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. E

    Energy current for field satisfying KG equation

    First to compute the time derivative of ##\mathcal{E}##,$$\mathcal{E}_t= \phi_t \phi_{tt} + c^2 (\nabla \phi_t) \cdot (\nabla \phi) + m^2 c^4 \phi \phi_t = \phi_t \left[ \phi_{tt} m^2 c^4 + \phi \right] + c^2 (\nabla \phi_t) \cdot (\nabla \phi)$$Then we switch out ##\phi_{tt} + m^2 c^4 \phi##...
  2. badluckmath

    Electric Field of two line charges

    I'm trying to solve this, but i don't really know how to start this problem. There are two line charges and i must find the E. Field on the center.
  3. J

    What effects does an electric field have on potential energy?

    Like an electric field is applying a sort of force on a particle. I was wondering if this at all impacts the potential energy of a particle. For instance, when the force of gravity does work on an object, its potential energy changes as a consequence. Would it be the same thing here?
  4. A

    B field around a wire, single wire electromagnet

    A quick description. A single straight wire and a second straight wire, both wires are electrically as well as physically separated, the physical separation distance assume is very small in order for the B field experienced by the second wire to be sufficiently strong. In all cases one of the...
  5. E

    What is the relationship between force lines and the stress tensor field?

    Force lines method is used in Solid Mechanics for visualization of internal forces in a deformed body. A force line represents graphically the internal force acting within a body across imaginary internal surfaces. The force lines show the maximal internal forces and their directions. But...
  6. D

    A The g_ij as potentials for the gravitational field

    The equation of motion for a particle in a gravitational field is ai = -Γijk vj vk In inertial coordinates the Lorentz force is mai = qFij vk So it seems like F corresponds to Γ. Just like F is expressed in terms of the derivatives of A, the christoffel symbols are expressed in terms of...
  7. Andrei0408

    Magnetic field of the planar wave

    I understand that because the vectors are perp, k x i = j, but why is k x j = -i? Why the minus? Could you please explain?
  8. A

    I Quantum Foundations: outlook on this research field?

    Recently, it has come to my attention a field called Quantum Foundations. This is exactly what brought me into Physics, even though back then I didn't know it was a research area. In my Physics classes, I got disappointed and unmotivated at the "Shut up and calculate!" attitude of my Physics...
  9. H

    I Magnetic Field Generated By Moving Charged Objects - Confusion Explained

    Suppose there is a charged line and near that line, there is a magnetic needle lying in the vertical plane of the line. The magnetic needle is radially placed. If the charged line and the magnetic needle are moving at a same constant velocity(parallel to the line, v<<c) towards an observer. I...
  10. J

    Field of a "pancake" coil in QI charger

    Attached is a photo the the primary transformer coil of a QI (I believe pronounced CHEE) wireless charger, as used for charging a cell phone. I know the fields of solenoids but what would be the magnetic field structure of this. By-the-way, a similar coil was used on old AM radio sets.
  11. entropy1

    B Is a photon an excitation of the electromagnetic field?

    Allow me to hijack this thread for a second: a photon is an excitation of the electromagnetic field, right? The photon does not exist until measured. So how can we send a photon in a particular direction, so it has a known position and momentum?
  12. S

    Line Integral to Verify the Magnetic Field B

    Hello folks, I'm working on a question as follows: I appreciate that there might be more sophisticated ways to do things, but I just want to check that my approach to the line integral is accurate. I will just give my working for the first side of the path. So I have set up the path as a...
  13. PhysicsTest

    Find the electric field from charge density

    There is a section in the BJT explanation the charge density and the corresponding electric field graphs. But i was not sure how the electric field is derived and hence i started deriving it. Please correct me if my understanding is wrong in posting the question It is an ##npn## BJT. My...
  14. B

    Understanding the electric field of a sphere with a hole

    Here's an image. O and O' are the respective centers, a is the distance between them, r is the distance from the center of the sphere to P, and r' = r - a, the distance from O' to P. The approach (which I don't understnad) given is to use Gauss' Law and superposition, so that we calculate the...
  15. DuckAmuck

