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. R

    Pressure versus stress in uniformly charged sphere

    Let the charge density be $\rho$, radius be $R$, total charge be $Q = \rho \frac{4}{3} \pi R^3$. We know from Gauss's law, $E (r) = \frac{Q r}{4 \pi \epsilon_0 R^3}$. We also know from Maxwell stress tensor $\sigma(r) = \half \epsilon_0 E^2$. We can compute the pressure due to the electric...
  2. Sagittarius A-Star

    I Understanding the Equivalence Principle

    Alan Macdonald claims in "4 Appendix: The Equivalence Principle" of his text "Special and General Relativity based on the Physical Meaning of the Spacetime Interval", that his calculation regarding a 2D-surface of a sphere proves, that the equivalence principle is violated. He defines the...
  3. Heisenberg7

    B Electric Field Inside a Hollow Sphere

    Let's assume that we have a hollow sphere with holes at opposite ends of the diameter. What would be the field inside the hollow sphere? I know that we can look at this as the superposition of the hollow sphere without holes and 2 patches with opposite surface charge density. For some reason, in...
  4. cianfa72

    I 2-sphere manifold intrinsic definition

    Hi, in the books I looked at, the 2-sphere manifold is introduced/defined using its embedding in Euclidean space ##\mathbb R^3##. On the other hand, Mobius strip and Klein bottle are defined "intrinsically" using quotient topologies and atlas charts. I believe the same view might also be...
  5. T

    Work done to construct Dielectric Sphere with Offset Hollow Cavity?

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  6. J

    The electric potential inside a conducting sphere with charge Q

    If there is no field inside the conductor, how can there be electric potential? I think of potential very similar to gravity, as how much energy would be required to move a particle of mass/charge against the gravitational/electric field. If there is no field at all, how would there still be...
  7. Vanadium 50

    I Spherical trig - sphere radius from 6 lengths

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  8. brotherbobby

    Depth of a basketball floating on water

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  9. T

    Charge on a grounded conducting sphere in a uniform electric field, after ungrounding and movement

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  10. lola9

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  11. SiRiVeon

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  12. C

    B Question about volume of a sphere

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  13. L

    Dipole Moment of a Hollow Sphere, simplify calculation

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  14. Theexploer

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  15. G

    Calculate the area intersected by a sphere and a rectangular prism

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  16. L

    Calculate the magnetic moment of a rotating sphere

    Ich wäre Ihnen sehr dankbar, wenn Sie sich meine Lösung der folgenden Übung ansehen: A sphere with radius ##R ## is spatially homogeneously loaded and rotates with constant angular velocity ##\vec{ \omega}## around the ##z ## axis running through the center of the sphere. Calculate the...
  17. Y

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  18. A

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  19. Q

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  20. P

    A How much thickness for sphere to withstand atmospheric pressure?

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  21. P

    Charge density in sphere that makes constant radial E-field inside

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  22. MatinSAR

    Change in volume of sphere with change in temperature

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  23. K

    B Two points of contact in rolling

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  24. T

    Line of charge and conducting sphere (method of images)

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  25. milkism

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  26. MatinSAR

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  27. B

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  28. B

    I Faraday's Nested Sphere Experiment

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  29. I

    A Finding Minimal Mean Distance Curves on the Unit Sphere

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  30. Wrichik Basu

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  31. P

    I Finite many Lattice Points in Sphere?

    Hello, I am wondering if in an n-ball the number of lattice points is finite. First, we have a ball which is bounded by the radius. The distance between two lattice points is given by the successive minimum. Theoretically, one could now draw a ball* around each lattice point in the (big)...
  32. Quasar100

    Normal Forces on a Sphere in a Non-vertical Groove

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  33. Lars Ph

    I Use the Ewald sphere to calculate h,k,l?

    How do I use the ewald sphere to calculate h, k and l?
  34. Lars Ph

    I Use the Ewald sphere to calculate h,k,l?

    I am unsure wether or not all I can use the Ewald sphere for is to calculate d_hkl for the diffracted wave vector. For cubic lattices for example d= a/sqrt(h^2+k^2+l^2). To determine the lattice constant "a" you would then need to know exactly what your h,k and l are or you use lattice-dependent...
  35. Twigg

    Reducing friction at interface between a sphere and a plane?

    I have a flat planar part made of crystalline sapphire (about ~2k weight, and polished to a mirror finish) that rests on three ball bearings, and I want to minimize the static friction at these 3 interfaces. The ball bearings are fixed so they cannot roll, and the sapphire part can only slip...
  36. Pushoam

    Calculating Volume of a Sphere Using Integration: What Mistakes Have I Made?

