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. Costweist

    Changes in electric field lines as a result of an oscillating charge

    The last couple of days I’ve been troubled with a specific part of electromagnetism. How will electric field lines be affected by an oscillating charge? More specific, what will happen with the “amplitude” of a wave in an electrical field line as the wave propagate away from the charge? 1. Will...
  2. Y

    Conservation of Energy on Current-Carrying Wire in Magnetic Field

    So force on a current carrying wire = ILxB. If I have a bunch of bar magnets making a uniform magnetic field of strength B, then a 1 meter long wire of 0 ohms carrying 1 Amp, the force on that wire is (1)(1)xB = 1B. If I let that force move the wire for a time T, let's assume the wire moved a...
  3. K

    B How the Earth's Magnetic field deflects the solar wind

    On a previous thread (now locked) I was wondering about how, precisely, the Earth's magnetic field protects us from the solar wind. Posting this here because what I wrote in that thread is very wrong, and I think it's an interesting topic. I had a hell of a time finding good information. I...
  4. A

    Electromagnetism - movement of a coil in a magnetic field

    So my idea was that to reach the equilibrium position, the final moment of force has to be 0 (so in the end the forces will “eliminate” each other). And I found the equation Fm=B*I*l*sinα, which should characterize the force, which affects wire with the current in a magnetic field, and Fleming’s...
  5. B

    How to create an Electric Field?

    Dear friends, First of all I have one question! As per Figure 1, how to implement electrical connection in real life which are seen inside Red Box? and what is the meaning of grounding the other terminal? Figure 1 And the second thing is that, I want to create and electric field on copper...
  6. B

    Do neuron electrical signals generate an electromagnetic field or wave?

    I'm not sure where this belongs, I'm guessing biomedical, but I'm interested from a physics perspective. Do neurons generate an electromagnetic field? In other words, all the neural activity in the brain, does it generate electromagnetic fields? If so, what are the details of these fields? I...
  7. A

    Charges, rod and magnetic field

    I have some difficulties in solving this problem. This is what I did. I wrote down the equation of motion for the masses. For the first point \begin{equation} m\ddot{\textbf{r}}_1=\textbf{F}_1=q\dot{\bar{\textbf{r}}}_1\times...
  8. H

    Question about the force exerted by a uniform electric field on a dipole

    I think the answe should be C as shown in the attachment.
  9. seabass101

    I Hubble deep field & ancient galaxies

    Hubble deep field allowed us to study galaxy evolution from 500 million years onward. Based on my (limited) understanding, I would expect ancient galaxies to contain fewer heavy elements and to have a more "juvenile" appearance, as compared to modern galaxies. Have we actually observed these...
  10. ilper

    Constant velocity in a central force field

    A body on a circular orbit in the field of a central force (satellites in gravity field of Earth; a charge in a magnetic field) is subjected to a force which is always perpendicular to its initial velocity v, hence in a time period dt it acquires an additional velocity dv, which is also...
  11. L

    Earth's Magnetic Field: Why Doesn't It Lose Its Properties?

    We know that when a magnet is exposed to high temperatures, it loses its magnetic properties. Why then does the Earth's magnetic field behave differently? That is, why doesn't the Earth lose its magnetic properties? According to BBC News Brasil, the core temperature is around 6000 ° C, higher...
  12. Ibix

    A Gravitational field of an infinite flat slab

    Off the back of a recently closed thread where there was some discussion about the gravitational field of an infinite flat slab, I decided to have a play at investigating that. I've found a few interesting things. It's fairly straightforward to solve for this situation. You use Cartesian-esque...
  13. Raydmajeed

    Gravitational potential energy from Einstein field equations

  14. M

    Magnetic field of a current loop

    This should be pretty straightforward and my take gives B = 0.000193*T for a) and B = 0.00000063 T for b). Or have I misunderstood something?
  15. F

    Understanding the Electric Field of Two Spheres: A Scientific Approach

    I am not quite sure how to present my answer in the form of a function with relation to the distance from the centre. What I got so far is the E1 and E2, for the internal and external sphere respectively. For internal sphere, the charge is volume * 𝜌, so it is $$ \frac{4\pi r^{3}}{3} * 𝜌$$...
  16. docnet

    I Precession of spin in a magnetic field

    In this lecture Lenny Susskind describes a spin in a magnetic field precesses around the axis of the direction of the magnetic field. This description is also frequently found in NMR theory which is a semi-classical theory. Lenny says if the magnetic field ##B_o## is applied in the ##z##...
  17. Steven Ellet

    Can a horseshoe magnet without separated poles still function as a magnet?

    If you take a horseshoe magnet and fuse the north and South Poles together (without destroying the magnetic field) would you have a “pole-less” magnet? And if so, what special properties would it have(other than other magnets)?
  18. R

    Is the magnetic field inside a solenoid stronger near the border?

