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. B

    Calculating Charge on a Spherical Surface with Varying Density

    Homework Statement Electric charge resides on a spherical surface of radius 0.3 centered at the origin with charge density specified in spherical polar coordinates by f(r,\phi, \theta) = 3 × 10^{-12} cos(\theta). Determine the total amount of electric charge on the sphere. Homework Equations...
  2. C

    Sign of Levi-Civita Symbol in spherical coordinates

    Hi, I am going through the derivation of an instanton solution (n=1) in Srednicki Chp. 93. Specifically, I went through eqn.s 93.29-93.38. However the sign of the Levi-Civita Symbol is bugging me: It says that in 4D Euclidean space, \epsilon^{1234}=+1 in Cartesian coordinates implies...
  3. M

    Gauss' Law: Spherical Symmetry

    Homework Statement Figure 23.52 gives the magnitude of the electric field inside and outside a sphere with a positive charge distributed uniformly throughout its volume. The scale of the vertical axis is set by Es = 5.0 x 10e7 N/C. What is the charge on the sphere? Homework Equations Net Flux...
  4. G

    Constructing an Atlas for ##S^2## with Spherical Coordinates

    Now, this is kind of embarrassing, but I've been trying to do this for too long now and failed: I want to construct an atlas for ##S^2##, but I want to use spherical coordinates rather than stereographic projection. Of course the first chart image is simply ##\theta \in (0, \pi), \varphi \in...
  5. V

    Electric Field inside concentric spherical shells

    Homework Statement I uploaded a file that gives the problem statement. Homework Equations I don't believe any equations are necessary. However, I could be wrong. I believe it to be a concept question. The relevant concept being that the electric field inside conducting materials in...
  6. P

    What is the Acceleration of a Spherical Shell Filled with Fluid on an Incline?

    Homework Statement A spherical shell of mass M and radius R is completely filled with a frictionless fluid, also of mass M. It is released from rest, and then it rolls without slipping down an incline that makes an angle θ with the horizontal. What will be the acceleration of the shell down the...
  7. pitbull

    Spherical Conductors: Voltage Calculation & Equilibrium | ρs=ρ0cos2theta

    Homework Statement Given two spherical conductors of radius R and tangent at O, both are charged and in equilibrium with surface charge density ρs=ρ0cos2theta. Calculate: a) Voltage of both spheres at O. (SOLUTION: V=2ρ0R/(3ε0) (...) Homework EquationsThe Attempt at a Solution So I tried to...
  8. H

    Spherical Charge distribution

    Homework Statement A Non-Uniform but spherically symmetric charge distribution has a charge density: \rho(r)=\rho_0(1-\frac{r}{R}) for r\le R \rho(r)=0 for r > R where \rho = \frac{3Q}{\pi R^3} is a positive constant Show that the total charge contained in this charge distribution is...
  9. allamsetty

    Need 2D plot of plane wave, cylindrical & spherical wave

    plane wave is represented as exp (ik.r) and the cylindrical wave as 1/sqrt(r) *exp(ik.r) and the spherical wave as 1/r*exp(ik.r) Have anyone tried to plot these waves? How to do it? Attempt: in Matlab assuming k=1 >> x=linspace(-1,1,100); >> for(ii=1:100) fp(ii)=exp(i*x(ii)); end >>...
  10. H

    Yaw Pitch & Roll to spherical Theta & Phi

    hi guys, can someone please tell me how to find Theta and Phi from Yaw Pitch and Roll? I use my smartphone orientation sensor on a project and i need to calculate the smartphone's normal vector projection.
  11. C

    Separation of Variables Spherical Coordinates

    Homework Statement So I'm doing a question from one of my past exams as attached, there are no copy right issues with this document that I know of and have asked my lecturer who wrote the exam and he said I am welcome to upload it. The question is 1)b)iv), my attempt is attached. I end up with...
  12. D

    Normalization coefficient for Spherical Harmonics with m=l

    Homework Statement Well it is not the problem itself that bothers me but the maths behind a part of it. As part of finding the coefficient I had to solve the integral of (Sin(x))^(2l+ 1). The solution given by the solution manual just pretty much jumps to the final answer...
  13. B

    Rotation of a Spherical Top

    Homework Statement A solid sphere of mass M and radius R rotates freely in space with an angular velocity ω about a fixed diameter. A particle of mass m, initially at one pole, moves with constant velocity v along a great circle of the sphere. Show that, when the particle has reached the other...
  14. A

