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.

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  1. J

    I Changing spherical coordinates in a Lagrangian

    In order to compute de lagrangian in spherical coordinates, one usually writes the following expression for the kinetic energy: $$T = \dfrac{1}{2} m ( \dot{r}^2 + r^2 \dot{\theta}^2 + r^2 \sin^2 \theta \dot{\phi}^2 )\ ,$$ where ##\theta## is the colatitud or polar angle and ##\phi## is the...
  2. T

    Vector Field Transformation to Spherical Coordinates

    I am trying to solve the following problem from my textbook: Formulate the vector field $$ \mathbf{\overrightarrow{a}} = x_{3}\mathbf{\hat{e_{1}}} + 2x_{1}\mathbf{\hat{e_{2}}} + x_{2}\mathbf{\hat{e_{3}}} $$ in spherical coordinates.My solution is the following: For the unit vectors I use the...
  3. F

    Does operator L^2 commute with spherical harmonics?

    My teacher said me this commutator is zero because the spherical harmonics are eigenfunctions of L^2. Actually, he said that any operator must commute with its eigenfunctions. I tried an example: [L^2,Y_20] expressing L^2 on spherical coordinates and I determined this commutator is not zero...
  4. C

    Parameterize Radial Vector of Electric Field due to Spherical Shell

    Homework statement: Find the electric field a distance z from the center of a spherical shell of radius R that carries a uniform charge density σ. Relevant Equations: Gauss' Law $$\vec{E}=k\int\frac{\sigma}{r^2}\hat{r}da$$ My Attempt: By using the spherical symmetry, it is fairly obvious...
  5. P

    Potential of a spherical shell

    a. This solution is i can consider the charge Q as a point charge and the electric potential at a distance r is ## V = Q/(4πεοr)## b. This is where the confusion starts again when r2>r>r1, my answer ## V = ρ*4*π(r^3 - r_1^3)/(3*4πεοr) \\ V = ρ*(r^3-r_1^3)/3εοr; ## I know i am making some...
  6. Terrycho

    Divergence of a position vector in spherical coordinates

    I know the divergence of any position vectors in spherical coordinates is just simply 3, which represents their dimension. But there's a little thing that confuses me. The vector field of A is written as follows, , and the divergence of a vector field A in spherical coordinates are written as...
  7. E

    I Velocity Vector Transformation from Cartesian to Spherical Coordinates

    Hi all, I can't find a single thing online that translates a cartesian velocity vector directly to spherical vector coordinate system. If I am given a cartesian point in space with a cartesian vector velocity and I want to convert it straight to spherical coordinates without the extra steps of...
  8. P

    Electric field in the Spherical Cavity

    a. For the question a the solution is If the uniform charge density is ρ then the charge of the sphere up to radius r is q = ρ * (4/3)*π * r3; Hence the electric field is E = (ρ *4π*r^3)/(3*εο*r^2); E = (ρ*r)/(3εο); b. I don't understand what is superposition? How to proceed? Please advise.
  9. Bilbo B

    Electric potential of a spherical conductor with a cavity

    Summary:: If the conductor is having a cavity and is provided with some charge, with the cavity too having some charge then how the potential will be affected on the outer surface of the conductor. The center of cavity and the center of hollow sphere does not coincide. As if their centers do...
  10. L

    A Volume element in Spherical Coordinates

    For me is not to easy to understand volume element ##dV## in different coordinates. In Deckart coordinates ##dV=dxdydz##. In spherical coordinates it is ##dV=r^2drd\theta d\varphi##. If we have sphere ##V=\frac{4}{3}r^3 \pi## why then dV=4\pi r^2dr always?
  11. naushaan

    Calculatingthe resultant gravitational force on spherical objects

  12. Ozen

    Does the Thickness Affect a Semi-Mirror Spherical Lens' Characteristics?

    Hello, I am trying to get a better understanding of optical lens's and came across a question I have not found an answer to. Say you have a lens made of either Polycarbonate or Trivex, and it is a spherical lens. A coating is applied to the side with the smaller radius to reflect certain light...
  13. P

    Is the Chain Rule Applied to Spherical Polar Coordinates Different?

    Ive found ##\delta x/\delta r## as ##sin\theta cos\phi## ##\delta r/\delta x## as ##csc\theta sec\phi## But unsure how to do the second part? Chain rule seems to give r/x not x/r?
  14. M

    A What are the uses of spherical harmonics?

