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.

View More On Wikipedia.org
Recent contents Top users
  1. 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?
  2. 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...
  3. 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...
  4. 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...
  5. 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...
  6. 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 +...
  7. M

    I Spherical Harmonics Axisymmetry

    I'm expanding a function in spherical harmonics. I want to conserve axisymmetry of the function. what harmonics would respect that? Should I only include m=0 terms?
  8. V

    B Solving an Integral on a Spherical Surface - Tips

    Hello. I ask for solution help from the integral below, where y and x represent angles in a metric of a spherical, 2-D surface. He was studying how to obtain the geodesic curves on the spherical surface, the sphere of radius r = 1, to simplify. The integral is the end result. It is enough, now...
  9. JD_PM

    Average electric field over a spherical surface

    Homework Statement I was working out problem 4, chapter 3 of Introduction to Electrodynamics by Griffiths: a) Show that the average electric field over a spherical surface, due to charges outside the sphere, is the same as the field at the centre. b) What is the average due to charges inside...
  10. T

    Calculating the drag force on a spherical object with holes

    I am looking for a bit of guidance on how one could calculate the drag force of a sphere with holes in the sphere falling through a fluid, in my case water. So I know for a low Reynolds number the drag force on a sphere is given by stoke law, but what I would like to do is calculate the drag...
  11. M

    Surface area of a shifted sphere in spherical coordinates

    Homework Statement find the surface area of a sphere shifted R in the z direction using spherical coordinate system. Homework Equations $$S= \int\int \rho^2 sin(\theta) d\theta d\phi$$ $$x^2+y^2+(z-R)^2=R^2$$ The Attempt at a Solution I tried to use the sphere equation mentioned above and...
  12. F

    B Is the inverse square law exact near a spherical body?

    I'm forking this off another thread where I brought it up but it was getting OT. It is good enough for a first approximation but it is certainly not exact. Consider a test mass one radius from a spherical body. Work out the contributions form two points diametrically opposed on the surface...
  13. M

    I Converting from spherical to cylindrical coordinates

    I have the coordinates of a hurricane at a particular point defined on the surface of a sphere i.e. longitude and latitude. Now I want to transform these coordinates into a axisymmetric representation cylindrical coordinate i.e. radial and azimuth angle. Is there a way to do the mathematical...
  14. cromata

    I Integration over a part of a spherical shell in Cartesian coordinates

    I am modeling some dynamical system and I came across integral that I don't know how to solve. I need to integrate vector function f=-xj+yi (i and j are unit vectors of Cartesian coordinate system). I need to integrate this function over a part of spherical shell of radius R. This part is...
  15. I

    Geometrical Optics - Light ray angles on a spherical mirror

    I can't see how the textbook produces the following relationships between angles: $$ \theta = \phi + \alpha \qquad (1)$$ $$ 2\theta = \alpha + \alpha ' \qquad (2)$$ My thinking is that the exterior angle theorem for triangles was used to create expression ##(1)##, but I am unsure as to how...
  16. A

    A How to find the displacement vector in Spherical coordinate

    Is there a way of subtracting two vectors in spherical coordinate system without first having to convert them to Cartesian or other forms? Since I have already searched and found the difference between Two Vectors in Spherical Coordinates as...
  17. PeroK

    Angular Momentum in Spherical Coordinates

    I've started on "Noether's Theorem" by Neuenschwander. This is page 35 of the 2011 edition. We have the Lagrangian for a central force: ##L = \frac12 m(\dot{r}^2 + r^2 \dot{\theta}^2 + r \dot{\phi}^2 \sin^2 \theta) - U(r)## Which gives the canonical momenta: ##p_{\theta} = mr^2...
  18. J

    Self adjoint operators in spherical polar coordinates

    Hi, I have a general question. How do I show that an operator expressed in spherical coordinates is self adjoint ? e.g. suppose i have the operator i ∂/∂ϕ. If the operator was a function of x I know exactly what to do, just check <ψ|Qψ>=<Qψ|ψ> But what about dr, dphi and d theta
  19. C

    I Force fields in curvilinear coordinate systems

    I am trying to solve problems where I calculate work do to force along paths in cylindrical and spherical coordinates. I can do almost by rote the problems in Cartesian: given a force ##\vec{F} = f(x,y,z)\hat{x} + g(x,y,z)\hat{y}+ h(x,y,z)\hat{z}## I can parametricize my some curve ##\gamma...
  20. beefbrisket

    I Computing inner products of spherical harmonics

    In this video, at around 37:10 he is explaining the orthogonality of spherical harmonics. I don't understand his explanation of the \sin \theta in the integrand when taking the inner product. As I interpret this integral, we are integrating these two spherical harmonics over the surface of a...
  21. R

    B Confusion about the radius unit vector in spherical coordinates

    If the radius unit vector is giving us some direction in spherical coordinates, why do we need the angle vectors or vice versa?
  22. M

    Convert a spherical vector into cylindrical coordinates

    Homework Statement Convert the vector given in spherical coordinates to cylindrical coordinates: \vec{F}(r,\theta,\varphi) = \frac{F_{0}}{arsin\theta}\bigg{[}(a^2 + arsin\theta cos\varphi)(sin\theta \hat{r} + cos\theta \hat{\theta}) - (a^2 + arsin\theta sin\varphi - r^2 sin^2\theta)...
  23. starstruck_

    Electric field of spherical shell

    Homework Statement Consider a spherical shell with uniform charge density ρ. The shell is drawn as a donut with inner (R1) and outer (R2) radii. Let r measure the distance from the center of the spherical shell, what is the electric field at r>R2, R1<r<R2, and r<R1. I am working on the r > R2...
  24. LarryC

    How Can Quantum Mechanics Explain the Eigenstates of a Spherical Pendulum?

