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. G

    Proof of image formation property of spherical mirrors

    Hi. I'm trying to proof the image formation property of a concave spherical mirror. I know you can do this easily with a particular choice of rays (namely one that hits the vertex and one that passes through the center of the sphere) but I would like to show that a generic ray yields the same...
  2. M

    How can the surface tension of a spherical balloon affect its radius over time?

    Homework Statement Consider the process of blowing up a spherical balloon. Measurements indicate that the “surface tension” of the balloon material is ##k## (assumed constant here with units of force per unit length). Assuming that an air compressor used to blow up the balloon can deliver a...
  3. W

    A Normal velocity to the surface in Spherical Coordinate System

    Let's say we have r=R( theta, phi, t) on the surface of the particle and need to find the normal vector in Spherical Coordinate system. We know that, the unit vector =grad(r-R( theta, phi, t)) / |grad((r-R( theta, phi, t))| where grad is Spherical gradient operator in term of e_r, e_\theta...
  4. JulienB

    Electrical potential of a conducting spherical shell

    Homework Statement Hi everybody! I would like to clear up some doubts I have about my electromagnetism homework: A positive point charge ##q## is placed in the center of an ideal conducting electrically neutral spherical shell, as shown in the attached picture. a) Calculate the electrical...
  5. G

    Finding the Angle of Cut for a Spherical Fruit Slice

    Homework Statement A spherical fruit has a radius of 1 inch. A slice has surface area of its peel equal to pi/8 . Determine the angle of cut for the slice. Homework Equations I'm sure there is a relevant equation here but I don't know it :-( The Attempt at a Solution So the radius is 1 inch...
  6. F

    Volume enclosed by two spheres using spherical coordinates

    Homework Statement Use spherical coordinates to find the volume of the solid enclosed between the spheres $$x^2+y^2+z^2=4$$ and $$x^2+y^2+z^2=4z$$ Homework Equations $$z=\rho cos\phi$$ $$\rho^2=x^2+y^2+z^2$$ $$dxdydz = \rho^2sin\phi d\rho d\phi d\theta$$ The Attempt at a Solution The first...
  7. A

    Gravitation energy of a spherical shell

    I am asked to find the total gravitational energy of a hollow sphere using the fact that the field energy density is given by ##u_g = \frac{-1}{8\pi G}g^2##. Now, ##g = \frac{Gm}{r^2}## in this case and substituting gives ##u_g = \frac{-GM^2}{8 \pi r^4}##. Integrating this over volume will give...
  8. G

    I Spherical Geometry Equilateral Pentagon

    Hey PF! I'm going through a textbook right now and it just said "obviously, you can't have an equilateral pentagon with 4 right angles in spherical geometry (Lambert quadrilaterals). However, I am not able to make the connection. can somebody help me understand why this is?
  9. P

    Three concentric spherical conductors -- Find the potential

    Homework Statement I am not sure whether to put this in the introductory level or advanced. It seems to be relatively introductory in an electromagnetism course. A spherical conductor of radius ##a## carries a charge ##q##. It is situated inside a concentric spherical conducting shell of...
  10. O

    A Ellipse of transformation from spherical to cartesian

    Hi, I have to resample images taken from camera, whose target is a spherical object, onto a regular grid of 2 spherical coordinates: the polar and azimutal angles (θ, Φ). For best accuracy, I need to be aware of, and visualise, the "footprints" of the small angle differences onto the original...
  11. kenok1216

    About charge inside a spherical metallic shell

    where are the charge =0? please help...no idea how to do
  12. i_hate_math

    Question on Sliding/rolling spherical ball

    Homework Statement A bowler throws a bowling ball of radius R = 11 cm along a lane. The ball (the figure) slides on the lane with initial speed vcom,0 = 7.8 m/s and initial angular speed ω0 = 0. The coefficient of kinetic friction between the ball and the lane is 0.13. The kinetic frictional...
  13. S

    I Properties of body with spherical symmetry

    I'm studing Gauss law for gravitational field flux for a mass that has spherical symmetry. Maybe it is an obvious question but what are exactly the propreties of a spherical simmetric body? Firstly does this imply that the body in question must be a sphere? Secondly is it correct to...
  14. I

