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. Calpalned

    Find an equation of the largest sphere

    Homework Statement Find an equation of the largest sphere with center (5, 4, 9) that is contained in the first octant. Homework Equations x2 + y2 + z2 The Attempt at a Solution [/B] (x - 5)2 + (y - 4)2 + (z - 9)2 = R2 I am under the impression that R must be no greater than 4, is this...
  2. ELB27

    Evaluating an integral for an expanding, charged sphere

    Homework Statement An expanding sphere, radius ##R(t) = vt## (##t>0##, constant ##v##) carries a charge ##Q##, uniformly distributed over its volume. Evaluate the integral Q_{eff} = \int \rho(\vec{r},t_r) d\tau with respect to the center. (##t_r## is the retarded time and ##d\tau## is an...
  3. G

    What's the distance a sphere travels from inclined plane?

    < Mentor Note -- thread moved to HH from the technical physics forums, so no HH Template is shown > So the problem that I have been assigned has formulas of rotational energy, momentum, trajectories, inertia, and inclined planes. A solid sphere is rolling down an inclined plane (that is placed...
  4. E

    Area of cylinder sliced by sphere

    Hi! Here is my task: Calculate area of cylinder $$x^{2}+y^{x}=ax$$ sliced by sphere $$x^{2}+y^{2}+z^{2}=a^{2}$$. Here is graph: How to do it? If problem was "Calculate area of sphere $$x^{2}+y^{2}+z^{2}=a^{2}$$ sliced by cylinder $$x^{2}+y^{x}=ax$$" I would solve it using double integrals...
  5. R

    Magnitude of an Electric Field of a Sphere at a Distance

    Homework Statement So I was given the equation that in a sphere, if at a distance r outisde a sphere of radius R (The sphere insulated, I guess it matters since it would prevent charge to leave?) The electric field E = kqR/r^3 I'm not sure how they got this Homework Equations Flux = Integral...
  6. P

    Understanding Vector Fields on a Sphere

    Homework Statement find the values of the integral \int_{S} \vec A\cdot\ d\vec a where, \vec A\ = (x^2+y^2+z^2)(x\hat e_{1}+y\hat e_{2}+z\hat e_{3}) and the surface S is defined by the sphere R^2=x^2+y^2+z^2 Homework Equations first i must evaluate the integral directly, so i don't...
  7. J

    Two charged nonconducting sphere shells

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  8. P

    Electric potential at surface sphere problem?

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  9. diracdelta

    How Does Permittivity Affect Energy Transmission in EM Waves?

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  10. B

    Closed-space sphere "displacement"?

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  11. C

    Ansoft Maxwell guidelines -- Simulate E(r) for charged sphere

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  12. ELB27

    Discharging a magnetized sphere

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  13. N

    Casimir Effect Force in a Collapsing Sphere

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  14. N

    Photon "escaping" from photon sphere in Schwarzchild space

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  15. M

    Flux Through a Non-Concentric Sphere

    Homework Statement Assume I want to calculate the electric flux through a spherical surface centred at point P with radius R which contains a point charge Q, that is not concentric with the spherical surface. Here, I can no longer assume that ∫∫sEdA = E.A, and I have to calculate the value of...
  16. C

    Calculating Force on a Sphere with 1 Bar Pressure

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  17. KingCrimson

    Why does a heavier sphere fall faster in a liquid ?

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  18. P

    Acceleration, Uniform Ball on Incline

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  19. V

    Integration constants, gravitational potential of sphere

    Homework Statement So I'm calculating the gravitational potential of a sphere at at point P. R = radius of sphere, r = distance from center of sphere to point P. I'm looking at two scenarios; r > R (1) and r < R (2). So I have the following integral: \begin{equation} V(r) = \int...
  20. B

    Uncovering the Mystery: 2D Creatures on Expanding Sphere

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  21. A

    Finding the Angle between Thread and Wall for Sphere A

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  22. DivergentSpectrum

    Charge on the surface of a sphere induced by a charge inside

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  23. W

    Pendulum Equilibrium: Can It Pass Balance Point Twice?