    I Einstein Field Eqns: East/West Coast Metrics

    My questions is: Depending on which metric you choose "east coast" or "west coast", do you have to also mind the sign on the cosmological constant in the Einstein field equations? R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} \pm \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu} For example, if you...
  16. greg_rack

    Period of a metal rod oscillating in a magnetic field

    This problem honestly got me in big confusion. I managed to find the angle ##\theta## at which the rod rests by equalling the components of weight and Lorentz's force... but from this point on I really don't know how to manage the harmonic oscillation part.
  17. steve1763

    A Exploring Free and Interaction Terms of L in Quantum Field Theory

    With free part L=-½(∂Φ)^2 -½m^2 Φ^2 and interaction term L=½gΦ^2Any help would be appreciated, thank you.
  18. greg_rack

    The electric field between two adjacent uniformly charged hemispheres

    here is the situation Hi guys, I should illustrate the motion of an electron in both cases, but I cannot really understand how will the field be like in the gap between the two(filled) hemispheres(conductor and non). Another thing is: for the conductive hemispheres, does it make any sense to...
  19. greg_rack

    Torque on a current loop caused by a magnetic field

    Okay, so, the magnetic field lying(parallel) to the plane of the coil is confusing me quite a bit. Usually, in this kind of problem, we have a magnetic field directed perpendicularly to the plane. Considering this orientation of the field, wouldn't the torque on this sort of "elementary brush...
  20. fluidistic

    Physics Looking for a postdoc in a different field than my PhD, among other things

    Hello PF, I am about to defend my PhD in some area of Physics/Materials Science. I am also currently working at a company, but I want to resign after I defend my PhD thesis. I am offered a postdoc position and a salary raise if I stay in my current company but I feel I would mostly apply some...
  21. Kaguro

    Poynting vector and electric field

    The Poynting vector $$\vec S=\frac{1}{\mu_0} \vec E \times \vec B$$ gives the power per unit area. If I need this in terms of electric field only,I should be able to write B=E/c (for EM wave) Assuming they're perpendicular, ##S =\frac{1}{\mu_0 c}E^2##. Now, ##c=\frac{1}{\sqrt{\mu_0 \epsilon_0}}...
  22. V

    A Adjoint representation and spinor field valued in the Lie algebra

    I'm following the lecture notes by https://www.thphys.uni-heidelberg.de/~weigand/QFT2-14/SkriptQFT2.pdf. On page 169, section 6.2 he is briefly touching on the non-abelian gauge symmetry in the SM. The fundamental representation makes sense to me. For example, for ##SU(3)##, we define the...
  23. wcjy

    Magnetic field due to infinite current carrying wire in the X and Y axes

    $$B = \frac {\mu_0 I}{2 \pi r} $$ By Right-hand Grip Rule, the direction of the magnetic field by wire in y-axis is into the paper (z) while the direction of the magnetic field by wire in X-axis is upwards (+i) The answer state the Magnetic field is in the (i - y) direction though. Next...
  24. M

    Finding Expression for Emitted Field: A Confused Student's Journey

    All I'm reallly confused on this problem is what the expression for the emitted field is. As long as I've got that, I'm good to go, but I just don't know what to use. I've tried looking for an expression for the emitted field but I've had no luck. Would appreciate any ideas or someone telling me...
  25. M

    I Separating particles in a zero point energy field

    From my understanding, quantum fluctuations create particle pairs that are usually annihilated. Is it possible to use some kind of force (eg: electromagnetism) to direct and separate antiparticles from normal particles? I believe experiments have proven that it is possible to store positrons...
  26. P

    Electrostatics - electric potential and field strength (dielectric)

    Could somebody check my solution?
  27. patric44

    Finding the magnetic field B given the vector potential A ?

    hi guys this seems like a simple problem but i am stuck reaching the final form as requested , the question is given the magnetic vector potential $$\vec{A} = \frac{\hat{\rho}}{\rho}\beta e^{[-kz+\frac{i\omega}{c}(nz-ct)]}$$ prove that $$B = (n/c + ik/\omega)(\hat{z}×\vec{E})$$ simple enough i...
  28. L

    I Charged Particle Free Fall in Grav Field: Does Anyone Know Answer?

