east of the antenna Homework Equations: electric field i've done E=##\frac A...
  21. ChrisXenon

    I E & B field phase relationship in EMR

    Common diagrams for the magnetic and electric field components of EMR show the fields at right angles in space with peaks aligned along the axis of propogation, for example Wikipedia here: https://en.wikipedia.org/wiki/Electromagnetic_radiation. However, Faraday's law says the E field depends...
  22. warhammer

    Classical Physics & Particle Motion Under Force Field

    The particle is moving under a force field with the potential energy equation described above. I find it logical that Newton's Laws can be used as in the question itself it is stated that the velocity is quite small and we could approximate its subsequent motion via the notions of Classical...
  23. bob012345

    Energy vs. Time in a Magnetic Field due to a Current Carrying Wire

    I want to know the total energy contained in a magnetic field due to a long wire (just consider a 1m segment) as a 1amp current is turned on starting at time zero. I'm assuming zero turn-on time for convenience. At t=0 the cylindrical field is formed and I wish to know the total energy as a...
  24. D

    Electric field in the narrow wire

    i've started from this I1=I2 then I1= JA1=##\frac {E l} R## I2= JA2=##\frac {E_2 l} R## but can't get anything useful relating them. Am i forgetting any other useful formula? I get as result E4
  25. A

    Electric Field for the circular path of a positively charged particle

    Here is picture. Answers is A. My attempt was that I thought if i were to place a positive test charge then it would go from top to bottom if there was a positive charge in the center it was avoiding and a positively charged particle at the top, but an electron at the bottom so it would avoid...
  26. D

    The distance where the magnetic field of two wires is zero

    Homework Statement: The distance between two parallel long wires, carrying currents equal to i and 3i respectively, is d as shown in the picture. What is the distance from the wire carrying the current i at which the magnetic field is zero Homework Equations: guess biot-savarat Being two...
  27. C

    Electric Field and Capacitance

    Question Part C Part D
  28. C

    Deriving an Expression for an Electric Field along the Z axis

  29. TheBigDig

    Derive an expression for the applied field of a superconducting wire

    So far the best I've been able to come up with is to use ##\vec{B} = \mu_0 \vec{H}## which gives me i_c = H 2\pi r j_c = \frac{H 2\pi r}{\pi r^2} = \frac{2H}{r} \therefore B = \mu_0 \frac{r j_c}{2} I'm fairly confident this is just terrible math and physics on my behalf but I'm struggling to...
  30. M

    How to characterize a power law field?

    Hello, as part of the study of fields with central forces, I came across with fields called power law, defined by F = - K/r ^ n u (u is radial vector passing through the origin O) I would like to dismiss case n = 2, which refers to the Newtonian fields whose study was exhaustively conducted in...
  31. S

    Find the speed of a copper loop falling in a magnetic field

    Hi all, so I had this problem and on the exam and I got a solution but I had an mass-term in there which wasn't given. I used Farraday's Law of Induction to get the Voltage induced. Then I used ##rho* \frac{A}{4a} ## for the resistance and divided the Voltage by that to get the current. I then...
  32. Yalanhar

    Electric Field due a charged disk

    Homework Statement: uniformly charged disk, radius r, with surface charge density ##\sigma## . I want to find the electric field along the axis through the centre of the disk at a h distance Homework Equations: ##dE=\frac {kdq}{r^2}## My Solution: ##dE=\frac {kdq}{r^2}## in this case r=s...
  33. Yalanhar

    Calculate the electric field due to a line of charge of finite length

    Homework Statement: A thin rod of length L and charge Q is uniformly charged, so it has a linear charge density ##\lambda =q/l## Find the electric field at point where is an arbitrarily positioned point. Homework Equations: ##dE=\frac{Kdq}{r^2}## A thin rod of length L and charge Q is...
  34. Q

    A Explicit form of annihilation and creation operators for Dirac field

    I'm unclear on what exactly an annihilation or creation operator looks like in QFT. In QM these operators for the simple harmonic oscillator had an explicit form in terms of $$ \hat{a}^\dagger = \frac{1}{\sqrt{2}}\left(- \frac{\mathrm{d}}{\mathrm{d}q} + q \right),\;\;\;\hat{a} =...
  35. AndresPB

    Electric Field from its Potential of a Half Circle along its Z axis

    So I figured out the potential is: dV = (1/(4*Pi*Epsilon_0))*[λ dl/sqrt(z^2+a^2)] . From that expression: We can figure out that since its half a ring we have to integrate from 0 to pi*a, so we would get: V = (1/(4*Pi*Epsilon_0))*[λ {pi*a]/sqrt(z^2+a^2)] In that expression: a = sqrt(x^2+y^2)...
  36. avnl

