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. J

    Write a triple integral in spherical coordinates

    Homework Statement Write a triple integral in spherical coordinates that represents the volume of the part of the sphere X^2+Y^2+Z^2=16 that lies in the first octant(where x,y, and z are coordinates are all greater than or equal to zero) Homework Equations So i know this is in...
  2. O

    Symmetrization of a tensor in spherical coordinate

    Hello, i don't know if my question is well posed, if i have a symmetric tensor Sij = (∂ixj + ∂jxi) / 2 with xi cartesian coordinates, how can i transform it in a spherical coordinates system (ρ,θ,\varphi)? (I need it for the calculus of shear stress tensor in spherical coordinate in fluid...
  3. Feodalherren

    From Cartesian to spherical integral

    Homework Statement Homework Equations The Attempt at a Solution Is this the correct setup? \int^{\pi}_{\frac{3\pi}{4}}\int^{2\pi}_{0}\int^{\sqrt{2}}_{0}\frac{1}{\rho^{2}} rho^2 Sin\phi d\rho d\theta d\phi I gave up on itex. It was either that or my computer flying out the...
  4. I

    Differentials of spherical surface area and volume

    please tell me if i did this correctly: task: I'm trying to divide the differential dA by dV where.. dA = differential surface area of a sphere, dV = differential volume of a sphere dA=8\pirdr dV=4\pir2dr so then dA/dV= 2/r Also, if i treat this as a derivative, then would...
  5. S

    Charge distribution between two spherical hollow conductors.

    Consider two spherical hollow conductors, charged to Q1 and Q2 coulombs respectively. What happens when one is placed within the other, and they are connected by a thin metallic wire? I do know that if they were placed at a distance from each other, the charge is distributed in the ratio of the...
  6. U

    Spherical coordinates choice for an electric field problem

    I am finding the electric field from a spherical shell at a point on the z-axis outside the shell. The shell is centered at the origin,and I am only allowed to use coulomb's law. I want to find dE in spherical coordinates first then transform it to Cartesian before integrating to get E. So I...
  7. A

    Electric Potential of a Spherical Shell

    Homework Statement A conducting spherical shell has inner radius a, outer radius b, and has a +Q point charge at the center. A charge of -Q is put on the conductor. a) What is the charge on the inner and outer surface of the shell? b) What is the electric field everywhere? c) What...
  8. M

    Lagrange qustion, a partilcle confined to a spherical cone

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  9. F

    Deriving E of a charged spherical shell from V

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  10. T

    Finding the volume using spherical coordinates

    Homework Statement Let V be the volume of the solid enclosed by the sphere x^2 + y^2 + z^2 - 2z = 0 , and the hemisphere x^2 + y^2 + z^2 = 9 , z ≥ 0. Find VHomework Equations Using spherical coordinates: x^2 + y^2 + z^2 = ρ^2 z = ρcos(ø) The Attempt at a Solution So I changed both of them to...
  11. S

    Transforming a vector in spherical coordinates

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  12. binbagsss

    Spherical surface area element

    See image attached. (I've had a google but can't find anything). I am trying to understand the expression : Rdθ.2∏Rsinθ Here are my thoughts so far: Rdθ is the width of a strip, θ being the variable changing/to integrate over, giving arise to the elements. 2∏Rsinθ must then...
  13. U

    Spherical coordinates length from differential length

    is it logical to ask this question in Spherical coordinates: Using the differential length dl , find the length where r=1 0<Θ<∏/4 ∏/2< θ <∏/4 where Θ is the azimuthal angle. What I mean by ∏/2< θ <∏/4 is that the line is a "diagonal" line which has an ascention of ∏/4 from the xy...
  14. binbagsss

    Quick EMF question - Spherical Shell- North/South Pole, Equator

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  15. baby_1

    Electric field a spherical surface

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  16. Spinnor

    B-mode plots, spherical harmonics?, fundamental modes?

    If the B-mode sky plots could be Fourier transformed what would be a plot of the lowest order B-mode harmonic plotted on a sphere look like? I guess we need two functions of spherical coordinates, one function for amplitude at points on a sphere and one function for the orientation at the...
  17. J

    Spherical Coordinates: Distance Between 2 Points

    Hello, I am looking into finding geodesic distances for an ellipsoid. I will designate two points then find the distance between them. This will be my geodesic distance. I have put together a schematic (attached) for reference. Ultimately I need to know the distance D as shown on the...
  18. P

    Spherical Capacitor: Calculating Capacitance in pF

    Homework Statement A spherical capacitor consists of a spherical conducting shell charge -Q concentric with a smaller conducting sphere of radius 4.0 cm and charge Q. The larger conducting shell has inner and outer radii of 11.0 cm and 13.0 cm, respectively. What is the capacitance of the...
  19. homer

    Concentric conducting spherical shells cut by a horizontal plane

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  20. V

    Derivative of Spherical Harmonic for negative m

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  21. Z

    Why Does <n',l',m'|\hat{z}|n,l,m> Equal Zero Unless m=m'?