    Transforming Spherical Angles to Cartesian Coordinates for Beam Dynamics

    Hello I have this problem - From a generator, I get a compton scattering with the electrons theta and phi angles. where I having the following equations for a particle px = E_particle * sin (theta) * cos (phi); py = E_particle * sin (theta) * sin (phi); pz = E_particle * cos (theta)...
  15. S

    Find the volume of the region D using spherical coordinates

    Homework Statement The problem and its solution are attached as TheProblemAndSolution.jpg. Homework Equations V(D) = ∫∫∫_D ρ^2 sinθ dρ dϕ dθ The Attempt at a Solution How exactly does the solution get cos α = 1/√(3)? Also, when the solution goes from the step with two integrals to the step...
  16. J

    Solving a PDE in spherical with source term

    Homework Statement I have a PDE and I need to solve it in spherical domain: $$\frac{dF(r,t)}{dt}=\alpha \frac{1}{r^2} \frac{d}{dr} r^2 \frac{dF(r,t)}{dr} +g(r,t) $$ I have BC's: $$ \frac{dF}{dr} = 0, r =0$$ $$ \frac{dF}{dr} = 0, r =R$$ Homework Equations So, in spherical coord. First...
  17. J

    Spherical coordinates of Partial Differential Equation

    Homework Statement I have a PDE and I need to solve it in spherical domain: $$\frac{\partial F(r,t)}{\partial t}=\alpha \frac{1}{r^2} \frac{\partial}{\partial r} r^2 \frac{\partial F(r,t)}{\partial r} +g(r,t) $$ I have BC's: $$ \frac{\partial F}{\partial dr} = 0, r =0$$ $$ \frac{\partial...
  18. S

    Curl of Z-unit vector in spherical coordinates

    Homework Statement There is a sphere of magnetic material in a uniform magnetic field \vec H_0=H_0\vec a_z, and after some calculations I got the magnetic moment vector \vec M_0=M_0\vec a_z, where M_0 is something that isn't important to my question. I am unsure if this magnetic moment vector...
  19. amind

    Magnification of a fish in a spherical bowl due to water

    Homework Statement A goldfish in a spherical fish bowl of radius R is at the level of the center of the bowl and at distance R/2 from the glass. What magnification of the fish is produced by the water of the bowl for a viewer looking along a line that includes the fish and the center, from the...
  20. CopyOfA

    Spherical Capacitor with Frequency Dependent Dielectric

    This is a long post. Sorry... 1. Homework Statement We are given a spherical capacitor with an inner conductor of radius ##a## and outer conductor of radius ##c##. The space between the conductors is half filled (##a<r<b##) with a dielectric with permittivity...
  21. M

    Spherical bubble rises to surface, Ideal Gas, Thermal Energy

    Homework Statement A spherical air bubble in a lake expands as it rises slowly to the surface. At the point it starts to rise, the pressure is 2.00 atm, the temperature of the water is 10.0 ∘C, and the radius of the bubble is 5.00 × 10^−3 m. At the surface, the pressure is 1.00 atm and the...
  22. P

    Bending disturbances in axisymmetrically loaded spherical

    Can anyone please give an example of the use of bending disturbances in axisymmetrically loaded spherical shell using Geckeler approximation
  23. gfd43tg

    How Can You Calculate the Drying Time of a Spherical Granule?

    Homework Statement Fig. below shows the cross-section of a porous spherical granule of radius a. The pores are initially saturated with water. The granule dries in air at pressure P and temperature T. The drying rate is controlled by diffusion of water vapor through the dry region B; the...
  24. gfd43tg

    Sherwood number of evaporating spherical droplet

    Homework Statement From Fick's Law of Diffusion, ##N_{A}C_{B} - N_{B}C_{A} = -DC \frac {dC_{A}}{dz}## Prove that Sh=2 for a volatile spherical droplet evaporating in a quiescent atmosphere. (Sherwood number for a sphere is given by ##Sh = \frac {k_{A}d}{D}##, where d = initial droplet...
  25. H

    The center of a 1.00 km diameter spherical pocket of oil

    Homework Statement The center of a 1.00 km diameter spherical pocket of oil is 1.00 km beneath the Earth's surface[/B]. Estimate by what percentage g directly above the pocket of oil would differ from the expected value of g for a uniform Earth? Assume the density of oil is 8.0*102 kg/m3...
  26. S