    Consider a pendulum in it's balance point hanging from ceiling. It can swing in all the directions in the space. The pendulum can only swing in a sphere(the string can't bend). Now, is it possible to release the pendulum in a particular height and with a initial condition that in the first...
  24. T

    Drag force on a sphere moving through a fluid

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  25. S

    Magnetic dipole at the centre of a sphere

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  26. M

    Testing Drag Force on a Sphere

    Hi all. This is my first question on these forums. I am given a task to test out whether or not FD=1/2 CDAρv2 is a good model to test drag force. Where CD is described by Reynold's Number. We have a balloon a small ball and a myriad of basic physics lab equipment.What is a experiment I could...
  27. H

    Ring and Sphere Linear Expansion

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  28. N

    Can a spinning dielectric sphere induce a magnetic field?

    Homework Statement http://web.phys.ntnu.no/~ingves/Teaching/TFY4240/Exam/Exam_tfy4240_Dec_2013.pdf http://web.phys.ntnu.no/~ingves/Teaching/TFY4240/Exam/Solution_tfy4240_Dec_2013.pdf problem 2g Homework EquationsThe Attempt at a Solution Hi, this is taken from problem 2g in the problem set...
  29. M

    Differentiation of a sphere -- raindrop evaporating as it falls

    < Moderator Note -- Thread moved from the technical PF Calculus forum > I can't seem to grasp the idea of this problem, any help is much needed. The problem reads, "As a spherical raindrop falls, it reaches a layer of dry air and begins to evaporate at a rate that is proportional to its...
  30. G

    Difference between a spinning sphere and rolling one

    Consider this: We have a sphere rolling down a slant, released from some height h with null velocity. At the end of the slant its potential energy will have been fully converted to kinetic energy, part translational and part rotational. Now consider this: at the end of the slant the ball enters...
  31. throneoo

    How Do You Calculate the Density of a Partially Submerged Sphere?

    Homework Statement a sphere of uniform density and radius R is floating on water , partially immersed such that the distance between the top of the sphere and the water surface is R/2 find the density of the sphere Homework Equations Archimedes Principle The Attempt at a Solution One can...
  32. C

    Finding potential for a sphere, when to use laplace equation

    Homework Statement This is example 3.9 in Griffiths Electrodynamics. "A specified charge density σ(θ) (inclination angle) is glued over the surface of a spherical shell of radius R. Find the resulting potential inside and outside the sphere." The problem suggests that although it is possible to...
  33. O

    How Long to Coat a Sphere with Chromium?

    Homework Statement [/B] A sphere 100 mm diameter is to be coated with chromium from a solution containing chromium in the six valent (hexavalent) state. How much time would be needed to produce a coating 20 μm thick if: • the current is 20 A • the cathodic efficiency is 15% • the atomic weight...
  34. H

    Quotient space of the unit sphere

    prove that the quotient space obtained by identifying the points on the southern hemisphere, is homeomorphic to the whole sphere.I am trying to define a homeomorphism between the quotient space and the sphere,and i need help doing it. Thank's in advance.
  35. gfd43tg

    Cooling of sphere with pseudo steady state condition

    Homework Statement A hot solid sphere of initial radius ##a## with a uniform initial temperature ##T_{0}## is allowed to cool under stagnant air at ambient temperature, ##T_{\infty}## . Assume the temperature within the sphere is uniform throughout the cooling process. Show that under...
  36. E

    Electric Potential with Non-conducting sphere and conducting shell

    Homework Statement A spherical nonconductor of radius a carries charge +Q uniformly spread through its volume. 2 hemispherical conducting shells of inner radius b and outer radius c are placed concentrically with the nonconducting sphere to form a single conducting sphere. The conducting shell...
  37. Spinnor

    Atlas of torus and sphere. Atlas of Calabi–Yau manifold.