    Homework Statement I want to show that <n',l',m'|\hat{z}|n,l,m> = 0 unless m=m', using the form of the spherical harmonics. Homework Equations Equations for spherical harmonics The Attempt at a Solution Not sure how to begin here since there aren't any simple eigenvalues for...
  22. M

    Potential of Concentric Spherical Insulator and Conductor

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  23. C

    Potential for a system of a solid sphere and spherical shell

    Homework Statement A metal sphere with radius a is supported on an insulating stand at the center of a hollow, metal spherical shell with radius b. There is charge +Q on the inner sphere and charge -Q on the outer shell. Take the potential V to be zero at infinite separation. Calculate...
  24. M

    Vector calculus: angular momentum operator in spherical coordinates

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  25. ShayanJ

    Magnetic field of a spherical capacitor

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  26. X

    Particle in Spherical Well : Sudden Approximation

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  27. D

    Derivation of Laplace in spherical co-ordinates

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  28. A

    Relation between charge and voltage in spherical capacitors

    1-If we had a spherical capacitor and the voltage across it is 1000 V, I need to know the charge on every plate, I know 2 ways to solve this either using C=Q/V or V=Q/r * 1/4ε0∏. So, which one should I use, and why? 2-Another thing, according to hyperphysics the formula for the capacitance of...
  29. E

    Derivation of Phi-Hat wrt Phi in Spherical Unit Vectors

    Homework Statement I just want to know how to get from this: ∂ø^/∂ø = -x^cosø - y^sinø to this: = -(r^sinθ+θ^cosθ) Homework Equations All the equations found here in the Spherical Coordinates section: http://en.wikipedia.org/wiki/Unit_vector The Attempt at a Solution I've...
  30. H

    Surface integral, spherical coordinates, earth

    Homework Statement Find the surface area of the Earth (as a fraction of the total surface of the earth) that lies above 50 degrees latitude North. Homework Equations $$A = \int_R\sqrt{|\det(g)|}d\theta d\phi$$ The Attempt at a Solution Hence I get $$\int_0^{360}...
  31. ShayanJ

    Spherical capacitor and vector potential

    There is a spherical capacitor with inner and outer radii of a and b respectively,has a dielectric material of small conductivity \sigma between its concentric conducting spheres.Calculate vector potential and magnetic field arising from this configuration. This is how I did it...
  32. BeBattey

    Conducting Spherical Shell Capacitor

    Homework Statement A conducting spherical shell is divided into upper and lower halves with a narrow insulating ring between them. The top half is at 10V and the bottom half is at -10V. Write down the appropriate expansion for Φ and use symmetry and the expected behavior at the origin to...
  33. A

    Solve for Focal Length of Convex Mirror: 1/50 + 1/20 = 1/-f

    Homework Statement An object is placed 50 cm in front of a convex mirror and its image is found to be 20 cm behind the mirror. What is the focal length of the mirror? Homework Equations 1/d0 + 1/di = 1/f The Attempt at a Solution 1/50 + 1/20 = 1/-f 70/1000 = 1/-f f = 14.28...
  34. Z

    Express wave function in spherical harmonics

    1. Problem: I have a wave function ψ(r) = (x + y + z)*f(r) and want to find the expectation values of L2 and Lz. It is suggested that I first change the wave function to spherical coordinates, then put that in terms of spherical harmonics of the form Yl,m. 2. Homework Equations ...
  35. B

    Electric field of a spherical metal shell

    Homework Statement A spherical metal shell has charge per unit area sigma and radius R. What is the magnitude of the electric field at a distance x from the surface of the sphere? You may include only these variables in your formula: sigma, R, epsilon_0, x Homework Equations...
  36. M

    Electric fields of a spherical shell

    My class hasn't delved into Gauss's Law much besides describing conductors at electrostatic equilibrium to have no net electric field or force within itself. For the picture, the question is: What are the magnitudes of the electric fields at: 1) r = a 2) r = 3/2 a 3) r = b...
  37. M

    Gauss's Law Problem - Spherical Shell with Non-uniform Charge

    Homework Statement Consider a spherical shell with inner radius r1=0.30 m and outer radius r2=1.00 m. The hollow inside the shell contains no charge; and charge is distributed on the inside surface of the shell and within the shell itself, such that the electrical field inside the shell itself...
  38. H

    What is the equation of motion for a launched spherical stone?