    From one point of view the charged particle is accelerating and should emit electromagnetic waves. But from the equivalence principle, I think, it should not. Does anybody know the answer?
  29. G

    I Problem about the usage of Gauss' law involving the curl of a B field

    I am trying to derive that $$\nabla \times B=\mu_0 J$$ First the derivation starts with the electric field $$dS=rsin\varphi d\theta r d\varphi $$ $$ \iint\limits_S E \cdot dS = \frac{q}{4 \pi \varepsilon_0} \iint\limits_S \frac{r}{|r|^3} \cdot dS $$...
  30. Peter Jones

    The oscillation of a particle in a special potential field

    I couldn't prove the first one but i tried to find the period F = -dU / dx = - d( U0tan^2( x / a ) ) / dx = - U0 ( ( 2 sec^2( x / a ) tan( x / a ) / a ) with F=d^2x/dt^2, tan(x/a)=x/a we have d^2x/dt^2 + U0 ( ( 2 sec^2( x / a ) ( x / a^2 ) =0 from there i don't know how to handle the...
  31. P

    Engineering Magnetic field strength of an electromagnet (coil wound around a bobbin)

    I am trying to design an electromagnet which consists of a copper PVC sheathed wire wound around a cylindrical plastic spool of Circumference (C) = pi x diameter. The spool has a hollow body of diameter D1. This wire has maximum length (L), cross sectional area A, resistivity P. The spool once...
  32. J

    I What is the Kalb-Ramond field?

    All String theories include the massless bosonic fields ##G_{\mu\nu}##, ##B_{\mu\nu}## and ##\Phi##. I understand that ##G_{\mu\nu}## is the spin-##2## field of the spacetime metric and ##\Phi## is the spin-##0## dilaton field. The ##B_{\mu\nu}## is called the Kalb-Ramond field and is said to...
  33. greg_rack

    Circular trajectory traveled by a charged particle in a magnetic field

    The Lorentz's force acting on a charged particle perpendicularly "hitting" a magnetic field will be directed upwards, and generally directed towards the center of the circumference traveled by this particle, and so will cause a centripetal acceleration to keep it in a circular motion. By...
  34. cestlavie

    Magnetic flux given magnetic field and sides (using variables)

    I know the answer is ##ka^3/2##. I got ##ka^2## and I don't know how to get the right answer. I saw an explanation using integrals, but my class is algebra-based. My attempt: ##Flux=ABcos\theta##. I figure ##cos\theta## is 1 becuase the angle between the magnetic field and the normal to the...
  35. DuckAmuck

    I EM Field Strength in Curved Spacetime: Is it Unchanged?

    It seems a gravitational field does not alter the electromagnetic field strength. Is this correct? My reasoning: With no gravity, field strength is: F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu Introduce gravity: \partial_\mu A_\nu \rightarrow \nabla_\mu A_\nu = \partial_\mu A_\nu +...
  36. Mr_Allod

    Magnetic Field, Field Intensity and Magnetisation

    Hello there, I've worked through this problem and I would just like to check whether I've understood it correctly. I found ##\vec H##, ##\vec B## and ##\vec M## using Ampere's Law and the above relations as I would for any thin current carrying wire and these were my answers: $$\vec H = \frac I...
  37. cestlavie

    Length of sides of a wire loop in a uniform magnetic field

    If ##\tau= 0.0727, N=60, i=1.3, B=1.0,## and ##\theta=15##, I tried the following calculation: ##\tau=NIABsin\theta## ##\tau=NIs^2Bsin\theta## ##s^2=\frac {\tau} {NIBsin\theta}=\frac {.0727} {60*1.3*1*sin(15)}=0.0632 m=6.32 cm## The answer is probably right in front of me, but I don't know what...
  38. SamRoss