    Hi PF! When solving the Laplace equation in spherical coordinates, the spherical harmonics are functions of ##\phi,\theta## but not ##r##. Why don't they include the ##r## component? Thanks!
  15. M

    I Understanding Spherical Tensors & Their Applications

    Hello! I came across spherical tensors, and I am a bit confused about the way they are applied. For example, Pauli matrices, can be grouped together to form a rank 1 (vector) spherical tensor as ##(\sigma_-, \sigma_z, \sigma_+)##, which are the raising operator, the z projection operator and the...
  16. A

    Fields inside charged rings vs spherical shells

    Hi. Since you can construct shells from a series of rings, why would there be an electric field inside a single ring but not inside a shell?
  17. C

    Element of surface area in spherical coordinates

    r,θ,ϕ For integration over the ##x y plane## the area element in polar coordinates is obviously ##r d \phi dr ## I can also easily see ,geometrically, how an area element on a sphere is ##r^2 sin\theta d\phi ## And I can verify these two cases with the Jacobian matrix. So that's where I'm at...
  18. S

    I Finding the Error in Computing Spherical Tensor of Rank 0 Using General Formula

    This should be a trivial question. I am trying to compute the spherical tensor ##T_0^{(0)} = \frac{(U_1 V_{-1} + U_{-1} V_1 - U_0 V_0)}{3}## using the general formula (Sakurai 3.11.27), but what I get is: $$ T_0^{(0)} = \sum_{q_1=-1}^1 \sum_{q_2=-1}^1 \langle 1,1;q_1,q_2|1,1;0,q\rangle...
  19. L

    I The free particle in spherical coordinates

    Hi! I'm studying Shankar's Principle of quantum mechanics I didn't get the last conclusion, can someone help me understand it, please. Where did the l over rho come from?
  20. ContagiousKnowledge

    General solution of the spherical wave equation

    Since the spherical wave equation is linear, the general solution is a summation of all normal modes. To find the particular solution for a given value of i, we can try using the method of separation of variables. $$ ψ(r,t)=R(r)T(t)ψ(r,t)=R(r)T(t) $$ Plug this separable solution into the...
  21. Moara

    Spherical gas distributed symmetricaly over space

    . We have that energy in a infinitesimal Spherical layer with number of mols dn is: dU=Cv.T.dn (1) But by the ideal gas law: PV=nRT (2) Differentiation gives: PdV+VdP=RTdn (3) (3) in (1) and using CV=3R/2 (monoatomic) gives: dU=3/2.(PdV+VdP) (4) Integration of (4) over the whole gas will...
  22. J

    Moment of Inertia: Thin Spherical Shell

    Homework Statement: Derive the formula for the moment of inertia of a thin spherical shell using spherical coordinates and multiple integrals. Homework Equations: Moment of Intertia is (2MR^2)/3 I = (2MR^2)/3
  23. E

    Electric field in a spherical shell

    Hi, been a while since I last asked here something. I am restudying electrostatics right now, and I am facing difficulties in the following question: My attempt: I tried to use Gauss' law, what I got is the equation in the capture but that doesn't lead me anywhere as I am unable to find a...
  24. Arman777

    I Deriving the area of a spherical triangle from the metric

    The metric for 2-sphere is $$ds^2 = dr^2 + R^2sin(r/R)d\theta^2$$ Is there an equation to describe the area of an triangle by using metric. Note: I know the formulation by using the angles but I am asking for an equation by using only the metric.
  25. snatchingthepi

    Max and min functions in spherical expansions

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  26. D

    Find the electric field inside and outside of a spherical shell superposition

    Hi! I need help with this problem. I tried to solve it by saying that it would be the same as the field of a the spherical shell alone plus the field of a point charge -q at A or B. For the field of the spherical shell I got ##E_1=\frac{q}{a\pi\epsilon_0 R^2}=\frac{\sigma}{\epsilon_0}## and for...
  27. D

    What happens when one of two cocentric spherical shells is grounded?

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  28. Jelsborg

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  29. Samanko

    Surface charge density of a conducting spherical shell

    The textbook says ' A conducting sphere shell with radius R is charged until the magnitude of the electric field just outside its surface is E. Then the surface charge density is σ = ϵ0 * E. ' The textbook does show why. Can anybody explain for me?
  30. jhongg7

    Equilibrium equations - spherical coordinates

  31. D

    How to show that the electric field inside a spherical shell is zero?

    Hi! I need help with this problem. I tried to do it the way you can see in the picture. I then has this: ##dE_z=dE\cdot \cos\theta## thus ##dE_z=\frac{\sigma dA}{4\pi\epsilon_0}\cos\theta=\frac{\sigma 2\pi L^2\sin\theta d\theta}{4\pi\epsilon_0 L^2}\cos\theta##. Then I integrated and ended up...
  32. Kaushik

    Stokes' Law: Why does viscous drag depend upon the radius of the spherical body?