    Homework Statement A spherical stone of mass 0.500 kg and radius 10 cm is launched vertically from ground level with an initial speed of 20.0 m/2. As it moves upwards, it experiences drag from the air as approximated by Stokes drag, F=6η∏Rv, where the viscosity of air is 1.002 mPa*s...
  39. R

    Schrodinger Equation in Spherical co-ordinates. Constants.

    When normalising the S.E. in spherical coordinates you split it up into 3 integrals, with respect to r, theta and phi. My question is, once you have found the constants for each, when writing out the normalised PSI do you simply place them as a product in the solution? i..e PSI...
  40. U

    Concentric conducting spherical shells

    Homework Statement Consider three concentric conducting shells, with potentials 0, ø_0, 0 and radius a, b ,c where a < b < c. (a)State conditions for Laplace to work and boundary conditions for E (b)Show ø is of the form: (c) Find ø and E everywhere. (d) Find the charge density and...
  41. K

    The title for this content could be What are Spherical Even Even Nuclei?

    Hello, I came across the name spherical even even nuclei in an exercise about the hyperfine structure. What does it refer to? That the number of protons and the number of neutrons are both even? So that there is no nuclear magnetic moment?
  42. O

    Does the Central Point Charge Affect Total Charge in a Spherical Shell?

    Just a quick general question in applying Gauss's law. Not exactly homework, more a general question so I can understand my other homeworks better. I have a spherical shell with inner radius R_1 and outer radius R_2 and a point charge Q in its center. It is NOT a conducting sphere. In the...
  43. O

    Potential of spherical charge distribution

    I want to derive this equation: V(r) = \frac{1}{\epsilon_0} [\frac{1}{r} \int_0^r \! r'^2 \rho(r') \, d r' + \int_r^{\infty} \! r' \rho(r') \, d r' ] of a spherical charge distribution. I can do it with the general integral definition of the electrostatic potential (which is basically...
  44. S

    Electric Potential Due to Spherical Shells

    Homework Statement A spherical shell of radius R0 has a non-uniform surface charge density: η=η0*cos(2θ), where θ is the angle measured from the positive z axis and η0 is a constant. This shell is inserted into another spherical shell (this one has a volume) with its inner lip at radius R1...
  45. GeorgeDishman

    Gravitational time dilation for a spherical body of finite radius

    I am considering the gravitational time dilation at the centre of a spherical, non-rotating body (such as the Earth). The usual formula for gravitational time dilation is √(1-r_s/r) where r_s is the Schwarzschild Radius and r is the radius of the clock compared to one at infinity, however, this...
  46. DrClaude

    Hamilton-Jacobi equation in spherical coordinates

    I was looking at the Wikipedia entry on the Hamilton-Jacobi equation, and was confounded by the equation at the beginning of the section on spherical coordinates: http://en.wikipedia.org/wiki/Hamilton–Jacobi_equation#Spherical_coordinates Shouldn't the Hamiltonian simply be $$ H =...
  47. Y

    Differentiation in spherical coordinates.

    1) If u(r,\theta,\phi)=\frac{1}{r}, is \frac{\partial{u}}{\partial {\theta}}=\frac{\partial{u}}{\partial {\phi}}=0 because u is independent of \theta and \;\phi? 2) If u(r,\theta,\phi)=\frac{1}{r}, is: \nabla^2u(r,\theta,\phi)=\frac{\partial^2{u}}{\partial...
  48. F

    Spherical harmonics and angular momentum operators

    When solving for the spherical harmonic equations of the orbital angular momentum this textbook I'm reading.. Does this mean that there must be a max value of Lz which is denoted by |ll>? Normally the ket would look like |lm>, and since m is maxed at m=l then |ll> is the ket consisting of the...
  49. MattRob

    Spherical Coordinates: Understanding Theta Equation

    So, I was curious about this and found more or less what I was looking for here: http://electron9.phys.utk.edu/vectors/3dcoordinates.htm Except, something is bothering me about those equations. At the very bottom, the equation for Theta in a spherical coordinate system; shouldn't it be...
  50. S

    How Do Different Notations Affect Spherical Harmonic Computations?

    Homework Statement For the spherical harmonics Umn; Vmn, compute the ones of orders 0,1, 2. Umn=cos(nθ)sinn(\varphi)Pmn(cos(\varphi)) Vmn=sin(nθ)sinn(\varphi)Pmn(cos(\varphi)) (b) How many non-zero spherical harmonics are there of order k? Homework Equations Equations of Umn; Vmn...
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