What is Spherical: Definition and 1000 Discussions
A sphere (from Greek σφαῖρα—sphaira, "globe, ball") is a geometrical object in three-dimensional space that is the surface of a ball (viz., analogous to the circular objects in two dimensions, where a "circle" circumscribes its "disk").
Like a circle in a two-dimensional space, a sphere is defined mathematically as the set of points that are all at the same distance r from a given point in a three-dimensional space. This distance r is the radius of the ball, which is made up from all points with a distance less than (or, for a closed ball, less than or equal to) r from the given point, which is the center of the mathematical ball. These are also referred to as the radius and center of the sphere, respectively. The longest straight line segment through the ball, connecting two points of the sphere, passes through the center and its length is thus twice the radius; it is a diameter of both the sphere and its ball.
While outside mathematics the terms "sphere" and "ball" are sometimes used interchangeably, in mathematics the above distinction is made between a sphere, which is a two-dimensional closed surface embedded in a three-dimensional Euclidean space, and a ball, which is a three-dimensional shape that includes the sphere and everything inside the sphere (a closed ball), or, more often, just the points inside, but not on the sphere (an open ball). The distinction between ball and sphere has not always been maintained and especially older mathematical references talk about a sphere as a solid. This is analogous to the situation in the plane, where the terms "circle" and "disk" can also be confounded.
I figured this would be a problem in some classical mechanics book but so far I can't find an answer anywhere. Assume there is no drag or lift, and since the planet is not rotating we don't have to worry about Coriolis effects. I'm working on a solution but I want to see if my work is correct...
For this problem,
I am confused my what they mean by ##\phi##. I have looked at the figure, but it is confusing. Makes it look like the x-axis and y-axis are not perpendicular, even thought I'm assuming they are since this is a right handed coordinate system. Does someone please know what...
I've worked out how to derive the formulas for a solid cylinder and a solid sphere rolling down a hill.
E.g., for a cylinder:
Emech = KE + PE
mgh = 1/2 mv^2 + 1/2 Iw^2
gh = 1/2 v^2 + 1/2 (1/2r^2) v^2/r^2
gh = 3/4 v^2
v^2 = 4/3 gh
I then performed a derivative with respect to time and found a...
This is most probably a dumb idea as I'm far from deep physics knowledge but I was thinking.
What if Earth is hot inside not because of the pressure and the radioactivity but because it's mass attracts particles (similarly to gravitational lensing) and they collide right in the Earth's center?
Let me start with my understanding of a photon. A source emits a single photon, which can be described as an excitation of the EM field. This excitation radiates outward, producing isochrons which in pure vacuum would be spherical. Then at some point the photon is absorbed by some atom. By...
I would assume that the droplet on the ceiling is spherical, since it is the shape that minimizes the surface energy for a given volume. The droplet is held by the surface tension force, which acts along the contact line between the droplet and the ceiling and is balanced by the weight of the...
Problem:
Solution:
When I looked at an example problem, they started writing the potential in terms of the Legendre polynomials.
The example problem:
This is what I did:
$$V_0 \alpha P_2 (\cos(\theta)) \Rightarrow \frac{\alpha 3 \cos ^2 (\theta)}{2} - \frac{\alpha}{2} \Rightarrow \frac{\alpha...
I host freely for the public a web app for determining the diameter of a sphere to resonate a given frequency and sound hole diameter and length, and then download a stl file for 3D printing. I've realized it has some issues and part of it is the equations i use to determine the sphere's...
If these point charges were placed in vacuum without any spherical shells in the picture, then the force between these charges would be ##F =\dfrac { k q_1 q_2} {d^2}##.
But, I am unable to reason how spherical shells would alter the force between them.
I do know that if charges were on the...
The water level in a spherical bowl has a diameter of 30 cm. If the horizontal diameter of the bowl is 10 cm below the water level, calculate the radius of the bowl and the depth of the water in the bowl.
I managed to draw a diagram below:
In my drawing, I am seeing the sphere ABCD as the...
I have been reading Wikipedia’s derivations of the Lorentz Transformations. Many of them start with the equation of a spherical wavefront and this reasoning:
- We are asked to imagine two events: light is emitted at 1 and absorbed somewhere else at 2. For a given reference frame, the distance...
