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.

View More On Wikipedia.org
Recent contents Top users
  1. D

    Coefficient spherical harmonics

    $$ Y_{\ell}^m = \sqrt{\frac{(2\ell + 1)(\ell - m)!}{4\pi(\ell + m)!}}P^m_{\ell}(\cos\theta)e^{im\varphi} $$ For ##\ell = m = 1##, we have $$ \sqrt{\frac{(2 + 1)(0)!}{4\pi(2)!}}P^1_{1}(\cos\theta)e^{i\varphi} = \frac{1}{2}\sqrt{\frac{3}{2\pi}}e^{i\varphi}\sin \theta $$ But Mathematica...
  2. F

    Electrostatic energy of spherical shell.

    Homework Statement Determine the electrostatic energy, W, of a spherical shell of radius R with total charge q, uniformly distributed. Compute it with the following methods: a) Calculate the potential V in spherical shell and calculate the energy with the equation: W = (1/2) * ∫σVda...
  3. D

    MHB Spherical Harmonics easy question

    $$ Y_{\ell}^m = \sqrt{\frac{(2\ell + 1)(\ell - m)!}{4\pi(\ell + m)!}}P^m_{\ell}(\cos\theta)e^{im\varphi} $$ For $\ell = m = 1$, we have $$ \sqrt{\frac{(2 + 1)(0)!}{4\pi(2)!}}P^1_{1}(\cos\theta)e^{i\varphi} = \frac{1}{2}\sqrt{\frac{3}{2\pi}}e^{i\varphi}\sin \theta $$ But Mathematica is telling...
  4. D

    MHB Solution of Laplace Spherical Equation with Boundary Conditions

    Consider Laplace's equation on a sphere of unit radius with the boundary condition $$ u(1,\theta,\varphi) = f(\theta,\varphi)\begin{cases} 100 & -\pi/4 < \varphi < \pi/4\\ 0 & \text{otherwise} \end{cases} $$ Here we will consider a three-term approximation to the solution, i.e., involving the...
  5. D

    MHB Boundary conditions spherical coordinates

    Laplace axisymmetric $u(a,\theta) = f(\theta)$ and $u(b,\theta) = 0$ where $a<\theta<b$. The general soln is $$ u(r,\theta) = \sum_{n=0}^{\infty}A_n r^n P_n(\cos\theta) + B_n\frac{1}{r^{n+1}}P_n(\cos\theta) $$ I am supposed to obtain $$ u(r,\theta) = \sum_{n =...
  6. H

    Spherical Conductor lies at the center of a uniform spherical charge sheet

    Homework Statement A ground spherical conductor of radius a lies at the center of a uniform spherical sheet of charge QB and radius b. a) How much charge is induce on the conductor's surface? Ans(-QBa/b) Evaluate V(r) at position between the conductor and the sheet and outside the sheet...
  7. P

    Electric Potential of a Spherical Charge Distribution

    Homework Statement A total charge q is uniformly distributed throughout a sphere of radius a. Find the electric potential in the region where r1<a and r2>a. The potential is defined anywhere inside the sphere. Homework Equations letting ρ = volume charge density and ε = permittivity...
  8. H

    Integral over spherical Bessel function

    Is there somebody who can help me how to solve this integral \int_{0}^{+\infty} dr r^{^{n+1}} e^{-\alpha r} j_l(kr)
  9. C

    Triple integration in spherical polars

    Homework Statement Determine the value of \int_{0}^{1} \int_{0}^{\sqrt{1-x^2}} \int_{0}^{\sqrt{1-x^2-y^2}} \sqrt{x^2+ y^2 + z^2} dz dy dx The Attempt at a Solution So in spherical polars, the integrand is simply ρ. \sqrt{1- x^2- y^2} = z = ρ\cos\phi = \cos\phi since we are on the unit...
  10. N

    Quick/general question about conducting spherical shell.

    If I have a +5 nC charge on the inside of the shell, the inside surface would be -5nC, the outside would be +5 nC and between those surfaces there would a 0 charge, right? So just to make sure I have it all straight, the INSIDE of the shell would actually be 0 because the INNER SURFACE is -5...
  11. E

    Gauss's Law - A nonconducting spherical shell

    1. Homework Statement A nonconducting spherical shell of inner radius R1 and outer radius R2 contains a uniform volume charge density ρ throughout the shell. Use Gauss's law to derive an equation for the magnitude of the electric field at the following radial distances r from the center of...
  12. E

    Gauss's Law and nonconducting spherical shell

    Homework Statement A nonconducting spherical shell of inner radius R1 and outer radius R2 contains a uniform volume charge density ρ throughout the shell. Use Gauss's law to derive an equation for the magnitude of the electric field at the following radial distances r from the center of the...
  13. pmd28

    A person slides downa spherical, frictionless surface.