    Magnetic field lines around electron and wire seem to contradict

    In the picture below, the direction of the magnetic field lines can be determined by using the right-hand rule with the thumb pointing in the direction of the current. If we use the right hand rule in the picture below, thinking of the yellow arrow as the current, we would not get the correct...
  39. Arman777

    I EDE - Solving the Klein - Gordon Equation for a scalar field

    Let us suppose we have a scalar field ##\phi##. The Klein-Gordon equations for the field can be written as \begin{equation} \ddot{\phi} + 3H \dot{\phi} + \frac{dV(\phi)}{d\phi} = 0 \end{equation} The other two are the Friedmann equations written in terms of the ##\phi## \begin{equation} H^2 =...
  40. G

    Problem about the derivation of divergence for a magnetic field

    Summary:: I am trying to derive that the divergence of a magnetic field is 0. One of the moves is to take the curl out of an integral. Can someone prove that this is addressable Biot Savart's law is $$B(r)=\frac{\mu _0}{4\pi} \int \frac{I(r') \times (r-r')}{|r-r|^3}dl'=\frac{\mu _0}{4\pi}...
  41. P

    Evanescent Waves in near field for aperture > lambda (diffraction)?

    We have two different accepted formulas for the far field and near field respectively. I want a numerical program that works for both, furthermore I want to use it to calculate power through the aperture after confirming it in the far field vs near field. I start off by treating the far field...
  42. G

    Find the electric field from polarization

    Attempt at solution: a) Since I need help with b) this section can be skipped. Results : ##ρ_{psa} = -Pa ## ##ρ_{psb} = Pb ## ##ρ_{p} = \frac {-1}{R^2} \frac {∂(R^2PR)}{∂R} = -3P ## b) This is where I am unsure (first time using gauss law for P) so I need some confirmation here: ## \int...
  43. Y

    Programs Associate of Applied Science Degree in non-career related field?

    Hello All, Does anyone here have degrees that aren't related to their careers? I was thinking of maybe taking some automotive courses at a community college towards an associates. This is merely for my own interests, and to allow me to work on my own car, knowing that I did the job correctly...
  44. madafo3435

    Why is the field within a solid conducting material zero?

    .
  45. C

    Characterizing Total Charge of Conductor A in an External Electrical Field

    Assume that a certain charge distribution ##\rho## generates an electrical field ##E_{ext}## in the surrounding space. We also note the corresponding generated potential ##V_{ext}##. Assume furthermore that a conductor A, with a definite shape and volume, is placed in field ##E_{ext}##, and is...
  46. PhysicsTest

    Understanding the Continuity of Current in a Rotating Magnetic Field

    I am analyzing the rotor magnetic field, i feel i understand the basic concept but have few clarifications. At pt1, the net mmf due to currents ##i_a = i_{max}; i_b = -\frac{i_{max}} 2 ; i_c = -\frac{i_{max}} 2## is ##\frac {3F_{max}} 2## Similarly i can do for Pt2. But my confusion is the...
  47. P

    Electrostatics: Calculate the Electric Field near a Charged Ring

    I have the problem with my solution. I don't know it is correct. Could somebody check it?
  48. docnet

    Finding the flow of a vector field

    In part c, plotting the vector field shows the vector field is symmetric in x and y in the sets {x=y}. in {x=y}, the variables can be interchanged and the solution becomes x = x°e^t y = y°e^tHowever, these solutions do not work for anywhere except {x=y} and don't satisfy dx/dt = y and dy/dt =...
  49. S

    Will EMF be induced in a coil that is accelerating in a uniform magnetic field?

    My answer will be no for both (a) and (b) because there is no change in magnetic flux experienced by the circular coil. Am I correct? Thanks
  50. huszarerik

    Electric field lines between a point-charge and a conducting sheet

    figure 1: → I don't understand how to approach this problem. Basically it asks for the distance r.I think I should use Gauss's law, but I've been thinking about the shape of the gaussian surface and I'm not sure about how it should look or where I should place it. Any help would be useful...
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