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.
I am unsure of my solutions and am looking for some guidance. a.)The real part of the wave in complex notation can be written as ##\widetilde{A} = A^{i\delta}##. Writing the Complex Wave equation, we have ##\vec E(t) = \widetilde{A}e^{(-kz-\Omega t)} \hat x##. Therefore the real part is ##\vec...
I was thinking of using the chain rule with
dF/dx = 0i + (3xsin(3x) - cos(3x))j
and
dF/dy = 0i + 0j
but dF/dy is still a vector so how can it be inverted to get dy/dF ?
what are the other methods to calculate this?
Bubbles of air were rising in some water in a rice field. The bubbles would float around at random for a while. If two bubbles got within a certain distance of one another they would very quickly merge. It looked like a predatory larger bubble pouncing on the smaller prey. Gotcha! Why is...
Let us consider relativistic particle (electron) which moves with relativistic speed ##v## in the Coulomb field (in the field of a fixed heavy nucleus). The main question is what is the potential energy of a particle in such a static field? Landau and Lifshitz in their book "Field Theory"...
"One way to ground everything in reality is to think purely about the records of experiments that are stored in computer memory. Very often, that's a list of times at which events happened."
-- Peter Morgan, old thread meaning-of-wave-function-collapse
"If we are to understand the relationship...
If there are two charges positive and negative and their electric field point in the same direction then the total electric field would be their sum of magnitudes. Why don't we consider the sign of the charges? For example, a parallel plate capacitor is inside the region where both the positive...
How does relativistic qft predict quantum fluctuations in the vacuum? We see this in the experiment proving the Casimir Effect so we know it's physical, but why?
Here is my attempt (Note:
## \left| \int_{C} f \left( z \right) \, dz \right| \leq \left| \int_C udx -vdy +ivdx +iudy \right|##
##= \left| \int_{C} \left( u+iv, -v +iu \right) \cdot \left(dx, dy \right) \right| ##
Here I am going to surround the above expression with another set of...
In a paper by Bain (2011), particles are left with little ontological value because of the Reeh-Schlieder theorem, the Unruh effect and Haag's theorem. The author claims (and here I am copying his conclusion):
First, the existence of local number operators requires the absolute temporal metric...
I tried gauss law.
And the fact that if alpha is less than pi/2 we can say that we have two parts with angle alpha and one other part which has a normal field at the center.
But non of them helped me answer.
The problem's solution says that we can use the fact that our section has longitudinal...
Why the area of the thin rings are ##2πasin\theta \, ds##? (a is the radius of the hollow sphere)
If we look from a little bit different way, the ring can be viewed as a thin trapezoid that has the same base length ( ##2πa sin\theta##), and the legs are ## ds##.
The angle between the leg and...
Gravity isn't a force in the strictest sense of the word, yet magnetism is exactly that: a force. As is strong, EW, etc.
Therefore, it's possible that the more massive magnetic object gets drawn to the center of a magnetic source at a faster rate than the less massive magnetic object. Discuss!
A piece of metal moving West to East in a North to South fixed magnetic field slows down...but how? Yes of course eddy currents are set up in the metal and these currents generate their own magnetic field which somehow slows down the moving metal piece...but how does this actually slow the...
Sorry if I post again about this topic (last time I promise!) but I still have some doubts regarding the concept of flux. This collection of problems I have quite standard but there are so many variations. Here is the circuit in question:
Something tells me that I could write a function that...
Statement: The magnetic field around a straight wire carrying a current can be explained Relativistically by changing the inertial frame of reference to the frame of the moving electrons - i.e., a Lorentz contraction of the positive charges in the wire will give a denser concentration of the...
The amplitude of ##\vec{B}## is given by:
$$B(x) = B_{0} - B_{0} \frac{x}{2l}$$ for ##0 \leq 0 \leq 2l##
This was my attempts at finding the flux of B:
$$\Phi(B) = (B_{0} - B_{0} \frac{x}{2l})(2l-x(t))l = B_{0}2l^2-2B_{0}x(t)l+ B_{0}\frac{x(t)^2}{2}$$
and the current: $$ i = -\frac{d...
$$B(t) = B_{0} \frac{t^2}{T^2}$$
for ##0 \leq t \leq T##
The issue here is more conceptual, because once I find the flux of B I know how to proceed to find the current. I got velocity (but it seems to me that it is the initial velocity), I could use it to find the time in function of space...
Faraday's law tell's you about the line intergal of the electric field, but you have to know the direction of the induced electric field first in order to properly apply it. How can I find its direction? Is it in the same direction as the induced current?
What is it the we detect in the first instance?
Is it the particle |wave or is it the field?
Is the former more fundamental than the latter in any sense or are we just talking the opposite sides of the same coin?
For instance does the em field create the photon and the electron or could...
Hi all. I am stuck with a seemingly silly doubt all of a sudden.
The direction of Electric Field is taken from Positive to Negative (because Field Lines originate from a Positive Charge and terminate at Negative Charge).
We know that direction of Dipole Moment is from Negative Charge to a...
A standard textbook problem features a constant B field and a conducting loop that increases in area at constant rate.
It is easy to work out the induced EMF and the associated electric field magnitude and direction (CW or CCW). The magnitude of the E field
is E = B v where v is a velocity...
This question appeared in a university entrance exam.Basically, if magnetic flux passing through a surface of a loop changes over time ,only then e.m.f will be induced to that loop.But here only a straight line is used and there's no chance of forming any area.So by definition there's no chance...
Hi all experts!
