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. guyvsdcsniper

    Determining Electric and Magnetic field given certain conditions

    I am unsure of my solutions and am looking for some guidance. a.)The real part of the wave in complex notation can be written as ##\widetilde{A} = A^{i\delta}##. Writing the Complex Wave equation, we have ##\vec E(t) = \widetilde{A}e^{(-kz-\Omega t)} \hat x##. Therefore the real part is ##\vec...
  2. S

    Differential equation of vector field

    I was thinking of using the chain rule with dF/dx = 0i + (3xsin(3x) - cos(3x))j and dF/dy = 0i + 0j but dF/dy is still a vector so how can it be inverted to get dy/dF ? what are the other methods to calculate this?
  3. H

    B Rice Field Bubbles: Gotcha! Why?

    Bubbles of air were rising in some water in a rice field. The bubbles would float around at random for a while. If two bubbles got within a certain distance of one another they would very quickly merge. It looked like a predatory larger bubble pouncing on the smaller prey. Gotcha! Why is...
  4. R

    A Potential Energy of Relativistic Particles in Coulomb Field

    Let us consider relativistic particle (electron) which moves with relativistic speed ##v## in the Coulomb field (in the field of a fixed heavy nucleus). The main question is what is the potential energy of a particle in such a static field? Landau and Lifshitz in their book "Field Theory"...
  5. P

    Engineering Electric field in spherical shell - trying to understand the location

    Nevermind.
  6. Fra

    A Peter Morgan (QM ~ random field, non-commutative lossy records?)

    "One way to ground everything in reality is to think purely about the records of experiments that are stored in computer memory. Very often, that's a list of times at which events happened." -- Peter Morgan, old thread meaning-of-wave-function-collapse "If we are to understand the relationship...
  7. H

    Why Does the Electric Field Sum Instead of Cancel with Opposite Charges?

    If there are two charges positive and negative and their electric field point in the same direction then the total electric field would be their sum of magnitudes. Why don't we consider the sign of the charges? For example, a parallel plate capacitor is inside the region where both the positive...
  8. J

    I Field fluctuations in the vacuum

    How does relativistic qft predict quantum fluctuations in the vacuum? We see this in the experiment proving the Casimir Effect so we know it's physical, but why?
  9. P

    Evaluating the Integral of a Vector Field Using Cauchy-Schwarz Inequality

    Here is my attempt (Note: ## \left| \int_{C} f \left( z \right) \, dz \right| \leq \left| \int_C udx -vdy +ivdx +iudy \right|## ##= \left| \int_{C} \left( u+iv, -v +iu \right) \cdot \left(dx, dy \right) \right| ## Here I am going to surround the above expression with another set of...
  10. lindberg

    I Unruh, Haag et al.: No Room for Particles in Quantum Field Theory?

    In a paper by Bain (2011), particles are left with little ontological value because of the Reeh-Schlieder theorem, the Unruh effect and Haag's theorem. The author claims (and here I am copying his conclusion): First, the existence of local number operators requires the absolute temporal metric...
  11. ermia

    Electric field needed to tear a conducting sphere

    ..
  12. lindberg

    I Haag's Theorem: Explain Free Field Nature

    What is the main reason for a free field staying free according to Haag's theorem?
  13. ermia

    Electric field of a part of a hemisphere

    I tried gauss law. And the fact that if alpha is less than pi/2 we can say that we have two parts with angle alpha and one other part which has a normal field at the center. But non of them helped me answer. The problem's solution says that we can use the fact that our section has longitudinal...
  14. Rikudo

    Gravitational field of a hollow sphere

    Why the area of the thin rings are ##2πasin\theta \, ds##? (a is the radius of the hollow sphere) If we look from a little bit different way, the ring can be viewed as a thin trapezoid that has the same base length ( ##2πa sin\theta##), and the legs are ## ds##. The angle between the leg and...
  15. B

    EMF Induced by Changing Magnetic Field

    Can anyone show me where I'm wrong?
  16. C

    I Do objects of differing mass fall at the same rate in a magnetic field?

    Gravity isn't a force in the strictest sense of the word, yet magnetism is exactly that: a force. As is strong, EW, etc. Therefore, it's possible that the more massive magnetic object gets drawn to the center of a magnetic source at a faster rate than the less massive magnetic object. Discuss!
  17. thedubdude

    I Why does metal moving though a magnetic field slow down?

    A piece of metal moving West to East in a North to South fixed magnetic field slows down...but how? Yes of course eddy currents are set up in the metal and these currents generate their own magnetic field which somehow slows down the moving metal piece...but how does this actually slow the...
  18. besebenomo

    Magnetic flux with magnetic field changing direction

    Sorry if I post again about this topic (last time I promise!) but I still have some doubts regarding the concept of flux. This collection of problems I have quite standard but there are so many variations. Here is the circuit in question: Something tells me that I could write a function that...
  19. warrenchu000

    Why is magnetic field B along a straight wire circular not radial?

