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. 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...
  2. 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...
  3. 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...
  4. 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...
  5. P

    Electrostatics - electric potential and field strength (dielectric)

    Could somebody check my solution?
  6. 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...
  7. 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?
  8. 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 $$dS=rsin\varphi d\theta r d\varphi $$ $$ \iint\limits_S E \cdot dS = \frac{q}{4 \pi \varepsilon_0} \iint\limits_S \frac{r}{|r|^3} \cdot dS $$...
  9. 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...
  10. 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...
  11. 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...
  12. 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...
  13. 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...
  14. 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 +...
  15. 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...
  16. 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...
  17. 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...
  18. 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 =...
  19. 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}...
  20. 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...
  21. 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...
  22. 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...
  23. madafo3435

    Why is the field within a solid conducting material zero?

    .
  24. 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...
  25. 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...
  26. 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?
  27. D

    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 =...
  28. 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
  29. 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...
  30. J

    I Is the electric field of an atom a superposition or mean of electron positions?

    We usually think about atomic orbital as wave(function), but it was created from e.g. electron and proton approaching ~10^-10m (or much more for Rydberg atoms), and electron has associated electric field. This wavefunction also describes probability distribution for finding electron (confirmed...
  31. yyfeng

    Simple Electric Field due to a Charged Disk

    My attempt at a solution is shown in attached file "work for #10.png". I used Desmos Scientific online calculator to obtain my final answer.
  32. M

    Placing an AC magnetic field inside a static DC field

    Hello, I am looking for some information on how 2 different types of magnetic fields interfere with each other. And i don't mean, 2 magnets, but let me be specifically: Lets say that you have a very strong static magnetic field, from a huge magnet. (for instance, the strength of the magnet of...
  33. jamalkoiyess

    Do you feel stuck in your research field?

    Does any researcher/scholar/professor here ever feel stuck with the field they have chosen to pursue? Do you ever feel it's too late to switch or regret not choosing something else while studying?
  34. F

    I What is an interpretation of wave of field?

    What is an interpretation of wave of field(wave that its medium is field) in QFT?Is it correct that the trajectory that wave propagates is the "classical trajectory" of particle created by the field?
  35. S

    Why is the Force on the Wire Directed from Stronger to Weaker Field?

    I understand how the weaker and stronger field occurs but I don't understand why the force on the wire is directed from stronger to weaker field. I mean why not from weaker to stronger? I also want to ask about Newton's 3rd law in this case. Newton's 3rd law states that for every action there...
  36. E

    The electric field inside a hole inside a conductor is still 0?

    This is not a homework question but something that bugs me a bit. My professor has stated that the electric field inside a conductor is 0. This I understand. However, he has also said that even if the conductor has some hole in it, the electric field inside this hole is also 0 Now, two...
  37. P

    Relativistic momentum through a magnetic field

    p=Bqr and del p = p del theta
  38. Haorong Wu

    Quantum Anyone tried "Problem Book in Quantum Field Theory" by Radovanovic?

    It is a wonderful book for learning QFT. Interesting problems with detailed solutions. I have tried the problems from chapter 1 to chapter 7. In most chapters, I could at least solve some part of the problems. But I got stuck in chapter 4, the Dirac equation. I could not solve any of the...
  39. willDavidson

    Electric field of a point charge

    I am trying to understand what a point charge is. Is it just an electron? Or is it just an idea?
  40. willDavidson

    Electric field in a second dielectric given a 2 dielectric system

    I tried approaching this by finding the tangential and normal electric fields. Is this the correct approach? I've attached a drawing of the surface provided. ##\oint_S E \cdot dl=0## ##E_{tan1}\Delta x-E_{tan2}\Delta x=0## We know that ##E_{tan1}=E_{tan2} Next, we can find the normal...
  41. PhysicsTest

    The net magnetic field at the center

    The current direction is as follows I think so much and do the right hand rule i get 0 at the center, but not sure why the answer is non zero. I have shown the directions of the magnetic fields, i have not shown the magnitudes of equal length but they all are equal. Why the answer is non zero...
  42. phywithAK

    How can I find conserved current for a Lagrangian involving vector fields?

    Untill now i have only been able to derive the equations of motion for this lagrangian when the field $$\phi$$ in the Euler-Lagrange equation is the covariant field $$A_{\nu}$$, which came out to be : $$-M^2A^{\nu} = \partial^{\mu}\partial_{\mu}A^{\nu}$$ I have seen examples based on the...
  43. P

    Position for maximum electric field between two wires

    For the first part, since $$ E(r) \propto \frac{1}{r} \hat{r}$$ by the principle of superposition the maximal electric field should be halfway in between the two wires. Then I'm not sure how to go about the second part of the question. I understand that the total potential due to the two wires...
  44. LCSphysicist

    Field with rot = 0 but non conservative.

    x and y are different than 0 As you can check, rot (del x F) is equal zero. Immediately someone could imagine this is a conservative field. But it is not, it is not path independent in certain occasions. I just know one way, involving polygonal, but there is not another way to check if F is...
  45. J

    Modeling Uniform Field Gap Electrodes, Rogowski or others

    Hi, I am trying to design uniform field gap electrodes using the Rogowski profile. I've read some papers on it, but I'm completely new to this and I am just having a hard time translating what's in the paper to my Matlab code to produce the same results. I'm not sure if this is the right place...
  46. PhysicsTest

    Magnetic field of a straight current-carrying conductor

    It is not a direct home work problem, i was thinking if a sine wave current passes through the straight current carrying conductor, what will be the magnetic field. For the DC current I know the formula as below. ##B = \frac {\mu_0 I 2a} {4\pi x\sqrt{x^2 + a^2}}## Let the current be ##I =...
  47. M

    Find the electric field intensity from an infinite line charge

    what I've done so far? -i've determined the vector between the point (4, 0, 0) and the point P. (4, 6, 8) - (4, 0, 0) (0, 6, 8) -The norm of this vector is the radial distance of the line to point P (the value of “ρ” in the formula) √(0^2 + 6^2 + 8^2) = 10 -> ρ = 10 -and its unit vector is...
  48. wcjy

    Electric field problem using Gauss' law: Point charge moving near a line charge

    F = qE ma = (2*10^-6) * (λ / (2pi*r*ε0) ) ma = (2*10^-6) * (4*10^-6 / (2pi*4*ε0) ) => I am not certain what to put for r ( But I sub in 4 because dist is 4) a = ( (2*10^-6) * (4*10^-6 / (2pi*4*ε0) ) )/ 0.1 a = 0.35950 v^2 = U^2 + 2 a s v = 0 u^2 = -2 a s => Can't sqrt negative so...
  49. JD_PM

    Minimal substitution on the Lagrangian of the complex KG field

    a) I think I got this one right. Please let me know otherwise We have (let's leave the ##x## dependence of the fields implicit :wink:) $$\mathscr{L} = N \Big(\partial_{\alpha} \phi \partial^{\alpha} \phi^{\dagger} - \mu^2 \phi \phi^{\dagger} \Big) = \partial_{\alpha} \phi^{\dagger}...
  50. C

    Help finding the equilibrium position of an electric field

    I seem completely lost at this. I barely know where to begin. I know that the forces will sum to 0 but the vectoral nature of the question is really confusing me. Best I have is that the distance between e and q2 has to be sqrt(2) times the distance between e and q1. I don't know where to go...
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