    I was reading some papers about calculating the magnetic field produced by a coil using the biot savart law and I saw some graphs that caught my attention. This one from a paper from Ravaud, et al. Titled "Calculation of the Magnetic Field Created by a Thick Coil". I saw similar graphs in...
  19. B

    Energy within an electric field

    I am trying to calculate the energy within an electric field that is generated between two plates by a pulse but am unsure of what voltage value to use. The pulse is a sinc wave. I am assuming I can still use the equation ## E= \frac{1}{2}CV^2 ##. I know the ##V_{rms}## and ##V_{max}## which...
  20. F

    Finding Electric field E due to an arc

    I need to account for tension, weight, and repulsion. For the tension, I can draw the x and y component of Tmax and see that the x components of the 2 tensions Tmax will cancel out, and there are 2 y components of the Tmax to factor in. Weight is just F = mg, where g is acceleration due to...
  21. A

    Electric Field Shielding: Does a Conductor Shield Inside?

    Hi. I was reading about conductors in electrostatic equilibrium and how it makes sense that they have zero electric field inside the material even when an external charge is brought near. The charge density of the material just rearranges itself to cancel. Then I searched for hollow conductors...
  22. C

    Can't solve an equation (Deflection of electrons in electrostatic field)

    Hello everyone! I've tried everything but the equation (3) in "Deflection of electrons in electrostatic field" is impossible. Can someone at least hint me to a a way the composed it ?
  23. Kaguro

    Concept of electric field and hollow conductors

    I think: Due to charge q, there will be a field in region 1, very much dependent on position of q. The inner surface charge density of irregular conductor is also dependent on the position( so that it could cancel the field of charge and E=0 inside body of irregular conductor). The outer...
  24. P

    Electrical field between conductors of the same potential

  25. M

    Electric field of non-conducting cylinder

    a) I have calculated (1) λ = ρA = ρπr^2 = 2.49 * 10^-10 C/m and placed it into (2) yielding E = λ / (2πεx^2) = 106.73 N/C. This doesn't seem to be correct by the feedback, however. b) Here just to consider the proportion of the cylinder mass constrained by y.
  26. K

    I Pushforward/Pullback of Vector Field

    I am looking at the following document. In section 2.3 they have the formula for the pushforward: f*(X) := Tf o X o f-1 I am having trouble trying to reconcile this with the more familiar equation: f*(X)(g ) = X(g o f) Any help would be appreciated.
  27. zonde

    I Interpretation of Potential Energy as Field Property

    I will quote this statement from another thread: In that thread number of other posters seemed to agree with this statement. So I tried to analyze it a bit. For the sake of my questions let's say we limit GR to Schwarzschild spacetime and if there are problems with gravitational potential...
  28. A

    A charged conducting ring rotating B field -- referece frames

    Admittedly I found similar threads here already but due to my rather lacking math skills I wanted to go through this myself. As for the math side, I see various different equations with which this is treated can someone please provide the formulas for calculating B field from a rotating charged...
  29. sep1231

    Electric Field as potential gradient

    I know that the electric field is directed from Q to P, but I'm not sure which magnitud is greater, I tried this
  30. B

    Using Faraday's Law to calculate the magnetic field

    I got stuck near the beginning, so I tried working backwards. Starting from B = (k X E0)/ω * cos(k⋅r - ωt +φ) I found -∂B/∂t = -k X E0 sin(k⋅r - ωt +φ) So now I need to find ∇ X (E0 cos(k⋅r - ωt +φ)) and see that it is equal to the above result. This is where I'm stuck though, I'm not sure...
  31. greg_rack

    Induced current in a coil from a constant uniform magnetic field?

    Assume a solenoid coil(made up of ##N## windings) placed in the horizontal(##\hat{y}##) direction and in a constant uniform magnetic field. Would an induced current run through the(closed) coil if it spins around its central horizontal ##\hat{y}## axis? My guess is "no", since such a current is...
  32. F

    I Can we quantize a static EM field?

    Can we have a quantization static EM field?If not, how can we interpret static EM field in stand point of QM?
  33. E

    I Classical equivalent of scalar free field in QFT

    Hi there, In QFT, a free scalar field can be represented by the lagrangian density $$\mathcal{L} = \frac{1}{2}\left(\partial\phi\right)^2 - \frac{1}{2}m^2\phi^2$$ I would like to find a classical system that has the same lagrangian. If we consider the transversal motion of an elastic string...
  34. F

    Lagrangian for the electromagnetic field coupled to a scalar field

    It is the first time that I am faced with a complex field, I would not want to be wrong about how to solve this type of problem. Usually to solve the equations of motion I apply the Euler Lagrange equations. $$\partial_\mu\frac{\partial L}{\partial \phi/_\mu}-\frac{\partial L}{\partial \phi}=0$$...
  35. AndreasC

    Non-interacting gas in homogeneous gravitational field

    It even gives a hint, it says "consider two horizontal surfaces z1 and z2 and think about what thermodynamic equilibrium means for particles traveling from one surface to the other". This really trips me up because I am not sure what to do with this. Obviously in equilibrium the number of...
  36. fluidistic

    Permanent magnet's magnetic field calculation

    Hello people, in a near future I'd like to calculate (numerically, with finite elements) the magnetic field of several permanent magnets of various shapes. I am wondering which equation(s) I should solve, exactly. It's been a long time I dived into an EM textbook and I don't have one in hand...
  37. F

    I What is difference between EM wave and EM field?

