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.
Hello, I asked a question about superconductors in 2020 and I was now wondering what superconducting chemical/material can have the highest magnetic field strength before the superconductivity is destroyed by it? Secondly, What the is maximum magnetic field strength of said material in Tesla per...
I am thinking about how an electric field has energy associated with it. If a single electron exists alone in a remote vaccuum, I believe it has it's own electric field surrounding it, and that this field has an energy content associated with it. My question is; does this electric field store...
What is the relationship between the electromagnetic field and space-time? I am basically assuming that space-time is one big gravitational field.
Is there a relationship between space-time and the field (I presume) created by the strong force (however negligible it may be at any significant...
Given the equation ##\frac{xy} 3##. It is a fact that the gradient vector function is always perpendicular to the contour graph of the origional function. However it is not so evident in the plot above. Any thought will be appreciated.
In Sydney Coleman Lectures on Quantum field Theory (p48), he finds : $$D\mathcal{L} = e^{\mu} \partial _{\mu} \mathcal{L}$$
My calulation, with ##\phi## my field and the variation of the field under space time tranlation ##D\phi = e^{\mu} \frac{\partial \phi}{\partial x^{\mu}}## ...
I need to find the magnetic field of a permanently magnetized cylidner:
In calculating the magnetic field, i find that it should be $M_{0} \mu / 2$ and $H = M_{0} / 2$ inside. I just want to make sure that i understand the concepts in this type of problems.
Since $M = H \chi (1)$, does this...
First I wrote in ##S'##, by using Gauss theorem
$$
\int_{\Sigma} \underline E' \cdot \hat n d\Sigma = \frac Q {\varepsilon_0} \rightarrow E'(r)2\pi rH=\frac{\lambda'H}{\varepsilon_0}
$$
$$
\underline E'(\underline r)=\frac{\lambda'}{2\pi\varepsilon_0r}\hat r
$$
Its components are...
I need to calculate the magnetic field generated by a static sphere at its center. On the surface of the sphere flows a constant current ##K \hat \phi##.
Now, my guess was that the field produced would be equal to the field produced by a lot of rings, that is, i will split the sphere in a lot...
When a charged particle moves in a magnetic field, the magnetic field will not work. But when a nail flying over a permanent magnet is attracted by the magnet, the magnet does work on the nail. Can anybody explain why this is so?
From what I understand, electrons are negatively charged, however, I have recently come to learn that electrons also have a spin which creates a magnetic field around each electron. I don't understand how the electron can be a negative monopole, yet have a completely independent magnetic field...
This is not a homework problem, I want to calculate the equation of the magnetic field intensity from the 3 phase currents separated by 120Degrees. The 3 currents are
##I_{{aa}^{'}} = I_M\sin \omega t ; -> eq1 \
I_{{bb}^{'}} = I_M\sin(\omega t - 120); -> eq2\
I_{{cc}^{'}} = I_M\sin(\omega t...
I can calculate the fields generated by the cylinder and the wire but I don't know how to calculate their vector sum to evaluate it at point A.
Cylinder field inside: ρR^2/2rε
Cylinder field outside: ρr/2ε
Field generated by the wire: λ/2πr
I should break the fields into components but I don't...
Hello! So I need to find the potential function of this Vector field
$$
\begin{matrix}
2xy -yz\\
x^2-xz\\
2z-xy
\end{matrix}
$$
Now first I tried to check if rotation is not ,since that is mandatory for the potentialfunction to exist.For that I used the jacobi matrix,and it was not...
I used the concept of electrostatic induction, which would cause the charges in metal ball near the ebonite rod to have +ve charges on end next to rod and a -ve charge on the end touching the other ball.
What confuses me is how charges separate on the second ball. The only way these balls can...
Hello! I am suspossed to write (sketch) this particular vector field.
$$V2(r) = \frac{C}{\sqrt{x^2+y^2+z^2})^3} * (x,y,z) $$ Note that the x y z is suspossed to be a vector so they would be written vertically (one over the other) but I don't know how to write vectors and matrices in LaTeX,so...
[New poster has been reminded to show their best efforts to work the schoolwork problem when starting a homework thread]
My question is : An electron beam with velocity vector v = (0; 0.6x10^8 ;0) m.s enters between two oppositely charged plates parallel to the xz plane.
- How large is the...
I'm given an ultra-high energy cosmic ray with energy 10^20 eV. It is coming from a source 10 Mpc away with an extragalactic magnetic field with strength B = 10^-9 G. I am to determine the maximum angular deflection of this cosmic ray, so it hits Earth.
I don't have an attempt of the solution...
While classical mechanics uses single action optimizing trajectory, QM can be formulated as Feynman ensemble of trajectories.
As in derivation of Brownian motion, mathematically it is convenient to use nonphysical: nowhere differentiable trajectories - should it be so?
Can this connection be...
Summary:: Can a moving object cause disruptions in a magnetic field that could be detectable?
Hello,
I was hoping someone could assist me on a query I have regarding disruptions in a magnetic field. For some context, I am creating a science fiction story which features a non-humanoid alien...
Hi, I tried to solve this exercise but I'm not sure about the process.
First of all, I imposed that "K = E":
so that "v = √ ( (2q ∆V)/m))"
then I replaced in "r = m v / (| q |B)", v with "√ ( (2q ∆V)/m))", and found out that R = (2√(2)) r.
Then for the second point,
I imposed Lorenz Force...
Hi, I was practicing some problems on the magnetic field and the electromotive force, when I got stuck on these two exercises. Could you help me figure out how to proceed?
In the first problem, I tried to find the magnetic field flux by multiplying the induced current for ∆t and R. Should I now...