    Homework Statement A person starts from rest at the top of a large, frictionless, spherical surface, and slides into the water below. At what angle θ does the person leave the surface? (Hint: When the person leaves the surface, the normal force is zero.) Homework Equations Don't Know...
  14. M

    Express a wave function as a combination of spherical harmonics

    Homework Statement An electron in a hydrogen atom is in a state described by the wave function: ψ(r,θ,φ)=R(r)[cos(θ)+eiφ(1+cos(θ))] What is the probability that measurement of L2 will give 6ℏ2 and measurement of Lz will give ℏ? Homework Equations The spherical harmonics The...
  15. soothsayer

    Surface area of a spherical cap

    Homework Statement Calculate the area of a circle of radius r (distance from center to circumference) in the two-dimensional geometry that is the surface of a sphere of radius a. Show that this reduces to πr2 when r << a Homework Equations Surface area of a spherical cap = 2πah = π(r2 +...
  16. H

    Spherical Charge Ball, Gauss law

    Homework Statement A spherical charged ball of radius a has total charge Q; there is no charge outside the ball and no sheet-charge on its surface. The (radial) field inside the ball has the form Er(r) = constant x r2 for r between 0 and a. Use Gauss's Law in integral form to evaluate the...
  17. N

    Spherical Capacitor with Charge, Potential Difference and More

    Homework Statement A spherical capacitor contains a charge of 3.10nC when connected to a potential difference of 220V . If its plates are separated by vacuum and the inner radius of the outer shell is 5.00cm . Part A) Calculate the capacitance. Homework Equations C = Q/V C = Aε0/d...
  18. T

    Consider a spherical shell with radius R and surface charge density σ = σ0 cosθ

    Homework Statement Consider a spherical shell with radius R and surface charge density σ = σ0 cosθ (a) What is the total charge carried by the shell? (b) Please evaluate the charge carried by the upper hemisphere, in terms of σ0. Homework Equations Q=∫σ0 cosθ da The...
  19. R

    Gravitational potential using spherical harmonics (WGS84)

    Hi, I am looking to use the definition from WGS84 to calculate Earth's gravitational potential using spherical harmonics, however I am having some difficulty finding the definition of one of the variables. Gravitational potential is given as the following: V = \frac{GM}{r}\left [ 1 +...
  20. D

    Solving for Net Charge on Inner Surface of Conducting Shell

    Homework Statement (1)A conducting sphere w/ charge +Q is surrounded by a spherical conducting shell. What is net charge on inner surface of the shell? (2) A charge is placed outside the shell. What is the net charge on the inner surface now? (3) What if the shell and sphere are not...
  21. R

    Solving Spherical Symmetry in Hydrogen Atom

    I have a problem; I am trying to show the spherical symmetry in a hydrogen atom, for a sum over the l=1 shell i.e the sum over the quadratics over three angular wave equations in l=1, |Y10|^2 + |Y11|^2 + |Y1-1.|^2 . This should equal up to a constant or a zero to yield no angular dependence...
  22. S

    Hamiltonian in spherical coordinates

    Homework Statement The total energy may be given by the hamiltonian in terms of the coordinates and linear momenta in Cartesian coordinates (that is, the kinetic energy term is split into the familiar pi2/2m. When transformed to spherical coordinates, however, two terms are angular momentum...
  23. S

    Conversion of energy expression from Cartesian to spherical coordinates

    A text I am reading displays the attached image. Can someone explain the general method for obtaining the velocity analogues of those terms (in parentheses) in 1.5? I know the second and third terms in parentheses in 1.6 and 1.7 are the squares of angular velocities, but can a general procedure...
  24. T

    Charged ball within spherical shell charge problem

    Homework Statement A small charged ball lies within the hollow of a metallic spherical shell of radius R. Here, for three situations, are the net charges on the ball and shell, respectively: 1 +4q, 0 2 -6q, +10q 3 +16q, -12q (a) Rank the situations according to the charge on the...
  25. T