I've tried writing the curl A (in spherical coord.) and equating the components, but I end up with something that is beyond me:
\begin{equation}
{\displaystyle {\begin{aligned}{B_r = \dfrac{1}{4 \pi} \dfrac{-3}{r^4} ( 3\cos^2{\theta} - 1) =\frac {1}{r\sin \theta }}\left({\frac {\partial...
The issue is that the singularity is not in the center of the sphere.
So how would I calculate it?
I have a few questions:
1. Can I calculate the terms separately like so:
$$A = grad(a+b) = grad(a) + grad(b)$$
2. If I use a spherical coordinate system with the center being at the singularity I...
"Spherical symmetry requires that the line element does not vary when##\theta## and##\phi## are varied,so that ##\theta##and ##\phi##only occur in the line element in the form(##d\theta^2+\sin^{2}\theta d\phi^2)##"
I wonder why:
"the line element does not vary when##\theta## and##\phi## are...
The picture above shows the integral that needs to be evaluated, and the associated picture ## \cos\alpha ## can be obtained via the law of cosines. I'm simply confused as to where the ##\cos\alpha ## comes from in the first place. I just don't see why ##\cos\alpha ## is necessary in this...
The electric field inside a charged spherical shell moving inertially is, per Gauss's law, zero.
If the spherical shell is accelerated, the field inside is not zero anymore, but it gains a non-null component along the direction of the acceleration, as mentioned, for example, in this paper.
The...
I am finding the potential everywhere in space due to a point charge a distance 'a' on the z-axis above an infinite xy-plane held at zero potential. This problem is fairly straight forward; place an image charge q' = -q at position -a on the z-axis. I have the solution in cartesian coordinates...
Hello,
Suppose I have a spherical hole in a elastic infinite space. I apply a time-dependent pressure to the inner surface of the spherical hole.
I know p = f(t).
If I only consider this as an elastic problem, no failure happened, how can I calculate the work done by p during the time from 0...
I am post-processing a simulation.
A spherical is meshed by many little triangles. A time-dependent pressure (p=10*t) is equally applied to the inner surface of a spherical in the normal direction all the time. After t1=0.1s, the spherical is broken, and each little triangle is disconnected...
What combination of resources can you recommend for introducing people to spherical harmonics? Assume that the audience has the mathematical maturity of first-year grad students in mathematics, and will want a decent introduction to the theory and constructions. But also assume that this is part...
Using the data given and recalling that in this configuration ##R<0## I get: ##\frac{1.33}{0.5}+\frac{1.5}{q}=\frac{1.5-1.33}{-0.2}\Rightarrow q\approx -0.427 m=-42.7 cm## so the image is virtual and is ##42.7\ cm## to the left of vertex ##V##. The magnification is ##M=\frac{n_1 q}{n_2...
In general relativity, rotation of mass gives rise to framedraging effects, just like linear motion does, because of the off-diagonal components in the mass-energy-momentum tensor. So around Bonnor beams there is framedragging, as well around a rotating mass.
Now imagine a spherical rotating...
I was asked to derive the relation $$p = u/3$$ for photon gas. Now, i have used classical mechanics and symmetry considerations, but the book has solved it in a interisting way:
I can follow the whole solution given, the only problem is the one about the probability to colide the sphere!. Where...
In this paper in NASA
https://www.giss.nasa.gov/staff/mmishchenko/publications/2004_kluwer_mishchenko.pdf
it claims (at page 38) that the defined spherical waves (12.4,12.5) are solutions of Maxwell's equations in the limit ##kr\to\infty##. I tried to work out the divergence and curl of...
In classical EM, consider an EM plane wave traveling in free space. The ratio of the amplitude of the electric field to the amplitude of the magnetic field is the velocity of the wave, the speed of light.
Is the above also true if the wave is spherical, expanding from a point source, as in a...
this formula in the picture is the average electrostatic field over a spherical volume of radius R. It is the same expression of the electrostatic field, at the (position) of the point charge, of a volume of charge of uniform density whole entire charge is equal to (negative)q.
My question is...
Textbook examples usually involve a plane monochromatic wave that is diffracted by a plane grating.
If one places an ideal focusing lens behind the grating one will get a diffraction pattern in the back focal plane of the lens.
The geometric size of this diffraction pattern is proportional to...