    A Calculating the propagator of a Spin-2 field

    Nieuwenhuizen uses a method for calculating the propagator by decomposing the field ## h_{\mu\nu}, ## first into symmetric part ## \varphi_{\mu\nu} ## and antisymmetric part ## \psi_{\mu\nu} ##, and then by a spin decomposition using projector operators. Using this he writes the dynamical...
  37. M

    A Gravitational dressing in quantum field theory

    In quantum field theory, a dressed particle is a particle ("bare particle") considered in combination with certain secondary effects that it produces (e.g. the virtual pair creation involved in vacuum polarization). The dressed states are regarded as more physical, hence closer to reality. Axel...
  38. V

    Solid State Textbook on crystal field theory and degeneracy breaking

    In particular I would like to have a resource for the relation between group theory, crystal field symmetries and breaking of degeneracies of orbitals. I've taken a graduate condensed matter course and graduate quantum mechanics courses. I have some basic knowledge of group theory but can learn...
  39. N

    Find the Electric Field E using Gauss' Law

    I tried to work out both a) and b), but I am not sure if I am correct. I drew a picture with a sphere around q first with radius r and then with radius 3r. For a) ##E.A=\frac {q}{ε_°}## (when using Gauss' Law) Since ##A=4πr^2##, I substituted this in the equation and solved for E giving me...
  40. K

    I Field theories in 4 dimensions

    Hello! I know this is a very general question (and I am really a very beginner in the field) so I am sorry if it is dumb, but here it goes. In Schwarz book on QFT, at the end of Section 14.4 (path integrals chapter) he says: "We do not know if QED exists, or if scalar ##\phi^4## exists, or even...
  41. Dan13

    How to interpret magnetic field diagrams?

  42. M

    I Why does quantum mechanics believe that gravity is a field?

    According to general relativity, gravity is simply the side-effect of bending the geometry of space-time. As a thought experiment imagine a 3D image being projected from a 2D hologram - the distance between the actual 2D pixels in the 2D plane always remains constant, yet depending on the shape...
  43. filip97

    I Higher Spin Field Rep: Tr(n,n)=0, Why Symmetric Tensor of Rank 2n?

    Representation (n,n)=nxn=2n+(2n-1)+...+2+1 , where + is direct sum, and x is tensor product. Trace of (n,n) is zero because tr(k) is zero, where k=1 to n. But why tensor (n,n) is symmetric tensor of rank 2n? I read Weinberg book Vol. 1 page 231. But he don't clearly says this .
  44. D

    Does an AC Magnetic field induce movement in a DC Magnetic Field?

    Summary: Can a rotating AC Magnetic field induce movement in a static DC Magnetic flux? I'm designing a control panel, and the customer has asked us to reduce the EMC as much as possible; there are no drives, or other noise creating devices, just AC circuits. I thought a good starting point...
  45. R

    Derivation of ideal magnetic dipole field strength

    For reference, this is from Griffiths, introduction to quantum mechanics electrodynamics, p253-255 When deriving the ideal magnetic dipole field strength, if we put the moment m at origin and make it parallel to the z-axis, the book went from the vector potential A $$ A=...
  46. B

    Electron helix in a magnetic field

    I thought that a nearly parallel entry path would result in a helix of very small, but constant, radius. I would not expect the electrons to focus at a point, but continue along the infinite helix. What have I missed?
  47. adamaero

    By using Gauss' law, can the electric field be p/(pi*ε*r^2)

    Which is better to use? The equation for the area or the circumference of a circle? Schaum's Electromagnetics (4 ed) by Edminister vs http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elecyl.html
  48. S

    Electric field due to two positive charges

    I am having trouble solving the following problem. I am given two positive charges on the x axis: I know that the electric field strength at point P is ##E=150 \frac{V}{m}##, ##d=1.8m## and ##a=2.5m##. I want to find the charge of ##Q##. As far as I know, the electric field on the y-axis...
  49. Alfredo Tifi

    Synchrotron Radiation: Charge Loss of Relativistic Particles in Magnetic Fields

    Problem Statement: It is possible to describe synchrotron radiation as caused by a loss of electrical charge of relativistic particles that are moving in a magnetic field? Relevant Equations: E = mc2 An Italian expert of black hole M87 (Elisabetta Liuzzo) explains that the expected axial...
  50. R

    Electric field on a ring's axis

    The contribution coming from a little segment of the ring is ##d\vec{E}=\frac{dQ}{r^2}cos\theta \hat{z}##, assuming that the horizontal components cancel out. But how can we show that?
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