For a real scalar field, I have the following expression for the field operator in momentum space.
$$\tilde{\phi}(t,\vec{k})=\frac{1}{\sqrt{2\omega}}\left(a_{\vec{k}}e^{-i\omega t}+a^{\dagger}_{-\vec{k}}e^{i\omega t}\right)$$
Why is it that I can discard the phase factors to produce the time...
Hi guys ! I am taking phys102 course .I figured out that i didnt fully understand the concept that is the Voltage. Please look at this question. In this case we can see that there is no voltage in point C and there is no electric field is in point D and as you know V=U/q .So i always thought...
In QFT where the electromagnetic field is mediated by virtual photons, is it possible to describe the larmor precession of an electron as a series of emission and absorption of virtual photons? how does the spin angular momentum "evolve" over a series of events? This feels like a challenging...
I am only asking about part (b)(i) and (b)(ii).
Below is the explanation for (b)(i).
What is going on in the above? I understand up till the 3rd line, about the left/right hand circular motion. What is the "upward motion" the solution mentioned? Is it suggesting that ions are moving...
Hello!
Lately I've been experimenting with the ways an electromagnet effects a Rare Earth magnet. The electromagnet we used was taken from a vibrator massager, probably 50s vintage. The resistance of the coil is 96 ohms and consumes about 1.25 amps when operated on 120 volts AC. When the...
I am a little confused why my answer is wrong... Briefly, my answer is as follow: $$P = \epsilon_{0} X E = (\epsilon-\epsilon_{0}) E; \space \space \space \space \space \space \space E = P/(\epsilon-\epsilon_{0})$$
So what I'm not sure on, is calculating the matrix elements for part (iii) with Pauli spinors and Pauli matrices, and then finding the form of the corresponding states. As I don't see how using the hint helps.
The following is using the eigenvalues of the spin-operators.
Provided what I...
For a solenoid, magnetic field at the centre = ##\mu_0nI##.
I see the argument on why at the opening at the ends of the solenoid, the B-field is ##\frac12\mu_0nI##.
Apparently, B-field is ##\frac12 \mu_0nI## at the sides of the solenoid too. (ie at/within the wires that make up the solenoid)...
In Theoretical Minimum: Quantum mechanics, Leonard Susskind describes an electron in the higher energy spin state in a magnetic field radiates a photon of energy ##\hbarγ|B_0|## and flips into the lower energy spin state. I am wondering if this photon is related to the "virutal photon" that...
I understand that negative charges create electric fields pointing inwards, and positive charges create electric fields pointing outwards, but what does this have to do with field stength? What is the relationship between field strength (flux?) and direction?
weight/mass = gravitational field strength.
my working is ->
weight = 150kgx10m/s² = 1500N
mass = 150kg
gravitational field strength= 10N/kg.
is this correct?
A massless scalar field in a curved spacetime propagates as $$(-g)^{-1/2}\partial_\mu(-g)^{1/2}g^{\mu\nu}\partial_\nu \psi=0 .$$
Suppose the gravitational field is weak, and ##g_{\mu\nu}=\eta_{\mu\nu}+\epsilon \gamma_{\mu\nu}## where ##\epsilon## is the perturbation parameter. And let the field...
Will translation parallel to x-axis work ?
Else please suggest the symmetry?
And does symmetry here refer to the symmetry of the sheet which causes the symmetry of the field or something else?
Please be kind to help.
Our class modified an experiment to measure the magnetic field strength in mT between 5cm and 30cm, and I have plotted data and found that the relationship resembles a power relationship (using a log vs log graph). In order to find the percentage uncertainty for the whole experiment I need the...
I used a couple ways to do this question, but I got neither correct. Can someone help, please? Thank you.
1. E= V/r = 700 / (60*10^-3) = 11667 (very far from the given answer)
2. E = (-kQ/r)⋅ dr
= kQ/r^2
= kQ/ [( 1/20/ 10^-3)^2 - (1/80/10^-3)^2]
(For this method, I stuck...
In string theory, physical states satisfy QBΨ = 0, where QB is the BRST operator. This equation of motion can be obtained from an action
S = ∫ QBΨ*Ψ + Ψ*Ψ*Ψ
There is a gauge invariance under δΨ = QBΛ. what is the framework in which the role of the BRST operator QB is understood in open string...
I find a exercise in Leonard Susskind's book Classical Mechanics
the Hamiltonian of a charged particle in a magnetic field(ignore the electric field) is $$H=\sum_{i} \left\{ \frac{1}{2m} \left[ p_{i}-\frac{e}{c}A_{i}(x) \right]\left[ p_{i}-\frac{e}{c}A_{i}(x) \right]...
Homework Statement:: The solution to the KG equation is assumed to take the form$$\Phi = \sum_{l=0}^{\infty} \sum_{m=-l}^{l} \frac{1}{r} \phi_{lm}(t,r) Y_{lm}(\theta, \phi)$$
Relevant Equations:: N/A
To first show that $$\left[ \frac{\partial^2}{\partial t^2} - \frac{d^2}{dr_*^2} +...
Hi!
I have been looking at differential forms, manifolds and de Rham cohomology. Now I'm trying to figure out the connection from cohomology and equations of motions and topological field theory. Problem is that I am only looking at abelian field theories and I only find introductions into...
I have been analyzing a set of data from a lab activity on the Zeeman effect. The data (i.e. images) gathered can be previewed via this Google drive link here.
While I am provided with the numerical data on the current (##I##), I am not provided with any data on the magnetic field. With the...
https://www.physicsforums.com/attachments/282201
Are we using this equation above to explain this question? The magnetic field is definitely in sinusoidal form but how does it proportional to the frequency of the source?