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. Y

    Spherical GR dust shell with vacuum inside and outside

    In various other threads we have been kicking around various equations for a spherical shell and discussing the implications. In this thread I would like to present what (I think) I have worked out about how the shell metric relates to the vacuum metric inside and outside the shell. I hope to...
  2. N

    Moment of inertia of a spherical shell

    Hey. There's one thing I've always been wondering about when it comes to deriving the expression for the moment of inertia of a spherical shell. Namely, why is the length of the infinitesimal cylinder used in the derivations (like here ) equal to ##R d \theta##, instead of ##R d \theta...
  3. A

    Evaluating triple integral with spherical coordinates

    Homework Statement Evaluate the iterated integral ∫ (from 0 to 1) ∫ [from -sqrt(1-x^2) to sqrt(1-x^2) ] ∫ (from 0 to 2-x^2-y^2) the function given as √(x^2 + y^2) dz dy dx The Attempt at a Solution I changed the coordinates and I got the new limits as ∫(from 0 to pi) ∫(from...
  4. H

    The magnetic field of a spinning spherical shell of uniform charge

    In chapter 5, magnetostatics, of Griffiths' Introduction to Electrodynamics (third edition), there's a problem in the back of the chapter that asks you to calculate the force of attraction between the northern and southern hemispheres of a spinning charged spherical shell. The problem in its...
  5. A

    Integral in spherical coordinates

    I recently had to do an integral like the one in the thread below: http://math.stackexchange.com/questions/142235/three-dimensional-fourier-transform-of-radial-function-without-bessel-and-neuman The problem I had was also evaluating the product and I am quite sure that the answer in the thread...
  6. A

    What is the Physical Meaning of Voltage on a Spherical Grain?

    I have been given the exercise above which relates to an article we are reading. I can calculate all the results but I can't interpret them physically. I think my major problem is that I don't see how self capacitance is a physically measureable quantity. For a recap the self capacitance is...
  7. I

    Neutron Flux Profile in a Spherical Moderator

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  8. M

    I don't understand the ranges of the angles in spherical coordinates

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  9. Philosophaie

    Double Integral Surface Area of Spherical Ball

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  10. Philosophaie

    Spherical Ball with uniform Charge Density

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  11. skate_nerd

    MHB Showing relationship between cartesian and spherical unit vectors

    I am asked to show that when \(\hat{e_r}\), \(\hat{e_\theta}\), and \(\hat{e_\phi}\) are unit vectors in spherical coordinates, that the cartesian unit vectors $$\hat{i} = \sin{\phi}\cos{\theta}\hat{e_r} + \cos{\phi}\cos{\theta}\hat{e_\phi} - \sin{\theta}\hat{e_\theta}$$ $$\hat{j} =...
  12. Philosophaie

    Electrical Field around Spherical Ball at origin

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  13. L

    Curl in spherical coords with seeming cartesian unit vector

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  14. M

    Angle of Incidence on a spherical lens

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  15. PsychonautQQ

    Evaluating a triple integral Spherical

    Homework Statement z(x^2+y^2+z^2)^(-3/2) where x^2+y^2+z^2 ≤ 4 and z ≥ 1 The Attempt at a Solution So spherically this comes down to cos∅sin∅dpdθd∅ p goes from 0 to 2, theta goes from 0 to 2pi, but I don't know how to figure out what ∅ goes from? I'm trying use trig identities but...
  16. Avatrin

    Coulomb's law and spherical charge distribution

    Homework Statement Find the E produced by a spherical charge distribution with uniform charge density at a point inside the sphere, using triple integration. Homework Equations E = 1/4πε ∫f(x,y,z)/r^2 dV The Attempt at a Solution f(x,y,z) = p Radius of sphere = R Position of...
  17. PsychonautQQ

    Question on spherical integration

    Homework Statement So if you integrate over a spherical area, say a ball of radius 1, then 0≤p≤1, 0≤θ≤2∏, and 0≤∅≤∏. My question is why don't you integrate ∅ between 0 and 2∏? I mean if you are integrating over a sphere then you have to go around it vertically AND horizontally completely...
  18. H