    I consider a disc of thickness ## R d\theta ## as shown in the figure. Then, $$ dV = \pi R^2 sin^2 \theta R d\theta $$ ( Area of the disc * its thickness) Hence, $$ V = \int^{\pi}_{0} \pi R^2 sin^2 \theta R d\theta $$ $$ V = \frac 1 {2} {\pi}^2 R^3 $$ ....(1) While $$ V =...
  37. dom_quixote

    B Geometric Issues with a line, a plane and a sphere...

    I - A point divides a line into two parts; II - A line divides a plane into two parts; III - Does a smaller sphere divide a larger sphere into two parts, like layers of an onion? Note that the first two statements, the question of infinity must be considered. For the third statement, is the...
  38. Addez123

    Calculate surface integral on sphere

    I'm supposed to do the surface integral on A by using spherical coordinates. $$A = (rsin\theta cos\phi, rsin\theta sin\phi, rcos\theta)/r^{3/2}$$ $$dS = h_{\theta}h_{\phi} d_{\theta}d_{\phi} = r^2sin\theta d_{\theta}d_{\phi}$$ Now I'm trying to do $$\iint A dS = (rsin\theta cos\phi, rsin\theta...
  39. I

    Flux through top part of sphere

    Flux=$$\iint(xz, -yz, y^2)\cdot(x,y,z)/\sqrt{2} dA=\int_0^{2\pi}\int_0^1 r^2\cos^2\theta \sqrt{1-r^2/2} rdrd\theta$$. Integrating this doesn't give the correct answer of ##\pi/4##.
  40. N

    Find the charge of the sphere (q2)

    I drew a diagram using all of the information however, I am stuck and not to sure how to get one of the charges
  41. J

    Hollowed out sphere exerting gravitational force

    I solved that the hollowed out mass is M/8, which is correct. I don't understand why it is incorrect to substitute the remaining mass (7M/8) back into the F = G*m1m2/r to produce the force. Why is the solution the force of the whole lead sphere minus the force of the “hole” lead sphere, which is...
  42. T

    Electric Potential of a Sphere: A Puzzling Problem

    I can calculate the electric field strength at any point above the plane with Gauss' Law (##E = \frac{\eta}{\varepsilon_0}##) and so the electric potential at any point a perpendicular distance ##z## above the conducting plane (##V=−\frac{\eta}{\varepsilon_0}z##). But I'm having trouble taking...
  43. M

    I Effect of time on density distribution+shape of uniformly dense sphere

    I agree with Doc. Al that, "For the simplified case of a uniform density spherical planet, the gravitational field varies linearly from 0 at the center to its full value at the surface." But, what is the effect, if any, on the shape and density distribution of such a sphere over time (e.g...
  44. squeekymouse

    B How to understand the steradians equation for measuring a sphere of light?

    Hello and please know I am very greatful for your help! I am wanting to learn how to measure light. I have chosen a specific light for this to help me better understand. Lm- 7800 CD-620.7 So, I got that far, lol. I don't really know how to input the numbers for the Steradians equation, I have an...
  45. lavalite

    Thin-walled sphere and fluid mechanics question

    Suppose you had a thin-walled sphere fully submerged in a liquid. The sphere is filled to the equator with a liquid of sufficient density to reach buoyant equilibrium. Will the lateral cross-sectional areas of the thin-walled sphere experience tensile stresses in the longitudinal axis? Why or...
  46. ermia

    Electric field needed to tear a conducting sphere

    ..
  47. Rikudo

    Gravitational field of a hollow sphere

    Why the area of the thin rings are ##2πasin\theta \, ds##? (a is the radius of the hollow sphere) If we look from a little bit different way, the ring can be viewed as a thin trapezoid that has the same base length ( ##2πa sin\theta##), and the legs are ## ds##. The angle between the leg and...
  48. tbn032

    B Rolling of non-deforming sphere on a non-deforming rough surface?

    According to my current understanding rolling friction rolling friction is the static friction (parallel to the surface on which the object is moving) applied by the frictional surface (rough surface) on the contact point or contact area of the object whose v≠Rw(v is translational velocity and...
  49. B

    Comp Sci Draw Longitude Lines of a Sphere on a Circle

    I'm trying to draw a 2D globe
  50. tbn032

    B Friction on pure rolling non deforming sphere?

    How will the friction work on a sphere which is purely rolling on a horizontal surface such that both the sphere and surface does not deform. The sphere at any time t will only have one point of contact, which would continuously changing as the sphere rolls. Will The friction be applied to the...
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