Spherical Definition and 1000 Threads

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. tanaygupta2000

    Electric field inside a spherical cavity inside a dielectric

  2. WMDhamnekar

    MHB How to find angle between two vectors, given their spherical co-ordinates?

    I know that $\arccos{(\cos{\phi_1}\cos{\phi_2}+\sin{\phi_1}\sin{\phi_2}\cos{(\theta_2-\theta_1)})}=\gamma$ But how can i answer the above question? If any member knows the proof of this formula may reply to this question with correct proof.
  3. Buzz Bloom

    I The proper Schwarzschild radial distance between two spherical shells

    For the purpose of this thread the metric is ds2 = - (1-rs/r) c2 dt2 + dr2 / (1-rs/r) where rs = 2GM/c2. (I modified the above from https://jila.colorado.edu/~ajsh/bh/schwp.html .) I assume that the two spherical shells are stationary. Therefore dt = 0. The r coordinate for the radii of the...
  4. G

    Capacitance of a spherical capacitor

    When I try to do Gauss, the permeability is not always that of the free space, but it varies: up to a certain radius it is that of the void and then it is the relative one. How can I relate them? I'm trying to calculate the capacity of a spherical capacitor. The scheme looks like this: inside I...
  5. Coltrane8

    I Spherical Harmonics Expansion convergence

    In the contex of ##L^2## space, it is usually stated that any square-integrable function can be expanded as a linear combination of Spherical Harmonics: $$ f(\theta,\varphi)=\sum_{\ell=0}^\infty \sum_{m=-\ell}^\ell f_\ell^m \, Y_\ell^m(\theta,\varphi)\tag 2 $$ where ##Y_\ell^m( \theta , \varphi...
  6. F

    Image charge of source charge in spherical cavity

    All are used to finding the image charge induced by a source charge outside a conducting sphere. The solution is supposed to also work for the case where the source charge is inside the conducting sphere, in which case the sphere is now a conducting cavity. But the solution suggests the image...
  7. J

    I Changing spherical coordinates in a Lagrangian

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  8. T

    Vector Field Transformation to Spherical Coordinates

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  9. F

    Does operator L^2 commute with spherical harmonics?

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  10. 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...
  11. 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...
  12. 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...
  13. E

    I Velocity Vector Transformation from Cartesian to Spherical Coordinates

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  14. 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.
  15. 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...
  16. L

    A Volume element in Spherical Coordinates

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  17. naushaan

    Calculatingthe resultant gravitational force on spherical objects

  18. 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...
  19. P

    Is the Chain Rule Applied to Spherical Polar Coordinates Different?

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  20. 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!
  21. M

    I Understanding Spherical Tensors & Their Applications

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  22. 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?
  23. C

    Element of surface area in spherical coordinates

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  24. S

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

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  25. L

    I The free particle in spherical coordinates

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  26. ContagiousKnowledge

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  27. Moara

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    . 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...
  28. J

    Moment of Inertia: Thin Spherical Shell

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  29. E

    Electric field in a spherical shell

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  30. 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.
  31. snatchingthepi

    Max and min functions in spherical expansions

    I'm trying to solve the vector potential of a solid rotating sphere with a constant charge density. I'm at a point where I'm performing the final integral that looks like $$ -\left( \frac {\mu_0 i} {3} \right) \sqrt{\frac 3 {2\pi}} \frac {q\omega}{R^3} Y_{1,1} \int_0^R (r')^3 \frac {r_<}...
  32. 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...
  33. D

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

    Hi! I need help with this problem. When the outer shell is grouded, its potential goes to zero, ##V_2=0## and so does it charge, right? ##-Q=0##. So the field would be the one produced by the inner shell ##E=\frac{Q}{4\pi\epsilon_0 R_1^2}##. When the inner shell is grounded, I think that...
  34. Jelsborg

    Electric field of a spherical conductor with a dipole in the center

    In a recent test we were asked to calculate the electric field outside a concentric spherical metal shell, in which a point dipole of magnitude p was placed in the center. Given values are the outer radius of the shell, R, The thickness of the shell, ##\Delta R## and the magnitude of the dipole...
  35. 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?
  36. jhongg7

    Equilibrium equations - spherical coordinates

  37. 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...
  38. 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...
  39. 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...
  40. 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...
  41. 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...
  42. 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...
  43. H

    I Spherical Symmetry & Electron Spin: An Exploration

    Can an electron in a spherically symmetrical potential energy function have non-zero spin angular momentum?
  44. 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...
  45. 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...
  46. 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?
  47. 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...
  48. 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...
  49. 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...
  50. 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.
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