    Divergence/flux of an E field for two spherical regions

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  19. S

    Electric Field of a hollow conducting spherical shell

    Homework Statement A hollow conducting spherical shell has radii of .80m and 1.20m. The sphere carries a net charge of -500 nC. A stationary point charge of +300 nC is present at the center origin. Calculate the electric field at points: a) 0.30m b) 1.00m c) 1.50m I have attached the image...
  20. J

    Spherical cylindrical and rectangular coordinates

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  21. M

    Charge and field distribution for spherical conductor with cavity

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  22. M

    Trigonometric function expanded in spherical harmonics

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  23. A

    Potential, field, Laplacian and Spherical Coordinates

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  24. R

    Electric fields and a spherical surface

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  25. P

    Work to remove point charge from center of spherical conducting shell

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  26. W

    Describing a Solid Ice Cream Cone with Spherical Coordinates

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  27. C

    Charged Metallic Sphere Touching Spherical Shell From Inside

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  28. Petrus

    MHB Triple integral, spherical coordinates

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  29. V

    Nabla Operator in Spherical Coordinates

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  30. Z

    Spherical Coordinate System Interpretation

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  31. D

    MHB Spherical Harmonics: Showing $\delta_{m,0}\sqrt{\frac{2\ell + 1}{4\pi}}$

    I am trying to show that \[ Y_{\ell}^m(0,\varphi) = \delta_{m,0}\sqrt{\frac{2\ell + 1}{4\pi}}. \] When \(m = 0\), I obtain \(\sqrt{\frac{2\ell + 1}{4\pi}}\). However, I am not getting 0 for other \(m\). Plus, to show this is true, I can't methodically go through each \(m\). How can I do this?
  32. V

    Gauss's Law for Spherical Symmetry

    Find the electric field for a non-conducting sphere of radius R = 1 meter that is surrounded by air in the region r > 1. The interior of the sphere has a charge density of ρ(r) = r. The answer is k(pi)/r^2, but I can't seem to get that. My problem is with finding the enclosed charge. I've tried...
  33. Philosophaie

    Solution of the Geodetic Spherical Equation

    I have calculated the Keplerian Elements for a particular Position and Velocity Vector for a Satellite around Earth. With a solution of the Geodetic Spherical Equation: u = \frac{1}{R} = \frac{μ}{h^2}(1+ e*Cos(\phi - \tilde\omega)) What does the "R" in this equation represent? where ω is the...
  34. PeteyCoco

    Line integral of a spherical vector field over cartesian path

    Homework Statement Compute the line integral of \vec{v} = (rcos^{2}\theta)\widehat{r} - (rcos\theta sin\theta)\widehat{\theta} + 3r\widehat{\phi} over the line from (0,1,0) to (0,1,2) (in Cartesian coordinates) The Attempt at a Solution Well, I expressed the path as a...
  35. N

    Gauss Law for non uniform spherical shell

    So i can see by symmetry arguments why The electric field inside a uniformly charged spherical shell would be zero inside. But what about a non uniformly charged spherical shell. Say most of the charge is located on one side, why is the electric field still zero? I can see that the flux...
  36. W

    Deriving potential distribution between concentric spherical electrode

    Homework Statement For some work I am doing I wish to be able to define the potential distribution as a function of the radius (ρ) between two concentric electrodes. Homework Equations One solution (from reliable literature) defines the varying radial potential as: V(ρ)=2V0(ρ0/ρ...
  37. R

    Solutions To Spherical Wave Equation

    If the solution to the electric part of the spherical wave equations is: E(r, t) = ( A/r)exp{i(k.r-ωt) What happens when t=0 and the waves originates at the origin, i.e. r=0 ... which I assume can't be right as you of course cannot divide by zero. Thanks!
  38. R

    Solutions To The Spherical Wave Equation

    If the solution to the electric part of the spherical wave equations is: E(r, t) = ( A/r)exp{i(k.r-ωt) What happens when t=0 and the waves originates at the origin, i.e. r=0 ... which I assume can't be right as you of course cannot divide by zero. Thanks!
  39. S