    Spherical well for deuteron

    Homework Statement How much of the time are the proton and neutron in a deuteron outside the range of the strong force? Suppose the strong force can be described by a spherical potential with parameters ##V_0 = 35 MeV##, ##R = 2.1fm##. The binding energy for deuteron is ##E_b = 2.22 MeV## and...
  15. K

    Sph. Solenoid Homework: Find Magnetic Field at Center

    Homework Statement A large number, N, of closely spaced turns of wire are wound in a single layer upon the surface of a wooden sphere of radius a. The planes of the sphere are perpendicular to the axis of the sphere (take axis to be horizontal), and turns are uniformly spaced per unit...
  16. S

    I Question about solution to Laplacian in Spherical Polars

    I was following this derivation of the solution to the Laplacian in spherical polars. I was wondering where the two equations ##\lambda_{1} + \lambda_{2} = -1## and ##\lambda_{1}\lambda_{2} = -\lambda## come from? Thanks.
  17. S

    Defining rho in spherical coordinates for strange shapes?

    Homework Statement The problem asks for a single triple integral (the integrand may be a sum but there must be a single definition for the bounds of the integral) representing the volume (in the first octant) of the shell defined by a sphere of radius 2 centered around the origin and a sphere...
  18. Dong Hoon Lee

    How to Convert Vectors to Spherical Coordinates at Given Points?

    Homework Statement transform the following vectors to spherical coordinates at the points given 10ax at P (x = -3 , y = 2, z=4) Homework Equations x y z can be chage into x = rsinθcosφ , y=rsinθsinφ , z=cosθ The Attempt at a Solution ax vector can be expressed ar,aθ,aφ so, I can change x ...
  19. Dong Hoon Lee

    Transform Vectors to Spherical Coordinates at P (-3,2,4)

    The problem is << transform the following vectors to spherical coordinates at the points given 10ax at P (x = -3 , y = 2, z=4)>> Actually, My first language isn't English, please understand that. x y z can be chage into x = rsinθcosφ , y=rsinθsinφ , z=cosθ ax vector can be expressed...
  20. E

    Radiation of a Spherical Body Inside a Black Body

    Homework Statement A spherical body of area A and emissivity 0.6 is kept inside a perfectly black body. Total heat radiated by the body at temperature T is (A) 0.4σAT4 (B) 0.8σAT4 (C) 0.6σAT4 (D) 1.0σAT4 Homework Equations Stefan-Boltzmann's Law: P = AεσT4The Attempt at a Solution If it wasn't...
  21. B

    Moment of Inertia for a Thick Spherical Shell

    Homework Statement A) [/B]Consider a hollow sphere of uniform density with an outer radius R and inner radius \alpha R, where 0\leq\alpha\leq1. Calculate its moment of inertia. B) Take the limit as \lim_{\alpha\to1} to determine the moment of inertia of a thin spherical shell. Homework...
  22. S

    Minkowski metric in spherical polar coordinates

    Homework Statement Consider Minkowski space in the usual Cartesian coordinates ##x^{\mu}=(t,x,y,z)##. The line element is ##ds^{2}=\eta_{\mu\nu}dx^{\mu}dx^{\nu}=-dt^{2}+dx^{2}+dy^{2}+dz^{2}## in these coordinates. Consider a new coordinate system ##x^{\mu'}## which differs from these...
  23. D

    Magnetic Field of Spherical Electromagnetic Wave

    1. The problem statemeent, all variables and given/known data The field electric's electromagnetic wave issued by a strut isotropic source is: \vec{E} = E_{0} r_{0}*cos(ωt − kr) \vec{θ} Find the magnetic field in spherical coordinates Homework Equations I think, i use the equation \vec{B} =...
  24. Mr. Rho

    I Limit of spherical bessel function of the second kind

    I know that the limit for the spherical bessel function of the first kind when $x<<1$ is: j_{n}(x<<1)=\frac{x^n}{(2n+1)!} I can see this from the formula for $j_{n}(x)$ (taken from wolfram's webpage): j_{n}(x)=2^{n}x^{n}\sum_{k=0}^{\infty}\frac{(-1)^{n}(k+n)!}{k!(2k+2n+1)!}x^{2k} and...
  25. Matt atkinson