    Stokes' Law gives us the value fo viscous force when a spherical body is under motion inside a fluid. ##F_{viscous} = 6\pi\eta av## (where ##a## is the radius of the spherical body and ##v## is the velocity with which it is moving) What is the reason for the Viscous drag to depend upon the...
  33. K

    B Intensity of Spherical Shell of Stars

    Given that L is the luminosity of a single star and there are n stars evenly distributed throughout this thin spherical shell of radius r with thickness dr, what is the total intensity from this shell of stars? My calculations were as follows: Intensity is the power per unit area per steradian...
  34. K

    Setup for Spherical Astronomy Problem

    My apologies for not detailing my attempts at a solution; I'm not sure how to to digitally illustrate or describe the various setups I attempted before looking at the solution to this problem. I am also ONLY asking about the setup, though I included the full question for context. The solution to...
  35. H

    MATLAB My Crank-Nicolson code for my diffusion equation isn't working

    I'm trying to solve the diffusion equation in spherical co-ordinates with spherical symmetry. I have included the PDE in question and the scheme I'm using and although it works, it diverges which I don't understand as Crank-Nicholson should be unconditionally stable for the diffusion. The code...
  36. L

    A Laplace transform in spherical coordinates

    Summary: A 1963 paper by Michael Wertheim uses a Laplace transform in spherical coordinates. How is the resulting equation obtained? In 1963, Michael Wertheim published a paper (relevant page attached here), where he presented the following equation (Eq. 1): $$ y(\bar{r}) = 1 + n...
  37. H

    I Spherical Symmetry & Electron Spin: An Exploration

    Can an electron in a spherically symmetrical potential energy function have non-zero spin angular momentum?
  38. P

    I Circumference of a circle on a spherical surface

    I was just reading an intro text about GR, which considers the circumference of a circle on a sphere of radius R as an example of intrinsic curvature - the thought being that you know you're on a 2D curved surface because the circle's circumference will be less than ##2\pi r##. They draw a...
  39. Dimitris Catzis

    MCNP4c2: Fission Reactions in a Spherical Subcritical Reactor

    Hi, i am new to simulation and for my thesis i have to make a simple simulation by using mcnp4c2. Is anybody familiar with this version of MCNP? I need to calculate the fission reactions per second in a geometry of a spherical sub critical reactor of Uranium with low percentage of U 235 with...
  40. T

    B The Mysterious Shape of Jupiter: Why Gas Planets Stay Spherical

    If Jupiter is made of gas, how can it maintain its spherical shape without being contained in a spherical shaped container?
  41. Noelani2306

    Electrical field outside a hollow spherical conductor

    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...
  42. mfb

    A General formula for lenses without spherical abberation

    Publication: Rafael G. González-Acuña and Héctor A. Chaparro-Romo, "General formula for bi-aspheric singlet lens design free of spherical aberration," Appl. Opt. 57, 9341-9345 (2018) Open access preprint: arXiv Given one surface of a lens, how does the other surface has to look like to avoid...
  43. J

    Pinhole Diffraction of a Spherical Wave

    Case (a) is the textbook of a planar incident wavefront and below it in the figure is the known simple formula for the central spot and fringes, or minima and maxima, angular distribution with respect to the optical axis. So, the question here is regarding case (b). The position (usually...
  44. alexmahone

    Average electric field over a spherical surface

    I'm sure the average is going to be an integral, but \displaystyle\frac{1}{4\pi R^2}\oint\mathbf{E}\cdot d\mathbf{a} gives me a scalar, not a vector.
  45. K

    Does FMESH in MCNP support spherical geometry with the TMESH tally?

    According to the manual FMESH only supports Cartesian or cylindrical geometry has there been any update to this or is that it?
  46. T

    Capacitance and induced charge of a spherical Capacitor + dielectric

    I) For the first part I used: ##V = - \int E ds = \int_a^c \frac{1}{4\pi\epsilon_0} Q /r^2 dr+ \int_c^{c+d} \frac{1}{k} \frac{1}{4\pi\epsilon_0} Q /r^2 dr + \int_{c+d}^b \frac{1}{4\pi\epsilon_0} Q /r^2 dr ## And by using ##C = Q/V## We get an answer which is somehow large for writing here...
  47. Celso

    Show that the terminal speed of a falling spherical object ....

    To write ##v## as a function of time, I wrote the equation ##m\frac{dv}{dt} = c_{2}v^2 + c_{1}v - mg \implies \frac{mdv}{c_{2}v^2 + c_{1}v - mg} = dt## To solve this, I thought about partial fractions, but several factors of ##-c_{1} \pm \sqrt {c_{1}^2 +4c_{2}*mg}## would appear and they don't...
  48. K

    Spherical capacitor with dielectrics

    Homework Statement Consider the following system: which consists of a conducting sphere with free charge , a dielectric shell with permittivity ##\epsilon_1##, another dielectric shell with permittivity ##\epsilon_2## and finally a conducting spherical shell with no free charge. Homework...
  49. solzonmars

    Electric field outside a spherical shell?

    If a charge is put inside a spherical shell, why is the electric field outside the shell independent of the location of the charge? Gauss's law could find that the net flux is independent, but not each individual field? Is this something about the surface charge density being independent of...
  50. Q

    I Deriving the spherical volume element

    I’m trying to derive the infinitesimal volume element in spherical coordinates. Obviously there are several ways to do this. The way I was attempting it was to start with the cartesian volume element, dxdydz, and transform it using $$dxdydz = \left (\frac{\partial x}{\partial r}dr +...
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