    I have trouble with finding the eigenstates of a spherical pendulum (length $l$, mass $m$) under the small angle approximation. My intuition is that the final result should be some sort of combinations of a harmonic oscillator in $\theta$ and a free particle in $\phi$, but it's not obvious to...
  25. G

    Spherical Capacitor Discharging Through Radial Resistor

    Homework Statement A spherical capacitor has internal radius ##a## and external radius ##b##. At time ##t = 0##, the charge of the capacitor is ##Q_0## Then the two shells are connected by a resistor in the radial direction of resistance ##R##. Find the Poynting vector and the energy...
  26. A

    Can a Moon-Sized Spinning Sphere Create Livable Artificial Gravity?

    Consider a hollow sphere roughly the size of the moon, spun up to produce 1g of centripetal acceleration along a band at its equator (about 15000 kph) Big stuff, I know. I have a few questions about the implication of such a system, and I hope someone can help me find some answers! - How tall...
  27. A

    A Fourier transform of outgoing spherical waves

    Please, can anyone explain how formula (5) is obtained in J.J. Barton article ''Approximate translation of screened spherical waves" . Phys.Rew. A ,Vol.32,N2, 1985. ? https://doi.org/10.1103/PhysRevA.32.1019 The same formula are given in the book Pendry J.B. "Low enrgy electron diffraction. The...
  28. Faizan Samad

    Sph Harmonics Homework: Find Potential for r>a, Contribution Using Superposition

    Homework Statement A sphere of radius a has V = 0 everywhere except between 0 < θ < π/2 and 0 < φ < π. Write an expression in spherical harmonics for the potential for r > a. For which values of m are there contributions? Determine the contributions through l= 2. How could you determine the...
  29. D

    Integration of Spherical Harmonics with a Gaussian (QM)

    Homework Statement I wish to solve this integral $$b_{lm}(k) = \frac{1}{2(\hbar)^{9/4}(2\pi)^{5/2}\sqrt{\sigma_{px} \sigma_{py} \sigma_{pz}}} \int_{\theta_k = 0}^{\pi}\int_{\varphi_k = 0}^{2\pi} i^l \text{exp}\left[ - \frac{1}{(2\hbar)^2}\left(\frac{(k_z - k_{z0})^2}{\sigma_{pz}^2} + \frac{(k_y...
  30. Domenico94

    I How does a fluid behave inside of a spherical cavity?

    Suppose you have a spherical cavity, with a flow of a fluid ( in particular water), entering it from one side. What will happen to this flow? Will it create turbulences? Will speed increase? Will eventual waves entering the cavity be reflected, so creating waves with bigger amplitude? Suppose...
  31. Felipe Lincoln

    Conservative force in spherical coordinates

    Homework Statement Is ##F=(F_r, F_\theta, F_\varphi)## a conservative force? ##F_r=ar\sin\theta\sin\varphi## ##F_\theta=ar\cos\theta\sin\varphi## ##F_\varphi=ar\cos\varphi## Homework Equations ##\nabla\times F=0## The Attempt at a Solution In this case we have to use the curl for spherical...
  32. M

    B Triple integral in spherical coordinates.

    While deriving the volume of sphere formula, I noticed that almost everyone substitute the limits 0 to 360 for the angle (theta) i.e the angle between the positive x-axis and the projection of the radius on the xy plane.Why not 0to 360 for the angle fi (angle between the positive z axis and...
  33. Krushnaraj Pandya

    Capacitance of spherical capacitor

    Homework Statement I calculated the capacitance of two concentric conducting spherical shells of radii a and b (a<b) when the inner shell has charge Q and outer has charge -Q correctly by using C=Q/(V(i)-V(o)), the problem I have is described below in "attempt at a solution" Homework...
  34. R

    MHB Find Angles of Spherical Triangle $\mathcal{P}$

    Consider the spherical triangle $\mathcal{P}$ with vertices $P_1 = (1,0,0)$, $P_2 = (0,1,0)$ and $P_3 = (1/\sqrt{3}, 1/\sqrt{3},1/\sqrt{3})$. Find the angles $\phi_1, \phi_2, \phi_3$ of $\mathcal{P}$ at $P_1, P_2, P_3$ respectively. I know the cosine angles are $\cos(\theta_1) = 0$...
  35. Arup Biswas

    I Are plane waves actually spherical waves at large distance?

