What is Sphere: Definition and 1000 Discussions

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. Jezza

    Magnetic field due to a spinning sphere

    Homework Statement We consider a sphere of radius R which carries a uniform surface charge density \sigma and spins with angular velocity \omega around a diameter. We use spherical coordinates (r, \theta, \phi) with origin at the centre of the sphere and the z-axis along the rotation axis...
  2. J

    What is the electrostatic potential energy of a sphere?

    Hello! I'm Steven, and I'm currently working on the following problem: The Earth can be seen as a conducting sphere with an electric field: E= -(150V/m)r (on its surface) and where r is the unit vector . The Earth has a radius 6371 km. So, I am asked to calculate the electrostatic potential...
  3. 1

    Center of mass of a sphere with cavity removed

    Homework Statement A solid sphere of density ##ρ## and radius ##R## is centered at the origin. It has a spherical cavity in it that is of radius ##R/4## and which is centered at ##(R/2, 0, 0)##, i.e. a small sphere of material has been removed from the large sphere. What is the the center of...
  4. F

    I Deriving the volume of a sphere using semi-circles

    Apologies if this question has already been asked, but is it not possible to derive the formula of a sphere by imagining a circle sliced in two, then rotating this semi-circle about the flat side (imagine the flat side is stuck to a skewer) by 2Pi, so as to sum up the semi-circles to "create"...
  5. FallenApple

    No Potential Energy due to Sphere of Mass?

    So we all know that it takes work to build up a sphere of charge by taking charge from infinity and piling it up into a sphere. Since the sphere wants to break apart under repulsion, its like a spring. It has intrinsic potential energy.However it doesn't seem the case with a sphere of mass, with...
  6. karush

    MHB 205.23 related rates sphere volume

    $\tiny {205.23} $ $\text{ The volume }\displaystyle V=\frac{4}{3}\pi{r}^{3} \text{ of a spherical balloon changes with the radius.} $ $\text{a) at what rate } \displaystyle \frac{ft^3}{ft} \\$ $\text{does the volume change with respect to the radius when $r=2 ft.$} $...
  7. nysnacc

    Triple integration over portion of Sphere

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  8. Destroxia

    Potential of Sphere, Given Potential of Surface

    Homework Statement [/B] The sphere of radius R has the potential at the surface equal to $$ V_0 = \alpha sin^2(\theta) + \beta $$ where ## \alpha, \beta ## are some constants. Find the potential inside, and outside the sphere.Homework Equations $$V(r,\theta) = \sum_{l=0}^{\infty}(A_l r^l +...
  9. E

    Two sphere collision -- What's the speed?

    Homework Statement Two solid spheres hung by thin threads from a horizontal support (Figure 1) are initially in contact with each other. Sphere 1 has inertia m1 = 0.040 kg , and sphere 2 has inertia m2 = 0.10 kg. When pulled to the left and released, sphere 1 collides elastically with sphere 2...
  10. D

    Sphere to donut mathematically

    I just read through this thread* where a mathematician tries to shutdown another guys use of the "sphere to donut" analogy relative to ripping spacetime because a sphere is a "2-dimensional manifold". That is so completely and entirely irrelevant when both are still just 3 dimensional shapes...
  11. W

    Electric field inside and outside a sphere (Not Gauss)

    Find E(r) inside and outside a uniformly charged spherical volume by superposing the electric fields produced by a collection of uniformly charged disks. a+b) Given equations, sketch of problem This is the equation in the handbook for a disk (but in the exercises the z becomes x, without loss of...
  12. TheDemx27

    How to Calculate Heat Current in a Spherical Shell?

    Homework Statement A spherical shell has inner and outer radii r_a and r_b, respectively, and the temperatures at the inner and outer surfaces are T_a and T_b. The thermal conductivity of he shell material is k. Derive an equation for the total heat current thought the shell in the steady...
  13. M

    Induced Charge on a Grounded Sphere

    Homework Statement A point charge q is located a distance d away from the centre of a grounded conducting sphere of radius R<d. I need to find the charge density on the sphere and the total induced charge on the sphere. This is very similar to example 2 here...
  14. J

    MHB Measuring Hollow Sphere Diameter with Compass & Ruler

    Using only a compass and a ruler, how can we measure the diameter of a hollow sphere?
  15. C

    I Why is the limit of θ from 0 to π in the formula for surface area of a sphere?

    I found this on the Internet . The formula is Surface Area = R^2 \displaystyle \int _0 ^ {2 \pi} \int _{0}^{\pi} \sin \theta d \theta d \phi I'm wondering why the limit of θ is from 0 to π only ? why not from 0 to 2π ? Because it's a perfect sphere...
  16. G

    Why does a single sphere have a capacitance?