    Is it true that the atlas for a torus can consist of a single map while the atlas for a sphere requires at least two maps? Can we ever get by with a single map for some Calabi–Yau manifolds assuming that question makes sense? If not is there some maximum number required? Thanks for any help!
  38. AwesomeTrains

    Electrostatic Self-Energy of a uniform charged sphere

    Homework Statement Hello, I have to calculate the self-energy of an uniform charged electron with radius R. The distributed charge is e. Homework Equations The SE is given as: E=\frac{1}{2}\int dV \int dV' \frac {\rho(\vec r)\rho(\vec r')}{ |\vec r - \vec r'|} according to the problem sheet...
  39. S

    Lagrangian mechanics: sphere inside a cylinder

    The problem goes by this: A sphere of radius ##\rho## is constrained to roll without slipping on the lower half of the inner surface of a hollow cylinder of inside radius R. Determine the Lagrangian function, the equation of constraint, and Lagrange's equations of motion. Find the frequency of...
  40. G

    A sphere of linear dielectric material surrounded by another dieletric material

    Homework Statement A sphere of linear dielectric material with permittivity ##\epsilon_1## and radius ##a## is surrounded by an infinite region of linear permittivity ##\epsilon_2##. In the spherical region, there is free charge embedded given by ##\rho_{free}=\beta r^2##, ##0<r<a##, where...
  41. deedsy

    How Do You Calculate Charge and Potential in a Polarized Dielectric Sphere?

    Homework Statement A dielectric sphere radius R is injected with free charge so that the resultant polarization can be described by ## \vec P = \frac{K}{r} \hat r_1## where ##\hat r_1## is the unit radial vector. a) Calculate the volume and the surface density of bound charge b) Calculate the...
  42. Satvik Pandey

    Finding Angle of Sphere Falling From Table Edge

    Homework Statement A solid spherical ball is placed carefully on the edge of a table in the position shown in the figure. The coefficient of static friction between the ball and the edge of the table is 0.5 . It is then given a very slight push. It begins to fall off the table. Find the angle...
  43. B0b-A

    Gold Fish-Eye View: YouTube Video

    youtu.be/9ZEdApyi9Vw?t=15s
  44. B

    Sphere rolling down an incline problem.

    Homework Statement A uniform solid sphere, of radius 0.20 m, rolls without slipping 6.0 m down a ramp that is inclined at 28° with the horizontal. What is the angular speed of the sphere at the bottom of the slope if it starts from rest? Homework Equations KE=1/2Iw^2 PE=mgh I don't know what...
  45. gangula pranav

    Exploring the Hubble Sphere & Hubble Constant

    why does the Hubble sphere increase amd why is the Hubble constant called a constant if the value keeps changing.
  46. C

    Moment of Inertia of a solid sphere

    Homework Statement Taylor, Classical Mechanics Problem 10.11 ** a) Use the result of problem 10.4 (derivation of the general integral for a moment of inertia of a continuous mass distribution in spherical coordinates, using point particles) to find the moment of inertia of a uniform solid...
  47. M

    Surface Area of Cylinder inside of sphere

    Homework Statement Find the area of that part of the cylinder x^2 + y^2 = 2ay that lies inside the sphere x^2 + y^2 + z^2 = 4a^2. Homework Equations [/B] If a surface S can be parametrized in terms of two variables u and v, then dS = Norm[dR(u,v)/du x dR(u,v)/dv]. The surface area is given...
  48. 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...
  49. K

    Determining Electric Potential

    Homework Statement Consider a nonconducting sphere of radius r2. It has a concentric spherical cavity of r1. The material between r1 and r2 has a charge density p (C/m3). Take V=0 and r=infinity. Determine the electrical potential V as a function of the distance r from the center for (a) r>r2...
  50. gfd43tg

    Max temperature inside sphere generating heat

    Homework Statement Radioactive wastes are stored in a spherical stainless tank of inner diameter 1 m and 1 cm wall thickness. Heat is generated uniformly in the wastes at a rate of 30,000 W/m3. The outer surface of the tank is cooled by air at 300 K with a heat transfer coefficient of 100 W/m2...
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