(a) Using Gauss's Law ##E_P=\frac{q_1+q_2+q_3}{4\pi\varepsilon_0(R_1+R_2+R_3+d)^2};(b) V_3-V_1=\int_{R_3}^{R_2}\frac{q_1+q_2}{4\pi\varepsilon_0 r^2}dr+\int_{R_2}^{R_1}\frac{q_1}{4\pi\varepsilon_0 r^2}dr=\frac{q_2}{4\pi\varepsilon_0}\left(\frac{1}{R_3}-\frac{1}{R_2}\right).##
(c)...
Electric field is 0 in the center of a spherical conductor. At a point P (black dot), I do not understand how the electric field cancels and becomes 0. Electric field is in blue.
I have found the total dipole moment of for this problem but am having trouble finding the electric field.
I believe my electric field when r>2R ( I mistakenly wrote it as r<2R on my work, but it is the E with a coefficient of 2/3) is correct as it fits the equation:
.
I don't believe this...
I've tried figuring out commutation relations between ##L_+## and various other operators and ##L^2## could've been A, but ##L_z, L^2## commute. Can someone help me out in figuring how to actually proceed from here?
Let ##|l,m\rangle## be a simultaneous eigenstate of operators ##L^2## and ##L_z## and we want to calculate ##\langle l,m|cos(\theta)|l,m'\rangle## where ##\theta## is the angle ##[0,\pi]##. It is true that in general ##\langle l,m|cos(\theta)|l,m'\rangle=0## ##(1)## for the same ##l## even if...
First draw a gaussian shape outside of the sphere (a larger sphere) with radius R. The total charge from the (inner) sphere will be:
$$Q = \sigma A$$
$$A = 4\pi r^2$$
$$Q = \sigma 4\pi r^2$$
Use Gauss's Law to derive electric field magnitude
$$\oint_{}^{} E \cdot dA = \frac{q_e}{\epsilon_o}$$...
The code I have for solving the diffusion equation on the spherical domain. The solution seems to differ drastically depending on the refinement of the mesh which obviously shouldn't be the case. I have checked and double checked my derivation and code and I can't seem to find an error. One...
For (a) this problem, the only thing I can see changing is the distribution of the negative charge on the inner wall of the cavity.
When the point charge is in the center of the cavity, you could say the induced charged is spread symmetrically on the inner cavity wall in order to oppose the...
I was recently looking for proven relations between focal length, radius of curvature, refractive index etc of a convex lens as I was working on an experiment, I did Find a relation, between Height from principal axis and focal length, and it was a huge relation!I did the experiment to verify...
From ray tracing I would say that the image is upright.
Using the equation ##\frac{1}{p}+\frac{1}{q}=\frac{1}{f}## with ##f=-\frac{R}{2}=-2## and ##M=-\frac{q}{p}=\frac{3}{4}## I get ##p=\frac{2}{3}cm\simeq 0.67 cm##.
Is this correct? Thanks
I'm making a program that generates lines in 3D space. One feature that I need is to have an incrementally increasing angle on a line (a bending line / curve).
The problem is simple if the line exists in the xy-plane, then it would be a case of stepping say 1m, increase the azimuthal angle φ...
The electric field is the one generated by the charge ##+Q## on the inner sphere of the capacitor, which generates a radial electric field ##\vec{E}=\frac{1}{4\pi\varepsilon_0}\frac{Q}{r^2}\hat{r}## which, due to the presence of the dielectric, become...
I wanted to know about spherical aberration in a biconvex and plano convex lens as I was planning an experiment with them.
I was reading about them and came upon the following passage.
I don't know whether the given equation is an empirical one or a derived equation.
Can anyone help me if you...
Greetings!
here is the solution which I undertand very well:
my question is:
if we go the spherical coordinates shouldn't we use the jacobian r^2*sinv?
thank you!
Each spherical shell will contribute to potential on the surface of inner shell and the same will apply to outer shell.
Due to inner shell ##V_1 = \frac {kQ} {{r_1}}## and due to outer shell ##V_1 = \frac {-kQ} {r_1}##. Therefore potential on inner surface is zero.
But the answers are ##V_1...
[Moderator's Note: Thread spin off due to topic and level change.]
For a spherically symmetric solution, if SET components were written in terms a single one of 4 coordinates, in a way plausible for a radial coordinate, the I believe solving the EFE would require spherical symmetry of the...