    Finding potential of a given wavefunction in spherical polar

    Homework Statement The ground state wavefuntion of a system in spherical polar coordinates is given by: Ψ (r,θ, φ)= (A/r) [exp (-ar) - exp (-br)] where a, b, A are constants. i) Determine A as a function of a and b, so as to normalize the wavefuntion. ii) From Schrödinger equation find V (r)...
  27. G

    Uniform Circular Motion of a Spherical Earth

    Homework Statement In this question, the Earth is modeled as a uniform sphere of radius 6400km. Objects are released from points just above the Earth's surface at the equator and at the North Pole. Which will fall to the Earth with the greater acceleration and by how much? Homework Equations...
  28. A

    Potential Energy of 2 Spherical Shells

    Homework Statement Find the energy required to assemble two uniform hollow spheres of charge q between radii a and b with a volume charge density roh-v. The shells are separated by a distance c. *description of picture* - two identical spherical shells with inner radius a and outer radius b...
  29. C

    Deriving spherical unit vectors in terms of cartesian unit vectors

    I'm trying to find the azimuthal angle unit vector \vec{\phi} in the cartesian basis by taking the cross product of the radial and \vec{z} unit vectors. \vec{z} \times \vec{r} = <0, 0, 1> \times <sin(\theta)cos(\phi), sin(\theta)sin(\phi), cos(\theta)> = <-sin(\theta)sin(\phi)...
  30. K

    Spherical coordinates confusion

    I am accustomed to ##x=rcos(\theta)sin(\phi)## ##y=rsin(\theta)sin(\phi)## ##z=rcos(\phi)## ##-\pi<\theta<\pi##, ##-\pi/2 < \phi < \pi/2## but see some people using these instead ##x=rcos(\theta)cos(\phi)## ##y=rsin(\theta)cos(\phi)## ##z=rsin(\phi)## ##-\pi<\theta<\pi##, ##-\pi/2 < \phi <...
  31. H

    Solving Electrostatic Potential of a Spherical Conducting Shell

    Homework Statement A spherical conducting shell of radius R is held at a potential V0. Outside the shell, the charge density is ρ(r) = ρ0sinθcosφ for R < r < 2R. Find the electrostatic potential everywhere outside the shell. Homework Equations Green's function in spherical coordinates between...
  32. CopyOfA

    Potential inside concentric spherical shells with non-uniform charge density

    Homework Statement We are given a two concentric spherical shells with small radius ## a ## and larger radius ## b ##. The inner and outer shells are made of conducting material and there is a volume charge density, ##\rho\left(r\right) ##, that exists between the shells,. The boundary...
  33. A

    Potential of Spherical Shell with Nonunifor Surface Charge

    Homework Statement A thin spherical shell of radius R carries a surface charge density of the form kcos 3 θ . Find the electric field inside and outside the sphere and demonstrate explicitly that its components satisfy the relevant boundary conditions at the surface Homework Equations The...
  34. Pezz

    Spherical EM Wave: Origin at t=0, S & S' Agree?

    Consider two frames: S and S', with S' moving to the right along the positive x-axis or S at a relative velocity v. The origins of S and S' coincide at t = 0. A spherical electromagnetic wave leaves the origin of S the moment S and S' coincide, or at t = 0. If we consider the transformation...
  35. A

    Non-Uniform Surface Charge Spherical Shell

    Homework Statement A thin spherical shell of radius R carries a surface charge density of the form kcos3 \theta Find the electric field inside and outside the sphere and demonstrate explicitly that its components satisfy the relevant boundary conditions at the surface. Homework Equations...
  36. G

    Spherical shape surface area calculation

    Homework Statement Homework Equations [/B] Surface area of sphere = 4*Pi*r^2, where r: is the radius of the sphere circleThe Attempt at a Solution Solution:[/B] 1. In terms of “r” and “R”, and the radius of sphere “S”, and “d”: Given that: · Surface area of both shaded area are equal...
  37. C

    Calculating The Area of a Spherical Cap

    1. Homework Statement Hi all, I am working through Gravity by James Hartle and have become stuck on a question asking me to calculate the area of a circle of radius r in the 2D geometry that is the surface of a sphere of radius a. A surface element on this sphere can be found to be...
  38. G

