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.
Hi,
I just got an IT Internship at a local nonprofit. It sounds like it will be a mid-high-level desktop support position. I will be imaging computers and that's all I know so far. I start ASAP and they're processing the paperwork as I speak. Anyway, I haven't worked in IT since 2017 and now I...
I want to render the Earth’s Magnetic field in a software and simulate solar wind electron interaction with it. How do I calculate the magnetic strength and vector orientation at each point around the Earth up to thousands of km?
Is there a formula?
Or do I need to download a vector field from...
Given a box made of perfectly conducting material. At some part of it at t=0 there is a localized magnetic field. It's sourceless and there aren't any dissipation. After t=0 it starts to spread and fill the box. What is the equilibrium state?
Thanks for the answer(s),
Robert
I suppose that that a spacetime geodesic of an object falling on Earth would a appear as straight line. But what I'd like to see is a whole bunch of relevant geodesics that would represent falling bodies all around the Earth such that one could zoom out and so see these straight line geodesics...
First, let me introduce the notation; given a Hamiltonian ##H## and a momentum operator ##\vec{P}##, and writing ##P=(H,\vec{P})##. Let ##|\Omega\rangle## be the ground state of ##H##, ##|\lambda_\vec{0}\rangle## an eigenstate of ##H## with momentum 0, i.e. ##\vec{P}|\lambda_\vec{0}\rangle=0##...
Hi, second problem in one evening, I'm sorry!
But I'm also not quite sure if I did this one right.
I had thought I need lenz's law but there is no current before entering the field so I just use the induced Voltage?
My approach:
## V = \frac {B*A}{t} ##
## IR = \frac {B*A}{t} ## and ## A = v*t...
I went with R=mv/qb, thus -> 6.64e-27*35.6e3/2*1.6e-19*1.8, and got 4.1e-4 m (metres), so diameter is 2R, 8.2e-4 m, as an answer, the reference site gives 3.95e+10 m as the answer, who's right here?
Hello everyone,
I was wondering which field of study teaches how to make electronic components. For example, I would like to learn how to make resistors, caps, led, transistors... I have more or less of the theory, such as what the Dirac gaps are, if we talk about semiconductor... But what I...
I have to find pμ(τ) of a particle of mass m and charge q with v(0) = (vx(0), vy(0), vz(0)) in a electric field E parallel to the y-axis and a magnetic field B parallel to z axis, both constant and uniform, with E = B.
Here follows what I have done (see pictures below):
I wrote 4 differential...
Hi, at this moment I'm trying to figure out one thing. I have a solenoid with a core that has an empty middle, the flux normally loops back around the outside of the solenoid to the other side where it enters back into the core. I need to route this field between the two ends of the solenoid...
I believe the answer is incorrect, reasons:
The answer assumes that electric field will exist .
But this is not the case , until and unless there is a bipolarity there cannot be an electric field ( in case of isolated charged objects, the field exists because the bipolarity is separated by a...
Hello! In several of the derivations I read so far in my QFT books (M. Schawarz, Peskin and Schroeder) they use the fact that "we can safely assume that the fields die off at ##x=\pm \infty##" in order to drop boundary terms. I am not sure I understand this statement in terms of QFT. A field in...
When a charge is at rest, it has an electric field only. When the charge starts moving , it is said to have accompanied a magnetic field. My question relates to its electric field while in motion. Does it still exist or not? I know in electron guns electrons are deflected while passing thru the...
Usually when setting up an energy equation I use the general form, (Initial KE) + (Initial PE) + (Any other work done to the body) = (Final KE) + (Final PE) ...
For this I said the initial GPE and KE are 0, and the work done by the field is GMm/x (derived by integrating a force of -GMm/r^2 from...
Do solar wind electrons turn left and positively charged ions turn right if they are interacting weakly due to long distance with Earth's magnetic field and fail to complete a loop for the electrons that are on the left of Earth and ions on the right? I assume electrons on the right and ions on...
Hello everyone,
There is an electrical field inside and outside (at the same time) the spherical hollow conductor when we place positive or negative charge inside, isn't it?
I know this is because of the induced charges on the inner and outer surfaces of the conductor. There is no field inside...