    Derivation of Laplace Operator in Spherical and Cylindrical Coordinates

    Hey Guys, Does anyone know where I can find a derivation of the laplace operator in spherical and cylidrical coordinates?
  26. T

    Cylindrical and Spherical Coordinates Changing

    Homework Statement Convert the following as indicated: 1. r = 3, θ = -π/6, φ = -1 to cylindrical 2. r = 3, θ = -π/6, φ = -1 to cartesian The Attempt at a Solution I just want to check if my answers are correct. 1. (2.52, -π/6, 1.62) 2. (-2.18, -1.26, 1.62)
  27. W

    Translate the rectangular equation to spherical

    Translate the rectangular equation to spherical and cylindrical equations. http://www.texify.com/img/%5CLARGE%5C%21x%5E2%2By%5E2%2B2y-3x%2Bz%5E2%3D25.gif
  28. R

    Spherical coordinates, vector field and dot product

    Homework Statement Show that the vector fields A = ar(sin2θ)/r2+2aθ(sinθ)/r2 and B = rcosθar+raθ are everywhere parallel to each other. Homework Equations \mathbf{A} \cdot \mathbf{B} = |\mathbf{A}||\mathbf{B}|\cos(0) The Attempt at a Solution So, if the dot product equals 1. They should be...
  29. B

    Spherical Percent Uncertainty and Violent Storm Question

    Q1: What is the percent uncertainty in the volume of a spherical beach ball whose radius is r = 3.85 plus or minus 0.06 m? I found the volume of the original sphere, as well as one with a radius of 3.91. I then subtracted the volumes to find the difference between the two, divided that by the...
  30. Z

    Trouble understanding meaning of triple integral in spherical coordinates

    Homework Statement Evaluate \iiint\limits_B e^{x^2 + y^2 + z ^2}dV where B is the unit ball. Homework Equations See above. The Attempt at a Solution Does this evaluate the volume of f(x, y, z) within the unit ball (i.e. anything falling outside the unit ball is discarded)...
  31. A

    Expressing Spherical coordinates in terms of cylindrical

    Homework Statement I'm trying to express spherical coordinates in terms of cylindrical and vice versa. I would appreciate it if someone could give me some feedback on my attempt at a solution. Thanks for the help! The Attempt at a Solution Spherical(cylindrical) r=(ρ^2+z^2)^(1/2)...
  32. R

    Qualitative Solid Spherical Conceptual

    Homework Statement For a charged solid metal sphere with total charge Q and radius R centered on the origin: Select "True" or "False" for each statement: 1.If the solid sphere is an insulator (instead of metal) with net charge Q, the net charge on the inside of the solid sphere is...
  33. E

    Volume integral of an ellipsoid with spherical coordinates.

    Homework Statement By making two successive simple changes of variables, evaluate: I =\int\int\int x^{2} dxdydz inside the volume of the ellipsoid: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=R^{2} Homework Equations dxdydz=r^2 Sin(phi) dphi dtheta dr The...
  34. B

    Charging a spherical shell by conduction

    Homework Statement An insulated spherical conductor of radius R1 carries a charge Q. A second conducting sphere of radius R2 and initially uncharged is then connected to the first by a long conducting wire. (a) After the connection, what can you say about the electric potential of each...
  35. H

    What Determines the Shape of an Atom?

    This isn't exactly homework, but I'm still in high-school and I feel guilty posting in the big guys' forums. I've recently learned about the shapes of spdf orbitals and the way they interact to form different bonds. This is completely different to the nice spherical atoms we were shown back...
  36. U

    Nonuniformly charged spherical surface

    A sphere of radius a in free space is nonuniformly charged over its surface such that the charge density is given by ρs(θ) = ρs0 sin 2θ, where ρs0 is a constant and 0≤θ≤∏. Compute the total charge of the sphere. So I know ρs = dQ/dS Integrating the surface charge density function will...
  37. J

    Electrical and magnetic fields are spherical right?

