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.
My understanding of this question is that, if you have a proton standing against a positive electric field, and move it in the opposite direction of the field, you're putting in work and therefore should have greater electric potential energy.
But that idea breaks down when you consider a...
My attempt at a solution:
Cylindrical coordinate system with ##r##, ##\theta##, ##z##. Conductivity ##\sigma## and permeability ##\mu_0##. Inner radius ##a## and outer radius ##b##. (##b>a##)
The external field is spatially uniform and driven at sinusoidally at frequency ##f##. The external...
Hi there,
The settling velocity is terminal velocity at which drag force is equal to the gravity. When the sphere is in rotating field the terminal settling velocity is reduced. What will be the expression for it in rotating field?
I did make the problem simpler by looking at the the part from d/2 down the upper plate
here are my initial parameters I am making my size step be h since lowering it may make calculating harder
I am especially getting weird results for the field and capacitance
R = 0.1; % Radius of the...
I have come across the following paper (arxiv preprint link, it looks like it is published in Phys Rev D) that claims to have found a consistent quantum field theory of tachyons, overcoming three issues which have been said in previous literature to make such a theory impossible...
I took a point P on the conductor with charge Q. We know that the field inside a conductor is zero in electrostatic equilibrium
Therefore I took induced charge on the neutral conductor to be x and gave the respective charges to the others. On the side facing conductor with charge Q the neutral...
My teacher said that gauss law may not accounts for the field due to the outside charges in the LHS in this expression
##\int E.ds## = ##\frac{q}{\epsilon}## as field lines coming in the surface leave it as well. Hence the total flux is 0
However i dont think thats very consistent with the...
First astrophotography was black and white. So then we colored it in. Then the light got filtered so people can see the universe better. When I ask why are there no photographs of magnetic field lines in space, or anywhere; I read that it's because our eyes cannot see them, but we can't see...
here is my attempted solution.
## d^2 = z^2 + \frac {L^2} {3} ##
## C ## is coulomb constant
since the point is symmetric, only the vertical component of the electric field remains. So,
$$ E = 3 E_y =3 \frac {C Q cos \theta} {d^2} $$
$$ E= 3 \frac {C Q z} {d^3} $$
thus part (a) is done ( i...
https://www.space.com/sun-magnetic-field-flip-solar-maximum-2024
I know theyre not the same physics. I just wondered if the the Suns 11 year polarity flip has any analogy to the flipping T-wrench effect.
The solenoid can be assumed to be a stack of rings of width ##dz## each where ##z:0\to-\infty## . $$i=nIdz$$ $$\sin\theta=\frac{R-r}{l}$$ $$\cos\theta=\frac{z+x}{l}$$ $$dB=\frac{\mu_{\circ}nIR d\varphi dz}{4\pi l^2}$$ Call ##\frac{\mu_{\circ}nIR}{4\pi}=\alpha## $$ B_y =\int dB\cos\theta ...
I understand the following .a conductor is made of atoms and atoms always strive to be at equilibrium and that's why the electric field inside a conductor is zero because the electros distribute themselves in such a way so that they are in equilibrium , yet they do produce an electric field...
There is a magnet in the shape of a hollow sphere. And the inner surfaces are all n-poles, and the outer surfaces are all s-poles. At this time, how is the magnetic field formed in the empty space of the sphere? There is no s pole for the magnetic force lines from the n pole to enter, so can...
If a vector ##V(x)## being transported down a path ##l##, The vector field is described with equation:
$$\partial_\mu V(x)=\Gamma_\mu V(x)$$
The solution of the equation can be described with parallel propagator ##P(x, x_0)##(in mathematics it is also called product integration):
$$V(x)=P(x...
I tried resolving the semi infinite rods into arcs of 90 degree each placed on the three axes but that doesnt take me anywhere....
Alternatively I tried finding out the field at the point due to each rod but im unable to find the perpendicular distance from the point to the rod...I dont think...
Lambda = charge density
I tried first taking out the field due to the circular arc and I got $$ (lambda / 4π (epsilon knot) ) (2 sin (theta)) $$
For reference this is the arc that was provided in the question of angle 2(theta) and the tangent
What I dont understand is how can the fields be...
Some axially symmetric star has two independent KVFs, ##T## and ##\Phi##. We don't know the expressions for these at all points -- the only thing we know is that as ##r^2 + z^2 \rightarrow \infty##, that ##T \rightarrow \partial/\partial t## and ##\Phi \rightarrow \partial/\partial \phi##. The...
We know that in electrostatics, there is path independency for line integral of E, so E is a conservative field and thus we have E=-gradV. Integrating this from ro(reference point of our choice) to the point r we are studying, along a random path, we get the solution of the above equation...
Using either H&R's Chapter 27 Example 3 or Problem 590 of the ##\mathbf{Physics Problem Solver}##, I've been unable to get the component ##E_x## or ##E_y##. There are now different angles at the charges. My thanks to berkeman for LaTeX advice, but any errors are of course my own. Thanks in...
I use FLHR and the plan diagram. This suggests that the left hand side of the coil is going down. I have added annotations to this, which agree with the mark scheme which then goes onto suggest that the coil turns clockwise. To me though this would make it go anti clockwise.