    Free Particle on spherical surface

    Homework Statement By finding the Lagrangian and using the metric: \left(\begin{array}{cc}R^2&0\\0&R^2sin^2(\theta)\end{array}\right) show that: \theta (t)=arccos(\sqrt{1-\frac{A^2}{\omega^2}}cos(\omega t +\theta_o)) Homework EquationsThe Attempt at a Solution So I got the lagrangian to be...
  26. L

    Momentum of a photon heading towards a spherical mass

    Homework Statement A distant observer is at rest relative to a spherical mass and at a distance where the effects of gravity are negligible. The distant observer sends a photon radially towards the mass. At the distant observer, the photon's frequency is f. What is the momentum relative to...
  27. tasleem moossun

    Finding the curl in spherical coordinates

    Hello I've been having trouble finding the curl of A⃗ = r^2[e][/Φ]. Could someone help me please?
  28. ElPimiento

    Work that must be done to charge a spherical shell

    1. Calculate the work that must be done on charges brought from infinity to charge a spherical shell of radius R = 0.100 m to a total charge of Q = 125 μC.2. V = k_e\int{\frac{dq}{r}} \triangle V = - \int{E \cdot ds} W = q\triangle V 3. I started with assuming the spherical shell produces an...
  29. F

    Calculating Angle Between Ecliptic and Horizon for Observer at 18 Degrees North

    Homework Statement Which is the angle between the ecliptic and the horizon in the moment that the point of Aries is hiding for an observer whose position is 18 degrees north? 2. Homework Equations None The Attempt at a Solution The first thing I have tried to is to do a drawing of the...
  30. J

    A Separating the Dirac Delta function in spherical coordinates

    The following integral arises in the calculation of the new density of a non-uniform elastic medium under stress: ∫dx ρ(r,θ)δ(x+u(x)-x') where ρ is a known mass density and u = ru_r+θu_θ a known vector function of spherical coordinates (r,θ) (no azimuthal dependence). How should the Dirac...
  31. Destroxia

    Spherical equation to Rectangular

    Homework Statement An equation is given in spherical coordinates ## \rho sin(\phi) = 8cos(\theta) ## Express the equation in rectangular coordinates a) ## (x-4)^2 + y^2 = 16 ## b) ## x^2 + y^2 + z^2 = 16 ## c) ## x^2 + (y-4)^2 = 16 ## Homework Equations ## x = \rho sin \phi cos \theta, y...
  32. V

    How to calculate phase difference for spherical waves?

    how to calculate phase difference for spherical waves?how to say whether they are in phase or out of phase? in sinusoidal we can easily say whether they are in phase or out of phase just by looking at it,but how to do the same for spherical waves?
  33. M

    Surface area of a spherical cap by integration

    hi guys, i have a question. i saw this picture, and i don't really understand how they derived with the formula. The aim is basically to find the formula for the surface area of a spherical cap. why do you differentiate the x=sqrt(rˆ2-yˆ2)? how does that help to find the surface? and then...
  34. K

    How Does Refraction Affect the Visibility of a Fish Inside a Spherical Bowl?

    Homework Statement A small tropical fish is at the center of a water-filled, spherical fish bowl 28.0 cm in diameter. (a) Find the apparent position and magnification of the fish to an observer outside the bowl. The effect of the thin walls of the bowl may be ignored. (b) Afriend advised the owner...
  35. emeriska

    Electrodynamic - Spherical cavity in dielectric

    First, sorry if something is not totally clear, I'm translating physics term the best I can! 1. Homework Statement A sphere or radius a of permittivity ε2 is placed in a dielectric ε1. Without the sphere, we would have E = E0. We want to find the solution to this problem when ε2 = 1...
  36. KostasV

    Parity and integration in spherical coordinates

    Hello people! I have ended up to this integral ##\int_{φ=0}^{2π} \int_{θ=0}^π \sin θ \ \cos θ~Y_{00}^*~Y_{00}~dθ \, dφ## while I was solving a problem. I know that in spherical coordinates when ##\vec r → -\vec r## : 1) The magnitude of ##\vec r## does not change : ##r' → r## 2) The angles...
  37. T

    Coordinate transformation from spherical to rectangular

    Iam having trouble understanding how one arrives at the transformation matrix for spherical to rectangular coordinates. I understand till getting the (x,y,z) from (r,th ie., z = rcos@ y = rsin@sin# x = rsin@cos# Note: @ - theta (vertical angle) # - phi (horizontal angle) Please show me how...
  38. Rumo