    Find the total electrostatic energy stored in the configuration

    Homework Statement A spherical conductor of radius ##a## carries a charge ##q## and also there is a jelly of constant charge density ##\rho## per unit volume extending from radius a out to radius ##b##. Find the electrostatic energy stored in the configuration. Homework Equations ##\oint...
  39. deedsy

    Spherical Charge Distribution - Electric Field Intensity

    Homework Statement A spherical charge distribution is given by p = p_0 (1- \frac{r^2}{a^2}), r\leq a and p = 0, r \gt a , where a is the radius of the sphere. Find the electric field intensity inside the charge distribution. Well I thought I found the answer until I looked at the back of...
  40. C

    Integrating a delta function with a spherical volume integral

    Homework Statement Integrate $$\int_V \delta^3(\vec r)~ d\tau$$ over all of space by using V as a sphere of radius r centered at the origin, by having r go to infinity. Homework EquationsThe Attempt at a Solution This integral actually came up in a homework problem for my E&M class and I'm...
  41. A

    Calculate Energy Density of Spherical Capacitor

    A capacitor is formed from two concentric spherical conducting shells separated by vacuum. The inner sphere has radius 10.5cm , and the outer sphere has radius 15.5cm . A potential difference of 110V is applied to the capacitor. What is the energy density at r= 10.6cm , just outside the inner...
  42. K

    Puzzled with the loxodrome ( spherical spiral ) equation

    Hi All! the mathworld website http://mathworld.wolfram.com/SphericalSpiral.html claims that the loxodrome is given by the parametric equations ##x=cos(t) cos(c)## ##y=sin(t) cos(c)## ##z=-sin(c)## Why so? Now, as far as I can see, since the spherical coordinates are ##x=sin\phi cos\theta##...
  43. P

    Spherical Aberration Estimation

    Homework Statement Estimate the size of the spherical abberation of a spherical mirror of 1m-diameter and a focal length of 2 meter. (Hint: Calculate the size of the smeared image of a star at the focal point and compare it to the size (in arc-sec) of an extended object)Homework Equations The...
  44. moriheru

    Concerning spherical Bessel and Neumann functions

    When transforming the Schrodinger equation into sphericall coordinates one usually substitutes psi(r,theta,phi) into the equation and ends up with something like this: -h(bar)^2/2m* d^2/dr^2*[rR(r)]+[V(r)+(l(l+1)*h(bar)^2)/2mr^2]*[rR(r)]=E[r R(r)] Question 1: How do I replace the Rnl(r) with...
  45. Z

    How Do You Determine the Ground State Energy in a Spherical Infinite Well?

    Homework Statement A particle of mass ##m## is constrained to move between two concentric hard spheres of radii ##r = a## and ##r = b##. There is no potential between the spheres. Find the ground state energy and wave function. Homework Equations $$\frac{-\hbar^2}{2m} \frac{d^2 u}{dr^2} +...
  46. moriheru

    Concerning spherical potential wells

    Can one work with spherical potential wells as square wells with an infinite amount of steppotentials of infinitly small size , thus integrating or summing the steppotentials? Would be great bunch of work, treating all the steppotentials and the different energys of the particle I mean for E>V...
  47. M

    Electric field of point charge within spherical shell.

    If there was a spherical shell with negative charge density and a positive point charge inside the shell, the electric field lines from the point charge would just be radially outward towards the shell right? What about the case where there's a positive charge density and a positive point...
  48. O

    Potential due to two oppositely charged half spherical shells

    Homework Statement Consider the spherical shell of radius R shown in the figure characterized by a hemisphere of surface charge density \sigma and total charge q in the upper half plane and a hemisphere of surface charge density -\sigma and total charge -q in the lower half plane. Find the...
  49. A

    Cross Products in Spherical Coordinates: Is A x B True?

    Is A x B = | i j k | also true for Spherical Coordinates? | r1 theta1 phi1 | | r2 theta2 phi2 | Or I have to convert them to Cartesian Coordinates and do the cross product and then...
  50. A

    Potential difference between two spherical conductors

    I'm working my way through MIT 8.02x on EdX (an archived course, so it's a bit lonely in there right now!). The problem statement: Two spherical conductors, A and B, are placed in vacuum. A has a radius rA=25 cm and B of rB=35 cm. The distance between the centers of the two spheres is d=225...
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