    Statement: The magnetic field around a straight wire carrying a current can be explained Relativistically by changing the inertial frame of reference to the frame of the moving electrons - i.e., a Lorentz contraction of the positive charges in the wire will give a denser concentration of the...
  20. besebenomo

    Moving bar enclosing a changing magnetic field generates a current

    The amplitude of ##\vec{B}## is given by: $$B(x) = B_{0} - B_{0} \frac{x}{2l}$$ for ##0 \leq 0 \leq 2l## This was my attempts at finding the flux of B: $$\Phi(B) = (B_{0} - B_{0} \frac{x}{2l})(2l-x(t))l = B_{0}2l^2-2B_{0}x(t)l+ B_{0}\frac{x(t)^2}{2}$$ and the current: $$ i = -\frac{d...
  21. besebenomo

    Magnetic flux of magnetic field changing as a function of time

    $$B(t) = B_{0} \frac{t^2}{T^2}$$ for ##0 \leq t \leq T## The issue here is more conceptual, because once I find the flux of B I know how to proceed to find the current. I got velocity (but it seems to me that it is the initial velocity), I could use it to find the time in function of space...
  22. Ahmed1029

    I How do I find the Direction of an induced electric field?

    Faraday's law tell's you about the line intergal of the electric field, but you have to know the direction of the induced electric field first in order to properly apply it. How can I find its direction? Is it in the same direction as the induced current?
  23. G

    B The relationship between the particle, the wave and the field

    What is it the we detect in the first instance? Is it the particle |wave or is it the field? Is the former more fundamental than the latter in any sense or are we just talking the opposite sides of the same coin? For instance does the em field create the photon and the electron or could...
  24. warhammer

    B Direction of Electric Field & Field Due to a Dipole

    Hi all. I am stuck with a seemingly silly doubt all of a sudden. The direction of Electric Field is taken from Positive to Negative (because Field Lines originate from a Positive Charge and terminate at Negative Charge). We know that direction of Dipole Moment is from Negative Charge to a...
  25. FFXT

    I Faraday induction in constant B field, with non-conduction wires

    A standard textbook problem features a constant B field and a conducting loop that increases in area at constant rate. It is easy to work out the induced EMF and the associated electric field magnitude and direction (CW or CCW). The magnitude of the E field is E = B v where v is a velocity...
  26. P

    Induced EMF due to motion of a wire perpendicular to a magnetic field

    This question appeared in a university entrance exam.Basically, if magnetic flux passing through a surface of a loop changes over time ,only then e.m.f will be induced to that loop.But here only a straight line is used and there's no chance of forming any area.So by definition there's no chance...
  27. StenEdeback

    I Best book for Lagrangian of classical, scalar, relativistic field?

    Hi all experts! I would like to read about the Lagrangian of a classical (non-quantum), real, scalar, relativistic field and how it is derived. What is the best book for that purpose?Sten Edebäck
  28. P

    Calculating eletric potential using line integral of electric field

    So, I am able to calculate the electric potential in another way but I know that this way is supposed to work as well, but I don't get the correct result. I calculated the electric field at P in the previous exercise and its absolute value is $$ E = \frac {k Q} {D^2-0.25*l^2} $$ This is...
  29. StenEdeback

    Deduction of formula for Lagrangian density for a classical relativistic field

    Hi, I am reading Robert D Klauber's book "Student Friendly Quantum Field Theory" volume 1 "Basic...". On page 48, bottom line, there is a formula for the classical Lagrangian density for a free (no forces), real, scalar, relativistic field, see the attached file. I like to understand formulas...
  30. D

    What is high breakthrough field?

    I have read in the following article the expression "high breakthrough field": https://link.springer.com/article/10.1557/PROC-871-I9.6 I tried to find out in the internet what is the definition of that and what it refers to in the transistors but I couldn't find anything! Thank you in advance!
  31. X

    I Magnetic field strength of a stack of magnets

    I know that for a single cylindrical neodymium magnet, the formula $$ \displaystyle{\displaylines{B(z)=\frac{μ_0M}{2}(\frac{z}{\sqrt{z^{2}+R^{2}}}-\frac{z-L}{\sqrt{(z-L)^{2}-R^{2}}})}} $$ shows the relationship between the magnetic field strength and the distance between the magnet. I was...
  32. warhammer

    I Electric Field & Interplay between Coordinate Systems | DJ Griffiths

    Hi. I believe I have what may be both a silly and or a weird query. In many Griffiths Problems based on Electric Field I have seen that a coordinate system other than Cartesian is being used; then using Cartesian the symmetry of the problem is worked out to deduce that the field is in (say) z...
  33. B

    Velocity of a relativistic particle in a uniform magnetic field

    d(ɣmv)/dt = qvB (dɣ/dt)mv + ɣm(dv/dt) = qvB Substituting gamma in and using the chain rule, it ends up simplifying to the following: ɣ^3*m(dv/dt) = qvB Now, I am confused on how to solve for v.
  34. V

    I Electric field of a moving charge that's abruptly stopped

    Hello everyone, This is in reference to fig 5.19 (screen shot attached - please read the paragraph which says "Figure 5.19 shows the..."). I don't get why the field outside of the sphere of radius ct acts as though the particle would have continued its motion. Author's words : "The field...
  35. Salmone

