Sphere 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. ChrisVer

    How is the S^3 metric defined by a unit vector and coordinate changes?

    We have the (I think FRW) metric in the coordinates y^{0}=t,~~y^{1}=\psi,~~y^{2}=\theta,~~y^{3}=\varphi g_{00}=1,~~g_{00}= - \frac{f^{2}(t)}{\alpha} ,~~ g_{00}= - \frac{f^{2}(t)}{\alpha} \sin^{2}\psi ,~~g_{00}= - \frac{f^{2}(t)}{\alpha} \sin^{2}\psi sin^{2}\theta Suppose we have define a...
  2. E

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  3. M

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  4. WannabeNewton

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  5. Satvik Pandey

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  6. G

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  7. I

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  8. U

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  9. Nathanael

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  10. V

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  11. B

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  12. A

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  13. M

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  14. U

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  15. U

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  16. Z

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  17. B

    Surface integrals to derive area of sphere

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  18. A

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  19. M

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  20. U

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  21. W

    How to quantize a particle confined to the surface of a sphere?

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  22. G

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  23. M

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  24. K

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  25. T

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  26. johann1301

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  27. Feodalherren

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  28. L

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  29. Feodalherren

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  30. K

    Barometric law for hard sphere sedimentation

    So I have the equation for Barometric law in terms of density as: \frac{\phi(z)}{\phi_{0}}=exp(-g.M.z/R.T) where R=Universal gas constant, z=height of sedimentation, T=Standard temperature, g=Gravitational acceleration, M=Molar mass. When this equation is used to calculate the height of the...
  31. Feodalherren

    Surface integrals - parametrizing a part of a sphere

    Homework Statement Find the area of the part of the sphere x^2 + y^2 + z^2 = 4z that lies inside the paraboloid x^2 + y^2 = z Homework Equations The Attempt at a Solution I solved for the intercepts and found that they are z=0 and z=3. The sphere is centered two units in the z-direction above...
  32. M

    Sphere with permanent radial magnetization

    Homework Statement We have a sphere of radius a with permanent magnetization \mathbf{M}=M\hat{e}_{\mathbf{r}}. Find the magnetic scalar potential. Homework Equations $$\Phi_M(\mathbf{x})=-\frac{1}{4\pi}\int_V \frac{\mathbf{\nabla}'\cdot\mathbf{M}(\mathbf{x}')}{|\mathbf{x}-\mathbf{x}'|}d^3...
  33. E

    A point charge inside a dielectric sphere

    Hello again I seen in this forum about this problem but not in the special case when the point charge is at the center of the sphere how do I solve the series legendre polynomials?
  34. N

    Finding charge on sphere using energy

    Homework Statement a) What charge would need to be placed on a metal sphere such that the electric field produced stored a total of 50 J if the radius of the sphere was 18 cm? b) How much work would need to be done to reduce the radius to 9 cm, assuming the sphere is (mechanically) easily...
  35. R

    Equidistant points on the suface of a sphere

    If we are asked to place 3 points on the surface of a sphere so that they are equidistant, it's easy to visualize that the result will be such that the three points form an equilateral triangle. If asked to place 4 points it's easy to visualize that the result is such that the points arrange...
  36. N

    Rolling sphere down an incline ramp

    Are there any good sources to understand the physics behind a rolling sphere down an incline plane?
  37. R

    Flow Past Sphere at High Re - Velocity Behavior

    Hello, I'm currently working on simulations in Solidworks of the flow of air (v = 450 m/s) around a sphere (radius = 1.5 cm). The results have been used in a cut plot representing the velocity in the x direction. The image is similar to this one...
  38. 8

    How to Find the Electric Field Inside a Partially Hollow Charged Sphere?

    Homework Statement A sphere with radius r has uniform charge density ρ within its volume, except for a small hollow sphere located at the center with radius R. Find the electrical field. Homework Equations ρ=Q/V ∫∫EdS=Q/ε The Attempt at a Solution With the spherical Gaussian surface...
  39. N

    A 20 kg sphere is at the origin and a 10 kg sphere is at (x,y) = (20,0

    Homework Statement a 20 kg sphere is at the origin and a 10 kg sphere is at (x,y) = (20,0). at what point or points could you place a small mass such that the net gravitational force on it due to the spheres is zero? Homework Equations g=GM/r^2 The Attempt at a Solution Can not...
  40. MarkFL

    MHB Radius of Sphere Inscribed in Square Pyramid: Michelle's Q&A

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  41. tom.stoer

    E.o.m. for a particle in a sphere of dust

    I am looking for the e.o.m. of a particle moving inside a sphere of homogeneous dust with density ρ. I start with the Lagrangian (in cartesian coordinates with i=1,2,3) L = \frac{m}{2}\dot{x}_i^2 - \frac{4\pi}{3}Gm\rho x_i^2 The e.o.m. and their solution are given by the harmonic...
  42. P

    About stretching a sphere by a radius of a.

    Homework Statement Copy-paste from my textbook: Let S_1 be the sphere of radius 1, centered at the origin. Let a be a number > 0. If X is a point of the sphere S_1, then aX is a point of the sphere of radius a, because ||aX|| = a||X|| = a. In this manner, we get all points of the sphere of...
  43. T

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    Homework Statement Q a) A charge is placed at a distance x from the center of a conducting sphere of radius R.Find the charge flown through the switch(from ground) when it is closed . b) In the above question replace the switch with an ammeter .What is the reading of ammeter if charge is...
  44. C

    What is the Electric Field Inside and Outside of a Sphere?

    hi guys, i have been following griffith's book on electrodynamics and i m stuck with probably one of the basic concepts on electric field. i did not understand what would be the electric field, 1) inside and 2)outside of a sphere with radius R ? also there are two problems which deals with...
  45. T

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    Homework Statement The region between two concentric spheres of radii a and b(>a) contains volume charge density ρ(r) = C / r where C is a constant and r is the radial distance, as shown in figure. A point charge q is placed at the origin, r= 0. Find the value of C for which the electric...
  46. C

    Falling Weight Attached to a Wheel and a Sphere

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  47. T

    Minimum speed of the bullet to penetrate a sphere

    Homework Statement A bullet of mass m and charge q is fired towards a solid uniformly charged sphere of radius R and total charge + q. If it strikes the surface of sphere with speed u, find the minimum speed u so that it can penetrate through the sphere. (Neglect all resistance forces or...
  48. B

    Solving a Charge at the Center of a Sphere Problem

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  49. K

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  50. V

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