    I am doubting that any plane wave is generated from a spherical wave. At large distance the radii of curvature becomes so large that we can think it as plane. Like we se Earth surface as plane though it is spherical. Is it true? I have a mathematical proof for my argument!
  36. Brilli

    Force on a part of a spherical shell on the other

    Homework Statement Given there is a conducting sphere which has a charge q on it. A plane cuts the sphere into 2 form a distance r from centre. How can we calculate the electrostatic force on one part on either side of the plane due ro the other part? Homework EquationsThe Attempt at a...
  37. K

    Optical Spherical glass vs parabolic acrylic

    Hello, I am trying to buy the parts to perform a Schlieren experiment (see an example here: https://bit.ly/2mwbzkl) It is suggested to use a Spherical Primary Telescope Mirror (glass), however when i look into getting larger than 160 mm versions, they start to get extremely expensive. So...
  38. Islam Hassan

    I Spherical Symmetry/Simultaneity of Supernova Explosions

    If the timing of detonation of nuclear weapons’ numerous implosive lenses must to be kept to within a microsecond or so in order to avoid asymmetrical detonation, does the timing of a supernova explosion similarly have tight constraints on its simultaneity and hence the sphericity of its...
  39. J

    I Convincing a Flat Earther: Proving the Earth is Spherical

    Because I must be a masochist I would like to know how I could convince a flat earther that the Earth is a sphere. The way I've chosen to do it is the following. I live in Honolulu and the person I am trying to convince lives in south america. At an agreed upon time we both take a 50 cm stick...
  40. icesalmon

    Electrical Potential of a uniformly charged spherical shell

    Homework Statement From Griffiths Third Edition: "Introduction to Electrodynamics" p.p. 81 ex. 2.6 "Find the potential inside and outside a spherical shell of radius R, which carries a uniform surface charge. Set the reference point at infinity. Homework Equations V(r) = -∫E⋅dl The...
  41. Jay Chang

    Why Is My Calculation of dS/dt for a Shrinking Spherical Raindrop Incorrect?

    Homework Statement A spherical raindrop evaporates at a rate proportional to its surface area with (positive) constant of proportionality k; i.e. the rate of change of the volume exactly equals −k times the surface area. Write differential equations for each of the quantities below as a...
  42. S

    Is there a B field in a charging spherical capacitor?

    Suppose a spherical capacitor is being charged. In this case the E field between the plates is growing with time which implies a displacement current which in turn implies a B field. How would one find this B field if it does exists? I'm guessing the B field is zero because of symmetry. I...
  43. W

    Question about Spherical Metric and Approximations

    Homework Statement This is Problem 2 from Chapter 1, Section V of A. Zee's Einstein Gravity in a Nutshell. Zee asks us to imagine a colony of "eskimo mites" that live at the north pole. The geometers of the colony have measured the following metric of their world to second order (with the...
  44. L

    I What is dx, dy and dz in spherical coordinates

    What is dx, dy and dz in spherical coordinates
  45. renec112

    QM: Issues with parity of spherical harmonics and Heisenberg

    Hi physics forms! I'm practicing to for an Quantum mechanics exam, and i have a problem. 1. Homework Statement I have two problems, but it's all related to the same main task. I have a state for the Hydrogen: ## \Psi = \frac{1}{\sqrt{2}}(\psi_{100} + i \psi_{211})## where ## \psi_{nlm}##...
  46. B

    Laplacian in spherical coordinates

    Homework Statement Hello at all! I have to calculate total energy for a nucleons system by equation: ##E_{tot}=\frac{1}{2}\sum_j(t_{jj}+\epsilon_j)## with ##\epsilon_j## eigenvalues and: ##t_{jj}=\int \psi_j^*(\frac{\hbar^2}{2m}\triangledown^2)\psi_j dr## My question is: if I'm in...
  47. Nabin kalauni

    An analytic expression to describe spherical aberration

    Homework Statement Derive an analytic expression for the distance from the vertex to the focus for a particular ray in terms of (i) the radius of curvature R of the concave mirror (ii) the angle of incidence θ between incident ray and radius of the mirror. Hence show that the focus moves closer...
  48. Another

    Vector potential in spherical coordinates

    in this problem i can solve v = ω x r = <0, -ωrsinψ, 0> in cartesian coordinates but i don't understand A in sphericle coordinates why? (inside) A = ⅓μ0Rσ(ω x r) = ⅓μ0Rσωrsin(θ) θ^ how to convert coordinate ?
  49. R

    Electric Flux Through Spherical Surface Centered at Origin

    1. A uniform linear charge density of 4.0 nC/m is distributed along the entire x axis. Consider a spherical (radius = 5.0 cm) surface centered on the origin. Determine the electric flux through this surface. Homework Equations L=2rπ φ=Q/ε λ=Q/L The Attempt at a Solution I found the charge by...
  50. R

    MHB Spherical coordinates and triple integrals

    Suppose $\displaystyle f = e^{(x^2+y^2+z^2)^{3/2}}$. We want to find the integral of $f$ in the region $R = \left\{x \ge 0, y \ge 0, z \ge 0, x^2+y^2+z^2 \le 1\right\}$. Could someone tell me how we quickly determine that $R$ can be written as: $R = \left\{\theta \in [0, \pi/2], \phi \in [0...
Back
Top