    I How an induced electric dipole vibrates with EM field

    If we have an electromagnetic wave like the one in the picture and a molecule which is, in the image, the small black ball with electron cloud being the part with "minus sign" in it, does the molecule with its cloud start to oscillate, once the EM wave hits it, as an induced electric dipole...
  36. arjun_ar

    Calculate the magnetic field from the vector potential

    I am trying to derive radial and axial magnetic fields of a current carrying loop from its magnetic vector potential. So far, I have succeeded in deriving the radial field but axial field derivation gives me trouble. My derivation of radial field (eq 1) can be found here. Can anyone point out...
  37. G

    I What kind of tensor is the gradient of a vector Field?

    (1,1)or(2,0)or(0,2)?And does a dual vector field have gradient?
  38. JandeWandelaar

    I Uniform gravitational field possible in GR?

    Mentors’ note: this thread is forked from https://www.physicsforums.com/threads/free-fall-in-curved-spacetime.1016510/ But what if the gravity field is homogeneous? Like that of an infinite massive plane? The objects in the ship will stay where they are. An infinite massive plane is quite...
  39. L

    A Vector analysis question. Laplacian of scalar and vector field

    If we define Laplacian of scalar field in some curvilinear coordinates ## \Delta U## could we then just say what ##\Delta## is in that orthogonal coordinates and then act with the same operator on the vector field ## \Delta \vec{A}##?
  40. Tesla In Person

    Electric field strength at a point due to 3 charges

    I got E. 13q as the answer. That is what i did: The electric field due to +q at origin 0 should equal the electric fields of charges -3q and the new charge placed at 2x. So applying the equation above like this; k*(q) / (2^2) = -3q*k + (k*C)/ 4 solving for C the new charge added, gives 13q. I...
  41. Salmone

    I Two molecules with different polarizability in an EM field

    If I have two separated and non-interacting molecules with different constants polarizabilities ##\alpha_1## and ##\alpha_2## and I send an EM field of frequency ##\omega## first on the molecule no.##1## and then on the molecule no.##2## so that the two molecules will have a dipole moment...
  42. JandeWandelaar

    A What is the cause of the Mexican hat potential of the Higgs field?

    The Higgs mechanism is an ingenious mechanism inspired by condensed state physics. The famous Mexican hat potential ensures a VEV value of about two times the mass of the Higgs particle (which, as an aside, is of comparable order as the W and Z vector bosons, the difference though is that its a...
  43. Delta2

    Potential energy of a sphere in the field of itself

    My attempt was to consider spherical shells of radius ##r## (##r\leq R##))and thickness ##dr## and then the potential energy of this shell would be in the field only of the "residual" sphere of radius ##r## (a result also known as shell theorem) $$U_{dr}=G\frac{\rho\frac{4}{3}\pi r^3 \rho 4\pi...
  44. WMDhamnekar

    Computing##\displaystyle\int_C f\cdot dr ## for the given vector field

  45. Tesla In Person

    Flux of Electric field through sphere

    My attempt: We have 3 charges inside 2 +ve and 1 -ve so i just added them up. 4 + 5 +(-7) = 2q Then there is a -5q charge outside the sphere. I did 2q + (-5q)= -3q . The electric field flux formula is Flux= q/ E0 . So i got -3q/E0 which is obviously wrong : ) . After quick googling , I...
  46. Tesla In Person

    Electric Field Inside a Conducting Sphere: Is it Always Zero?

    Is the electric field inside a sphere always 0? Even if we have charges on the surface?
  47. Eclair_de_XII

    B Can the continuity of functions be defined in the field of rational numbers?

    I argue not. Let ##f:\mathbb{Q}\rightarrow\mathbb{R}## be defined s.t. ##f(r)=r^2##. Consider an increasing sequence of points, to be denoted as ##r_n##, that converges to ##\sqrt2##. It should be clear that ##\sqrt2\equiv\sup\{r_n\}_{n\in\mathbb{N}}##. Continuity defined in terms of sequences...
  48. N

    Electric Potential Field Calculation

    I've already tried to calculate the potential with respect to the 3 segments and then apply superposition (V1+V2+V3). However, I was not very successful. My error I think is in the calculation of the radii, mainly of the line segment that is on the z axis. Can anybody help me? I need some light...
  49. HelloCthulhu

    Mathematically expressing field driven water autoionization

    I recently read a paper on using an electric field to drive water autoionizaton. I'm trying to figure out how to use the Laplace equation on pg 9; 4th paragraph; to solve for voltage. I'm also interested in how this equation would change if I replaced the hemispherical tip with a parallel plate...
  50. T

    B Notation for a "scalar absolute field"?

    The notation I think best describes it is ## F = \lVert\int^{space}_s|\vec{V}|ds\rVert ## So you have a vector field V in a 3d space. For each point you integrate over all of space (similar to a gravitational or electromagnetic field) *but* vectors in opposite directions do not cancel, they...
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