    Weyls representation of a propagating (z-v*t) spherical wave

    Hello! The following wave solves the 3D wave equation: $$ \frac{\sin\left(k\sqrt{x^2+y^2+\frac{(z-vt)^2}{1-\frac{v^2}{c^2}}}\right)}{\sqrt{x^2+y^2+\frac{(z-vt)^2}{1-\frac{v^2}{c^2}}}}\cos\left(w\frac{t-\frac{vz}{c^2}}{\sqrt{1-\frac{v^2}{c^2}}}\right) $$ This is a propagating standing spherical...
  39. D

    How to derive formula for capacitance of spherical capacitor

    Please explain in detail how to derive formula for capacitance of spherical capacitor?
  40. M

    Spherical vectors and rotation of axes

    I have a velocity vector as a function of a latitude and longitude on the surface of a sphere. Let us assume I have a point V(lambda, phi) where V is the velocity. The north pole of this sphere is rotated and I have a new north pole and I have a point V'(lambda, phi) in the new system. I have...
  41. Andreol263

    Green's Function for Spherical Problem(Jackson)

    Hello guys, here's my question is how the book managed to solve this boundary value problem?? can anyone explain it to me in detail? thanks in advance.
  42. J

    Electric field from spherical charge distribution

    Hello. I have a problem calculating the electric field from spherical charge distribution. The exercise is: 1. Homework Statement Homework Equations To solve the problem for $$ 0\le R < a$$ i tried 2 ways: $$ \vec{E} = \frac{\vec{a_R}}{4\pi\epsilon_0}\int_v\frac{1} {R^2}\rho dv $$ and the...
  43. M

    Velocity in spherical polar coordinates

    I am looking at this derivation of velocity in spherical polar coordinates and I am confused by the definition of r, theta and phi. http://www.usna.edu/Users/math/rmm/SphericalCoordinates.pdf I thought phi was the co latitude in the r,θ,∅ system and not the latitude. Of course the two are...
  44. ShayanJ

    Three point charges inside a conducting spherical shell

    Homework Statement There are three point charges inside a conducting spherical shell of radius R. One of them of charge -2q is in the origin and the other two with charge q are in z=d and z=-d. Find the potential inside the sphere! Homework Equations ##\nabla^2\phi=4\pi \rho## 3. The attempt...
  45. O

    Newton's Shell theorem- Gravity inside spherical shell

    Hello all, Guys in my textbook they state that on a point mass at point outside spherical shell of uniform density, the gravitational force is just as if the entire mass of the shell is concentrated at the Centre of shell. The text also states, the force of attraction due to a hollow...
  46. gracy

    Spherical Capacitors: Large Radii & Close Together

    I thought any two concentric conducting spheres of radii a and b such that a<b form a spherical parallel plate capacitor.But according to my book A spherical capacitor behaves as a parallel plate capacitor if it's spherical surfaces have large radii and are close to each other. 1)it's spherical...
  47. ClaireBear1596

    Question related to a 3D finite spherical well

    I have a particle in a spherical well with the conditions that V(r) = 0 is r < a, and V(r) = V0 if r ≥ a. In this problem we are only considering the l=0 in the radial equation. After solving this I found that in the region 0<r<a, u(r)=Bsin(kr) (k=√2mE/hbar), and in the other region...
  48. Loonuh

    Azimuthally Symmetric Potential for a Spherical Conductor

    Homework Statement Homework Equations /The Attempt at a Solution[/B] I am trying to solve problem 2-13 from my textbook "Principles of Electrodynamics" (see image below). I believe that I should be solving the potential as \varphi(r,\theta) = \sum_{n=0}^\infty (A_n r^n +...
  49. P

    Calculating Charge at the Center of a Spherical Shell has me stumped

    Edit: Forgot to type "stumped" at the end of the title 1. Homework Statement Instead of typing it out, a link to a scanned document of the problem is here: http://imgur.com/Be3jSLp. Homework Equations The equations to use are stated in the problem here: http://imgur.com/Be3jSLp The Attempt...
  50. C

    Rec, Spherical and cylind coordinates.

    Homework Statement Homework EquationsThe Attempt at a Solutionhere is the setup for each, can someone check if they are correct before I evaluate the volume?
Back
Top