So... what am I...
Are there currently professional physicists who work on Einstein’s quest to unify the gravitational field and the electromagnetic field (as classical fields), or has this idea been completely abandoned? Is it simply a hopeless endeavour?
Consider a negatively charged spherical conductor. On the surface of it, what is the direction of its electric field? Well, the definition of the direction of an electric field is the direction a positive test charge would go if placed at that point. But... it wouldn't move anywhere! So is the...
This is the statement in question:
But if they were scalar fields, they would not transform at all. How could they contribute differently if they didn't change?
The Lagrangian, $$\mathcal L(x)= \frac 1 2 \partial^{\mu} \phi (x) \partial_{\mu} \phi (x) - \frac 1 2 m^2 \phi (x)^2$$ for a scalar field ##\phi (x)## is said to be Lorentz invariant and to transform covariantly under translation.
What does it mean that it transforms covariantly under translation?
I proceeded as follows
Current in sector ##d\theta=## is:
$$dI=\int_{x=0}^{x=R}{\frac{I}{\pi R^2/2}\times\frac{d\theta}{2\pi}\times 2\pi x dx}$$
Field due to sector ##d\theta## is therefore
$$dB=\int^{x=R}_{x=0}{\frac{\mu_○}{2\pi x}\times\frac{I}{\pi R^2/2}\times\frac{d\theta}{2\pi}\times 2\pi x...
The diagram is something like this, and I want to calculate the magnetic field at the center using the Biot-Savart law. In this case, do the magnetic fields formed by the quarter circles pairwise alternate with each other?
I'm trying to conduct an experiment where I calculate the magnetic field strength of a magnet, by comparing the levitation distances between two magnets. My experiment involves using different masses to anchor down magnetic repulsion between 2 magnets. Fg = Fm.
The formula I am using for this...
Sorry, I guess I should have remembered all of this from my school days, but right now I have forgotten so much that I need some help.
I am developing some simple experiments for school children (age ca. 12). This one involving magnets.
I am not asking for detailed calculations, that is way...
I'd like to hear your professional opinion on and experience with using Quantum Field Theory for the Gifted Amateur by Tom Lancaster and Stephen J. Blundell as a self-study textbook. Thank you.
I am not sure why latex is not rendering, but here is the question.
The answer is ##\frac{a^2}{8}## and for the love of my life, I don't know how. Can you please help me with this?
I've no idea how to solve this problem. The sign of the charge is not mentioned, so I'm assuming the charge is "+". The charge exerts an outward electric field. Since two lengths of the right-angle triangle are given, I use the Pythagorean to find the hypotenuse, which is the distance between q...
There are two identical spheres with the same charge that are the vertices of an equilateral triangle. ##+3 \mu C## will exert an outward electric field, which is drawn in the FBD below (see the attached pic), Since the horizontal force components (1x and 2x) are equal and opposite at point P...
First, my ignorance... I know there are classes of equations: Laplace, Poisson, Wave, Diffusion, etc.
(I suppose Laplace is a subset of Poisson, but that is not the issue).
Into what category of mathematical equations would you place the field equations of elasticity (stress/strain/displacement)?
I've been working on developing infinitesimal recursion (what I call continuous hierarchy), but I ended up arriving at "field series" instead. My searches didn't seem to come up with anything reasonable (battlefield the video game series), so I'm wondering what the official name for a field...
I have a simulated data of charged particles in a magnetic field. I have selected clusters, each cluster contains a set of points(x,z) and I want to perform RK4 between the first and second clusters and fill the positions in a histogram.
I have selected the clusters with the initial...
from the partition function - am trying to show that ##\langle \mu \rangle = \beta^{-1} (\partial \log Z / \partial B)## where ##Z## is the canonical partition function for one atom, i.e. ##Z = \sum_{m=-j}^{j} \mathrm{exp}(\mu_0 \beta B m)##, and ##\mu = \mu_0 m##. The average...
For this,
From the work kinetic energy theorem, if we assume that the book and the earth is the system, and that the finial and inital speed of the system is zero, then is the work KE theorem there is no net work done on the system. However, clearly there is work done on the system is shown by...
Hi,
I am confused about whether decreasing the magnetic field used for a generator could increase the generator's power output.
I used four equations:
1. Torque = Force x radius
2. Torque = NIAB (N = number of turns, I = current, A = area of armature, B = magnetic field).
3. emf =...
Hi,
I am trying to learn relativistic classical field theory as a preparation for studying quantum field theory.
I am currently reading chapter 13 i Herbert Goldstein's Classical Mechanics edition 3, but I think that this book is a bit too brief and does not fully derive and explain the...
How does an electric field of a moving charge, for example a moving electron, inside a wire looks like? Does it looks like this with distorted circular radial lines?
Why doesn't this work if the field is strong? Or does it work as long as there are no singularities?
Mentor's Note: Original thread title was, "Calculating rest mass by integrating T_{00} over a 3 volume for static metric"
My question is specifically with calculating the intensity. The book solution is
I=P/(4*pi*r^2)
but would this not give me a weaker electrical amplitude in the final calculation after plugging it in to
I=(1/2)*√(ε0/μ0)*(E02) ?