I would like to read about the Lagrangian of a classical (non-quantum), real, scalar, relativistic field and how it is derived. What is the best book for that purpose?Sten Edebäck
So, I am able to calculate the electric potential in another way but I know that this way is supposed to work as well, but I don't get the correct result.
I calculated the electric field at P in the previous exercise and its absolute value is $$ E = \frac {k Q} {D^2-0.25*l^2} $$ This is...
Hi,
I am reading Robert D Klauber's book "Student Friendly Quantum Field Theory" volume 1 "Basic...". On page 48, bottom line, there is a formula for the classical Lagrangian density for a free (no forces), real, scalar, relativistic field, see the attached file.
I like to understand formulas...
I have read in the following article the expression "high breakthrough field": https://link.springer.com/article/10.1557/PROC-871-I9.6
I tried to find out in the internet what is the definition of that and what it refers to in the transistors but I couldn't find anything!
Thank you in advance!
I know that for a single cylindrical neodymium magnet, the formula
$$ \displaystyle{\displaylines{B(z)=\frac{μ_0M}{2}(\frac{z}{\sqrt{z^{2}+R^{2}}}-\frac{z-L}{\sqrt{(z-L)^{2}-R^{2}}})}} $$ shows the relationship between the magnetic field strength and the distance between the magnet. I was...
Hi. I believe I have what may be both a silly and or a weird query. In many Griffiths Problems based on Electric Field I have seen that a coordinate system other than Cartesian is being used; then using Cartesian the symmetry of the problem is worked out to deduce that the field is in (say) z...
d(ɣmv)/dt = qvB
(dɣ/dt)mv + ɣm(dv/dt) = qvB
Substituting gamma in and using the chain rule, it ends up simplifying to the following:
ɣ^3*m(dv/dt) = qvB
Now, I am confused on how to solve for v.
Hello everyone,
This is in reference to fig 5.19 (screen shot attached - please read the paragraph which says "Figure 5.19 shows the...").
I don't get why the field outside of the sphere of radius ct acts as though the particle would have continued its motion. Author's words : "The field...
If we have an electromagnetic wave like the one in the picture and a molecule which is, in the image, the small black ball with electron cloud being the part with "minus sign" in it, does the molecule with its cloud start to oscillate, once the EM wave hits it, as an induced electric dipole...
I am trying to derive radial and axial magnetic fields of a current carrying loop from its magnetic vector potential. So far, I have succeeded in deriving the radial field but axial field derivation gives me trouble.
My derivation of radial field (eq 1) can be found here.
Can anyone point out...
Mentors’ note: this thread is forked from https://www.physicsforums.com/threads/free-fall-in-curved-spacetime.1016510/
But what if the gravity field is homogeneous? Like that of an infinite massive plane? The objects in the ship will stay where they are. An infinite massive plane is quite...
If we define Laplacian of scalar field in some curvilinear coordinates ## \Delta U## could we then just say what ##\Delta## is in that orthogonal coordinates and then act with the same operator on the vector field ## \Delta \vec{A}##?
I got E. 13q as the answer. That is what i did: The electric field due to +q at origin 0 should equal the electric fields of charges -3q and the new charge placed at 2x. So applying the equation above like this; k*(q) / (2^2) = -3q*k + (k*C)/ 4 solving for C the new charge added, gives 13q. I...
If I have two separated and non-interacting molecules with different constants polarizabilities ##\alpha_1## and ##\alpha_2## and I send an EM field of frequency ##\omega## first on the molecule no.##1## and then on the molecule no.##2## so that the two molecules will have a dipole moment...
The Higgs mechanism is an ingenious mechanism inspired by condensed state physics. The famous Mexican hat potential ensures a VEV value of about two times the mass of the Higgs particle (which, as an aside, is of comparable order as the W and Z vector bosons, the difference though is that its a...
My attempt was to consider spherical shells of radius ##r## (##r\leq R##))and thickness ##dr## and then the potential energy of this shell would be in the field only of the "residual" sphere of radius ##r## (a result also known as shell theorem) $$U_{dr}=G\frac{\rho\frac{4}{3}\pi r^3 \rho 4\pi...
My attempt: We have 3 charges inside 2 +ve and 1 -ve so i just added them up. 4 + 5 +(-7) = 2q
Then there is a -5q charge outside the sphere. I did 2q + (-5q)= -3q . The electric field flux formula is Flux= q/ E0 . So i got -3q/E0 which is obviously wrong : ) . After quick googling , I...
I argue not. Let ##f:\mathbb{Q}\rightarrow\mathbb{R}## be defined s.t. ##f(r)=r^2##. Consider an increasing sequence of points, to be denoted as ##r_n##, that converges to ##\sqrt2##. It should be clear that ##\sqrt2\equiv\sup\{r_n\}_{n\in\mathbb{N}}##. Continuity defined in terms of sequences...
I've already tried to calculate the potential with respect to the 3 segments and then apply superposition (V1+V2+V3). However, I was not very successful. My error I think is in the calculation of the radii, mainly of the line segment that is on the z axis. Can anybody help me? I need some light...
I recently read a paper on using an electric field to drive water autoionizaton. I'm trying to figure out how to use the Laplace equation on pg 9; 4th paragraph; to solve for voltage. I'm also interested in how this equation would change if I replaced the hemispherical tip with a parallel plate...
The notation I think best describes it is
## F = \lVert\int^{space}_s|\vec{V}|ds\rVert ##
So you have a vector field V in a 3d space. For each point you integrate over all of space (similar to a gravitational or electromagnetic field) *but* vectors in opposite directions do not cancel, they...