We are given: q1 = +2.0 x 10-5 C, q2 = q3 = -3.0 x 10-5 C, r31 = r21 = 2 m
a) We start by finding the electric force between q3 to q1 and q2 to q1
FE31 = k * q1 * q3 / r312
FE31 = (9.0 x 109 Nm2/C2) * (+2.0 x 10-5 C) * (3.0 x 10-5 C) / (2 m)2
FE31 = 1.35 N
FE21 = k * q1 * q2 / r212
Since...
I know how to find the electric field of more traditional designs i.e. a sphere, through Gauss' Law but I don't think Gauss' Law applies to this scenario. I tried to separate each part of the electrode into simple spheres and rods and using Gauss' Law to find these individual elements. This...
Tried to find the resultant force, but I can't see how the magnetic field affects. I used Faraday's law to find the the diferece of potentials in the plate Wich should be B.d.v, where v is the vertical velocity of plate, but there were not given the resistance or resistivity to relate with the...
a) We can solve for acceleration by looking at FNETy
FNETy = FE (G is negligible)
FNETy = m * a
The mass (m) of an electron is 9.1093836 x 10-31 kg.
The elementary charge (q) of an electron is -1.60217662 x 10-19 C
a = ε * q / m
a = (4.0 x 102 N/C * 1.6022 x 10-19 C) / 9.1094 x...
As I said my goal is to derive the equation ##\tilde{B}^k(\vec{q})=-i\varepsilon^{ijk}q^i\tilde{A}^j_{cl}(\vec{q})##
As far as I know, the magnetic field is defined using the potential as ##\vec{B}=\vec{\nabla}\times\vec{A}##
Then in equation 6.6 they define ##A^\mu(x)=\int...
Problem Statement: Potential difference is the difference in the quantity of charge in both potentials. How does the magnetic field creates this difference of charges ?
Relevant Equations: None
Hi,
I was reading electricity and found that the difference in potential of both end is the...
I need some clarification on how the Higgs field work. The popular science explanation explains the effect on moving particles the way an object would be impacted by moving through a medium of maple syrup. I understand this is a very bad, inaccurate analogy. What I want to understand is does...
I have a proton and an electron at a certain distance from it. The proton exerts an electrostatic force on the electron. I then neutralize the proton's charge by firing another electron at it from behind. How long does it take for the first electron to sense the change?
Summary: Does a static magnetic field affect a weaker pulsed magnetic field close by?
If you have a static magnet close to a low level pulsed electromagnetic field, will the static magnetic field influence the pulsed field if the static magnetic field is the same strength or stronger than the...
If you place a zinc / copper battery it will create an electric field pointing from the copper to the zinc and my question is this what makes this electric field , the zinc pushes electrons in the circuit and never "stays" negatively charged for an electric field to be created same for the...
For the front wire, I got the magnitude of the magnetic field in terms of the magnitude of the magnetic force, the current, "l," and the "theta". I am unsure how to proceed because I thought that the magnetic force is independent of any other forces. I am also just lost in general. Any help...
Hi, so I was able to solve this problem by just equating the forces (Tension, mg, and EQ).
But I thought I could also solve this problem with Conservation of Energy.
However, I calculated it several times, and I never get the right answer this way.
Doesn't the Electric Field do the work to put...
Hi.
Are these two books complementary, or do they have too much in common?
https://www.amazon.com/dp/1107034264/?tag=pfamazon01-20
www.amazon.com/Quantum-Field-Theory-Standard-Model/dp/1107034736/
My problem is that I still don't quite understand the difference between university courses in...
If within a volume v ,there exists 10 velocity fields at different points then can anyone please suggest how to compute ##\int_v(\nabla•v)## within the volume?? using matlab
For exm if the velocity vector field be ##v=x\hat x+y\hat y+z\hat z## and for x=1 to 10,y=1 to 10 and z= 1 to 10 the 10...
Options are at the top of page as a) b) c) d)
Answer may more than one.
Now since 'a' is distance from the smaller surface of cone so as we move along the axis area will increase,So current charge density will decrease and as we know J=sigma E,E will decrease,but V will remain constant since...