    Hi. The capacitance of an ideal plate capacitor (coaxial cable) goes to zero as the plate distance (outer radius) goes to infinity. This doesn't happen with concentric spheres as we let the outer radius go to infinity, hence a single sphere has a nonzero capacitance. What's the exact reason...
  17. ReidMerrill

    Equation of a sphere in XYZ coordinates

    Homework Statement Show that the equation represents a sphere, and find its center and radius. 3x2+3y2+3z2 = 10+ 6y+12z Homework EquationsThe Attempt at a Solution 3x2+3y2-6y +3z2 -12z =10 My equation is how the constants in-front of the squared terms affect the sphere formula? Besides that...
  18. S

    Prove this is a right triangle in a sphere

    Homework Statement Let P be a point on the sphere with center O, the origin, diameter AB, and radius r. Prove the triangle APB is a right triangle Homework Equations |AB|^2 = |AP|^2 + |PB|^2 |AB}^2 = 4r^2 The Attempt at a Solution Not sure if showing the above equations are true is the...
  19. O

    Nonconducting spherical shell with uniform charge

    Homework Statement Suppose the nonconducting sphere of Example 22-4 has a spherical cavity of radius r1 centered at the sphere's center (see the figure). Assuming the charge Q is distributed uniformly in the "shell" (between r = r1 and r = r0), determine the electric field as a function of r...
  20. O

    I Why is the E-field inside a solid conducting sphere zero?

    The common explanation is this: If the conductor has a net charge, then the charges repel each other until they arrange themselves symmetrically around the outside of the sphere, and if you do the math the electric field will cancel out everywhere inside the conducting sphere. Alright, but what...
  21. N

    I Re-derive the surface area of a sphere

    Hey everyone, I've been stuck on this one piece of HW for days and was hoping someone could help me. It reads: The surface area, A, of a sphere with radius R is given by A=4πR^2 Re-derive this formula and write down the 3 essential steps. This formula is usually derived from a double...
  22. G Cooke

    Potential on the inner surface of a spherical shell

    Is there a potential on the inner surface of a charged spherical shell? I know that there is no electric field on the inner surface, as shown by Gauss's Law, but that isn't enough information to say that the potential (V) there is zero since E = dV/dr, so V could be a nonzero constant. If...
  23. JulienB

    Surface integral of vector fields (sphere)

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  24. dumbdumNotSmart

    Electric Field in a cavity of a charged sphere

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  25. K

    Particle confined to move on the surface of sphere

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  26. Turbodog66

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  27. Khatti

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  28. G

    Uniform E field for spherical shell.

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  29. Prasun-rick

    I Volume of sphere using integration?

    Is it possible to find the volume of a sphere(i know the formula) using definite integration ? And if possible how to proceed ?? Thanks in advance
  30. T

    Hollow sphere half submerged in a liquid, find density

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  31. C

    How to Prove that u and Gradient(f(u)) are Colinear on the Unit Sphere?

    Homework Statement Let be ##f : V \rightarrow \mathbb{R}## a ##C^{1}## function define on a neighbourhood V of the unit sphere ##S = S_{n-1}##(in ##\mathbb{R}^{n}## with its euclidian structure.). By compacity it exists u in S with ##f(u) = max_{x \in S}f(x) = m##. My goal is to show that ##u##...
  32. I

    Rotation around x-axis of a sphere

    I have posted this question earlier but I think it was ill-stated. I try to give it in a simpler fashion in this thread. In the problem it is stated that there a rotation around the x-axis of a (stereographic) sphere is given by $$\delta \phi = \cot \phi \cot \theta \delta \theta$$ where...
  33. I

    Rotations around the x and y axes of stereographic sphere

    Homework Statement Show that the equations $$ \delta \phi = \cot \theta \cot \phi \delta \theta, \quad \delta \phi =- \cot \theta \tan \phi \delta \theta$$ represent rotations around the x and y axes respectively of a stereographic sphere. Both these two rotations map the sphere on itself and...
  34. C

    Contact area of ideal sphere resting on flat surface

    Greetings All, I have a rather odd question which has been bothering me. If you have a perfectly round sphere sitting on a perfectly flat plane, what is the area of surface contact between the two? Is there an actual value, or is it something which can't be calculated. I'm assuming the diameter...
  35. NicolasPan

    Find the electric field of a uniformly polarized sphere

    Homework Statement We want to calculate the field of a uniformly polarized sphere of radius=R Homework Equations V(\vec{r}) = \frac{1}{4 \pi\epsilon_{0}} \oint_{S} \frac{\sigma_{b}}{r} da' + \int_{V} \frac{\rho_{b}}{r} d\tau' The Attempt at a Solution i)I know that \sigma_{b} = P...
  36. A

    I Calculating the Surface Area of a Sphere: How Does it Differ?

    please , I'm french , so i didn't quite get the meaning of this sentence.
  37. K