    They say wave function is different to quantum field. Then what is the difference between EM wave and EM field?(By the way :Is that EM wave the wave function of photons?).It seem to me EM wave is the wave of EM field?
  38. E

    Waveform produced by a collapsing magnetic field

    A thought experiment: A electron is moving in a straight line at velocity v. It instantly stops dead. It doesn't move another femtometer. Obviously its magnetic field collapses and produces light. What is the waveform of the light produced? Is it something like this...
  39. J

    Other How much does one's undergraduate field of research matter?

    I am currently an undergraduate sophomore at a US university that is very reputable for physics. I am majoring in physics, and would like to one day attend grad school, so I tried to start research early and was able to find a research position this fall semester. I emailed a couple of theory...
  40. TechieDork

    Find an expression for a magnetic field from a given electric field

    Here this is my attempt : Reference Textbook : Zangwill's Modern Electrodynamics I stuck at the last step , I really have no idea what to do next.
  41. sergiokapone

    Law of motion for orbiting particle in a uniform magnetic field.

    Hi all, I interested in how can I get low of motion in for orbiting particle in a uniform magnetic field $$\frac{d\vec{r}}{dt} = \vec{\omega}\times\vec{r},\qquad \vec{\omega} = \frac{e\vec{B}}{mc},$$ Of course, rotating about z' axis is very simple. \begin{equation}\label{eq:K}...
  42. Y

    How does the saturation flux density affect magnetic field strength?

    So I'm confused what the Saturation Flux Density is referring to. Defintion says it is when you no longer get an increase in H-field when increasing external B-field. So, does the satuation flux mean the core can only create fields UP TO that saturation flux, or that it can make a stronger...
  43. Dale

    I Number of photons in an arbitrary EM field

    Correct me if I am wrong. I understand that if ##\vec E## and ##\vec B## are solutions to Maxwell’s equations then ##\Psi= \vec E + i \vec B## is a solution to Schrodinger’s equation. Is there an easy way to calculate the statistical distribution of the number of photons, or at least the...
  44. namo99

    A charged particle entering a magnetic field -- find the position

  45. D

    How do I generate a magnetic vector field using equations?

    I am considering using a pair of point charges: positive and negative electric charge to model a magnetic dipole's magnetic field by just average the electric field vectors between the two charged particles where they overlap. Will that work? In this case the + field will be vectors pointing...
  46. Philip Koeck

    The magnetic field just above a lens in an electron microscope

    I'm looking for an estimation or simulation of the magnetic field in the horizontal plane just above a typical lens in a transmission electron microscope. A rough cross section of such a lens can be seen here: electron lens - Bing images . The lens is cylindrically symmetric around the vertical...
  47. A

    Electric Field from Non-Uniformly Polarized Sphere

    Solving for the volume and surface bound charge densities was easy using equations 1) and 2). The polarization only has an r component so ##ρ_b=-\frac 1 {r^2} \frac {d} {dr} (r^2 \vec P)=-α(n+2)r^{n-1}##, and ##\hat n=\hat r## so ##σ_b=αa^n##. To find ##\vec E## I intend to use equation 3)...
  48. E

    What is the Constant of Motion in a Rotating Potential Field?

    I'm getting a bit stuck here, the Lagrangian and equation of motion is$$\mathcal{L} = \frac{1}{2} m \dot{\mathbf{x}}^2 - V_0(R^{-\omega t} \mathbf{x}) \implies m\ddot{\mathbf{x}} = -\nabla_{\mathbf{x}} V_0(R^{-\omega t}\mathbf{x})$$as expected. To try and verify that the quantity ##E - \omega...
  49. waazwag

    Induced voltage difference in a magnetic field

    Hi everyone, I'm currently working on the problem listed above. I'm pretty new to electrodynamics, and I'm learning on my own through a book. I was wondering if someone can please help me through this problem. Here are my thoughts:I think I need to use Faraday's Law of Induction for part (a)...
  50. PhysicsTest

    To find the parameters of velocity in an electric field

    i have drawn the E field as below, hence the F will be in the upward direction for electron a. Using energy is constant, the velocity ##v_x## as it crosses A is ##0.5mv_x^2 = q*V_a## ##v_x = \sqrt{(\frac{2qV_a} m)} m/s## one doubt i have here is, the question mentions electrons, but i have...
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