I started with the first of the relevant equations, replacing the p with the operator -iħ∇ and expanding the squared term to yield:
H = (-ħ^2 / 2m)∇^2 + (iqħ/m)A·∇ + (q^2 / 2m)A^2 + qV
But since A = (1/2)B x r
(iqħ/m)A·∇ = (iqħ / 2m)(r x ∇)·B = -(q / 2m)L·B = -(qB_0 / 2m)L_z
and A^2 =...
All i could think is that the z component of velocity will remain unchnged as there is no force in that direction.And for the x and y component can we imagine the helical motion as a superposition of a circle and a straight line.So for x and y component can we solve for a particle moving in a...
I know that we need to use some boundry condition both on the a radius surface and the b radius surface and somehowuse the superposition on them both, the boundry condition most be for the tangential and the radial part,
the only things I got is that i don`t know how to produce a magnetisation M...
So I understand that when an electric field is produced in a conductor of length L, the net electric field in the conductor will be 0 because the rearrangement of electrons in the conductor results in the production of its own electric field which cancels out the one produced initially...
This is a very basic issue but really important as well.
The rectangular loop has length ##l## and width ##h##. I have seen the argument of neglecting the encircled sides of the loop because ##h << 1## while using Ampere's law to calculate the magnetic field flowing over a plane.
I find this...
The Hubble telescope was able to capture images of the edges of our visible universe in its deep space photos. These were among its most breathtaking pictures. They show galaxies from about 14 billion light years away, as well as in the past, from the very beginning of time and space in our...
Problem Statement: How does the Higgs field “give mass”
Relevant Equations: The exact way that particles interact with the Higgs field and therefore create mass.
I’m trying to figure out how the Higgs field works, one problem is that while I originally though of the Higgs field like a medium...
Ex =10.0Vm-1
dx= d2=x2+y2
x=4.00m
y=3.00m
d2=4.002+3.002
d=5.00m
Ex = -dV/dx
10.0Vm-1=-dV/5.00m
10.0Vm-1*5.00m=-dV
-50.0V=dV
So from origin at 100V-50.0V = 50.0V
But the solution I am given gives the answer at 50.5V and the information "directed at 45.0◦ " does not seem to have been used so may...
Hi there,
I'm just starting Zee's Einstein Gravity in a Nutshell, and I'm stuck on a seemingly very easy assumption that I can't figure out. On the Tensor Field section (p.53) he develops for vectors x' and x, and tensor R (with all indices being upper indices) : x'=Rx => x=RT x' (because R-1=RT...
Summary: Mathematical and Physical queries in regards to Electromagnetic Fields and their manipulation of Space-Time.
I recently started looking into Einstein's Field Equations, to get a better understanding of how mass distorts and curves the plane of Space-Time, however from doing this I...
Short summary: I'd like to receive some advice on changing my field of study, transitioning from theoretical hep to something like Statistical Physics applied to neurosciences, but I don't know anything about the latter. Thank you in advance for the patience!
Some background: I'm a first year...
I am wondering if there is a book with solved example problems I can follow so that I can carry out the following calculation:
I have a set of unperturbed electron Green functions $g^{r0},g^{a0},g^{<0},g^{>0}$ and phonon Green functions $d^{r0},d^{a0},d^{<0},d^{>0}$ which are for a central...
So, as it says in the title, I am trying to calculate overall voltage induced in a coreless coil in the cases of it being stationary and moving in an alternating magnetic field. To go more into detail, I would like to create a mathematical model of a coil in an alternating magnetic field that...
I bring a dielectric in a region with electric field ##\vec E_0##. Net electric field ## \vec E_{net} = \vec E_0 + \vec E_p ## , where ## \vec E_p ## is electric field due to polarization of dielectric.
For linear dielectric, ## \vec E_p ## is 0 outside the dielectric. So,
## \vec E_{net} =...
I tried to look once at the zy axis and saw a two infinite capacitors with fictive charge density of M on the upper plane, and -M in the lower with a distance of h from each other, the two capacitors saparated with d in the y axis,
but when I look in xy axis there was 2 another capcitors the...