    Mass of charged sphere suspended between charged plates

    Homework Statement A small sphere with charge 2.4 micro coulombs is suspended from a thread between 2 charged plates. The plates have a voltage of 62 and the distance between the plates is 3.1 cm. The sphere hangs at 18 degrees to the vertical. Homework Equations E = V/r FE = Eq Fnetx = FE -...
  38. Spinnor

    I Evenly pump a sphere of lasing material

    Qualitatively what is the nature of emitted light if we evenly pump a sphere of lasing material? Suppose there are no mirrors to favor one direction. Does the answer depend at all on the size of the sphere? Thanks!
  39. ChrisVer

    A Mean volume of sphere (normal distributed radius)

    I read in http://www-library.desy.de/preparch/books/vstatmp_engl.pdf page 29 (43 for pdf) that the mean volume is: <V> = \int_{-\infty}^\infty dr V(r) N(r| r_0,s) I have two questions. Q1: why do they take the radius to be from -infinity to +infinity and not from 0 to infinity? Q2: is there an...
  40. P

    Sphere striking an incline (not asking for solutions....)

    The sphere is released at a height H above a fixed inclined plane, as shown in the attached figure. The coefficient of restitution at impact is e>0 (that is the sphere leaves the surface just after impact), the coefficient of friction between the sphere and the plane is \mu. I need a...
  41. S

    A small solid sphere of mass M0, of radius R0, and of unifor

    Homework Statement A small solid sphere of mass M0, of radius R0, and of uniform density ρ0 is placed in a large bowl containing water. It floats and the level of the water in the dish is L. Given the information below, determine the possible effects on the water level L, (R-Rises, F-Falls...
  42. E

    I Fermi sphere and density of states

    Hello! When computing the density of states of electrons in a lattice, a material with dimensions L_x, L_y, L_z can be considered. The allowed \mathbf{k} vectors will have components k_x = \displaystyle \frac{\pi}{L_x}p k_y = \displaystyle \frac{\pi}{L_y}q k_z = \displaystyle \frac{\pi}{L_z}r...
  43. Z

    What is the moment of inertia of a sphere and how is it derived?

    I know the moment of inertia for both a solid sphere and a hollow sphere is , but my teacher has derived a moment of inertia of the sphere but am not sure about what axis she was deriving it , and she got this answer 3/5 MR^2
  44. L

    I Universe is a sphere that is centered on any observer? How?

    I read here, http://www.space.com/24781-big-bang-theory-alternatives-infographic.html , that, "What we call the "observable universe" (or the "Hubble Volume") is the spherical region, about 90 billion light-years in diameter, that is centered on any given observer. This is the only part of the...
  45. steroidjunkie

    Metal sphere on a thread in a horizontal electric field

    Homework Statement Charged metal sphere hanging on an isolated thread of negligible mass is put in a homogeneous horizontal electric field so that the thread makes a 45 degree angle with the el. field. What angle does the thread with the sphere close with the el. field after we remove 40% of...
  46. P

    Electrostatics problem- two sphere

    Homework Statement A spherical capacitor comprises two thin metal spheres of different radii but with a common centre. The following series of operations is completed: The spheres are mutually connected by an internal wire. The outer sphere is raised to potential +V with respect to ground. The...
  47. W

    Covariant derivative of vector fields on the sphere

    Homework Statement Given two vector fields ##W_ρ## and ##U^ρ## on the sphere (with ρ = θ, φ), calculate ##D_v W_ρ## and ##D_v U^ρ##. As a small check, show that ##(D_v W_ρ)U^ρ + W_ρ(D_v U^ρ) = ∂_v(W_ρU^ρ)## Homework Equations ##D_vW_ρ = ∂_vW_ρ - \Gamma_{vρ}^σ W_σ## ##D_vU^ρ = ∂_vU^ρ +...
  48. Guy ML

    Question about polarization density

    Homework Statement Given a sphere with radius R, centered at (0,0,0), it's dipole density given as ##P\left(\vec{r}\right)=\alpha\left(R-r\right)\hat{z}## where r is the distance from the center of the ball. I'm required to find: Bound charge density inside the sphere, bound charge density on...
  49. J

    Energy lost from sphere due to spark

    1. Homework Statement Homework EquationsThe Attempt at a Solution I did this : 1/2 (2.7x10^-6)(150x10^3) - 1/2 (1.575x 10^-6)(75x10^3) =0.143 J But the answer is 0.15 , where did I did wrong?
  50. B

    MHB Normal vector of a sphere

    I am given the sphere V= x^2 + y^2 + z^2 =< 1 I have converted it to spherical coordinates: x = rsin(t)cos(f) y = rsin(t)cos(f) z = rcos(t) where t ranges from 0 to pi, and f ranges from 0 to 2pi. I am unsure how to go about this problem from here. Any guidance would be really appreciated :)
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