    When i think of electromagnetic waves i think of a fast moving sphere of expanding or contracting fields,either magnetic or electrical depending on where its at in its cycle. So i guess I am picturing a single photon as a sphere. Is this a correct visualization?(i doubt it). If so, how does a...
  38. sophiecentaur

    The Sun is very spherical - allegedly

    I read this Scientific American link and I found it very interesting. I have a problem understanding it, though, because I had read that one of the explanations for the existence of Solar Systems is based on Angular Momentum. The argument goes that if all the mass were concentrated in the host...
  39. R

    Velocity of fluid through a spherical surface

    Fluid flows with velocity 2i - 3j m/s at point P having coordinates (1,2,4). Consider a plane through P which is normal to the vector b=-i+2k. What is the speed at which the fluid passes through the plane? Should I do the dot product of the position vector P=[1,2,4] and b vector, then...
  40. grav-universe

    Schwarzschild metric and spherical symmetry

    In deriving the Schwarzschild metric, the first assumption is that the transformation of r^2 (dθ^2 + sin^2 θ dψ) remains unchanged due to the spherical symmetry. What does that mean exactly? What is the logic behind it? Please apply any math involved in algebraic form. Thanks.
  41. G

    Mathematica [Mathematica] Solving Heat Equation in Spherical Coordinates

    Hello Folks, I have this equation to solve (expressed in LaTeX): \frac{\partial{h}}{\partial t} = \frac{1}{n} \left[ \frac{1}{r^2 \sin^2{\phi}} \frac{\partial}{\partial \theta} \left( K \frac{\partial h}{\partial \theta} \right) + \frac{1}{r^2 \sin \phi} \frac{\partial}{\partial \phi}...
  42. S

    Charge distribution over concentric spherical shells

    Homework Statement three concnetric conducting spherical shells are there with charges +4Q on innermost shell,-2Q on middle shell and -5Q on outermost shell. what is the charge distribution and charge on inner side of outermost shell? Homework Equations The Attempt at a Solution...
  43. F

    Volume enclosed by a spherical coordinate surface

    Homework Statement Find the volume enclosed by the spherical coordinate surface ρ = 2sin∅ Homework Equations dV = ∫∫∫(ρ^2)sin∅dρd∅dθ The Attempt at a Solution (Sorry about my notation!) Alright, here's what I've done so far... Since the region is a torus, centered...
  44. jfy4

    Spherical Tensor operators for half-integers

    Hi, There are, for example, lists of spherical tensor operators for l=\text{integer} steps, e.g. l=0,1,2,.... T_{k}^{q}(J)\rightarrow T_{0}^{0}=1, \quad T_{1}^{\pm 1}=\mp \sqrt{\frac{1}{2}}J_{\pm},\quad T_{1}^0=J_z and this continues forever. I was wondering if there are operators...
  45. R

    Solving Spherical Pendulum w/ Friction & Generalized Force

    Hi. I'm trying to make a small simulation of several simple physical systems (C++). I have the differential equation of a spherical pendulum with only the gravity force and without friction. \theta'' = \sin(\theta) (\cos(\theta) \phi'^2 − \frac{g}{L}) \phi'' = −2 \cot(\theta) \theta' \phi'...
  46. R

    Does a spherical wavefront thicken as it moves outwards ?

    If a flash of light is emitted spherically and this is measured in terms of its duration by two distant observers with one twice as far away from the source as the other, and the source and observers are all at rest with respect to each other, will the flash appear to have the same duration for...
  47. P

    Divergence in spherical coordinate system

    I have been studying gradience; divergence and curl. I think i understand them in cartesian coordinate system; But i don't understand how do they get such complex stuffs out of nowhere in calculating divergence in spherical and cylindrical coordinate system. Any helps; links or suggestion...
  48. nomadreid

    Negative energy: same in Casimir, Hawking rad, & spherical space?

    The term "negative energy" is used (a) for the energy below the vacuum energy between the two plates in the Casimir effect, (b) the energy carried by the sister particle to the radiated particle in Hawking radiation, that is, the particle from the matter-antimatter pair which goes into the...
  49. T

    Spherical co-ordinates with Implicit function thm

    So I'm asked to determine near which points of R^3 can we solve for ρ, δ, θ in terms of x,y,z: x = ρ sinδ cosθ y= ρ sinδ sinθ z= ρcosδ so the spherical co-ordinates using IFT. Attempt: Ok so in order to determine solutions, I need to first find where the determinant of the freceht...
  50. W

    Potential at Center of Insulating Spherical Shell

    Homework Statement The inner radius of a spherical insulating shell is c=14.6 cm, and the outer radius is d=15.7 cm. The shell carries a charge of q=1451 E−8 C, distributed uniformly through its volume. The goal of this problem is to determine the potential at the center of the shell (r=0)...
Back
Top