    Solve Sphere Charge: Find Electric Field 2.5m from Center

    Homework Statement I have a question about notation. My professor posted an older practice test with some different notation techniques than I am used to. "Sphere Charge Find the electric field 2.5 m from the center of a region of space with a charge density given by ro=5.5 E-15 R**(2.3)"...
  40. E

    Deriving equations of motion in spherical coordinates

    Homework Statement OK, we've been asked to derive the equations of motion in spherical coordinates. According to the assignment, we should end up with this: $$ \bf \vec{v} \rm = \frac{d \bf \vec{r} \rm}{dt} = \dot{r} \bf \hat{r} \rm + r \dot{\theta}\hat{\boldsymbol \theta} \rm + r...
  41. W

    Divergence in spherical coordinates

    Problem: For the vector function \vec{F}(\vec{r})=\frac{r\hat{r}}{(r^2+{\epsilon}^2)^{3/2}} a. Calculate the divergence of ##\vec{F}(\vec{r})##, and sketch a plot of the divergence as a function ##r##, for ##\epsilon##<<1, ##\epsilon##≈1 , and ##\epsilon##>>1. b. Calculate the flux of...
  42. W

    Stokes's theorem in spherical coordinates

    Problem: Say we have a vector function ##\vec{F} (\vec{r})=\hat{\phi}##. a. Calculate ##\oint_C \vec{F} \cdot d\vec{\ell}##, where C is the circle of radius R in the xy plane centered at the origin b. Calculate ##\int_H \nabla \times \vec{F} \cdot d\vec{a}##, where H is the hemisphere...
  43. H

    Derive the divergence formula for spherical coordinates

    Homework Statement The formula for divergence in the spherical coordinate system can be defined as follows: \nabla\bullet\vec{f} = \frac{1}{r^2} \frac{\partial}{\partial r} (r^2 f_r) + \frac{1}{r sinθ} \frac{\partial}{\partial θ} (f_θ sinθ) + \frac{1}{r sinθ}\frac{\partial f_\phi}{\partial...
  44. B

    Is a (hyper)sphere a (hyper)plane in spherical coordinates?

    Hi, can I say that a sphere is a plane, because in spherical coordinates, I can simply express it as <(r, \theta, \varphi)^T, (1, 0, 0)^T> = R? It does sound too easy to me. I'm asking because I'm thinking about whether it is valid to generalize results from the John-Radon transform (over...
  45. Y

    Electric Field with a Solid Spherical Conductor

    Homework Statement A spherical conductor has a spherical cavity in its interior. The cavity is not centered on the center of the conductor. If a positive charge is placed on the conductor, the electric field in the cavity A. points generally toward the outer surface of the conductor. B...
  46. E

    Spherical mirror area according to the amount of light

    Hello, I need some information about spherical mirrors that I can't find in internet or this forum. How to calculate the amount of light that is focused in the mirror's focus point depending on the mirror's area and the amount of light emited by the source? If that light is reflected by...
  47. Y

    Partial derivative in Spherical Coordinates

    Is partial derivative of ##u(x,y,z)## equals to \frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}+\frac{\partial u}{\partial z} Is partial derivative of ##u(r,\theta,\phi)## in Spherical Coordinates equals to \frac{\partial u}{\partial r}+\frac{\partial u}{\partial...
  48. L

    Rotation in spherical coordinates

    Hi guys, This isn't really a homework problem but I just need a bit of help grasping rotations in spherical coordinates. My main question is, Is it possible to rotate a vector r about the y-axis by an angle β if r is expressed in spherical coordinates and you don't want to convert r...
  49. andyrk

    Charged Spherical Shell and Solid Sphere

    A spherical shell and a conducting sphere each of radius R are charged to maximum potential. Which of the two has more charge? My attempt: Since in a conductor, no charge can reside inside the conductor so all charge is on the surface of the conductor just like the spherical shell. Now ...
  50. F

    Binomial formula for spherical tensors

    We know that the Newton binomial formula is valid for numbers in elementary algebra. Is there an equivalent formula for commuting spherical tensors? If so, how is it? To be specific let us suppose that A and B are two spherical tensors of rank 1 and I want to calculate (A + B)4 and I want...
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