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. T

    Moment of Inertia of a Sphere derivation?

    Hi, Is there a way to derive the moment of inertia of a sphere without using the M of I of a cylinder? In other words, is it possible to find a sphere's from scratch? Please include a derivation in your answer, unless there isn't one of course.
  2. L

    Electric field inside a dielectric sphere with cavity

    Original Problem: "A sphere of radius a is made of a nonconducting material that has a uniform volume charge density [PLAIN]http://jkwiens.com/2007/10/24/answer-electric-field-of-a-nonconducting-sphere-with-a-spherical-cavity/d2606be4e0cd2c9a6179c8f2e3547a85_2.gif. A spherical cavity of...
  3. Drakkith

    Calculating Error of Volume of a Sphere Using Differentials

    Homework Statement The circumference of a sphere was measured to be 90 cm with a possible error of 0.5 cm. Use differentials to estimate the maximum error in the calculated volume. Homework Equations Volume of sphere: V=4/3πR3 Circumference of Sphere: C=2πR ΔC = 0.5 cm The Attempt at a...
  4. Atlas3

    Sphere in Cube: Can it be Defined?

    Can it be defined a disfigured sphere to approximate a cube mathematically? 8 corners equilateral to some extent. Not exact.
  5. B

    Point and circle on sphere probability

    Homework Statement Given a randomly drawn circle on a sphere, calculate the probability that it will pass within a defined distance of a set point. To make it clear, imagine the example of the the Earth and Mt Everest. What is the probability that a randomly drawn circle will come within, say...
  6. W

    Angular Velocity Of A Sphere Rotating Under Gravity

    Homework Statement A solid sphere of mass m and radius a can rotate freely about a point A on its surface. The sphere is held initially at rest with the line OA through A and the centre of the sphere O horizontal and is released under gravity. Find the angular velocity of the system when OA...
  7. S

    + charged sphere and - charged sphere touch and separate

    Homework Statement A small metal sphere X is charged by losing 500 electrons. An identical metal sphere Y is charged by gaining 1000 electrons. The two spheres are first put in contact with each other and then separated. If -e is the charge on an electron, what is the charge on each sphere...
  8. CheesyPeeps

    Sphere on a Flat Plane: 3 Points of Contact?

    I've been reading about how much of a sphere actually touches a flat plane (spheres are very interesting things, it turns out!). Mathematically, a perfect sphere has only one point of contact, meaning that the area of this contact is infinitely small(?), but as physicists, we know that there...
  9. S

    PDE: Heated Sphere Homework Solution

    Homework Statement This is not really a school problem, it's actually something I am trying to figure out. So, we have a sphere with given radius. (Actually let's assume that all the parameters are known). The sphere has equally distributed heaters and is in the beginning at constant...
  10. U

    Pressure of a sphere on a regular surface

    Since the pressure a sphere exerts on a surface tends to infinity, how do you actually calculate it? My guess would be trying to see how many atoms of the surface (a straight line) and of the sphere collide. But this is very dependent on the materials and exterior factors. I have searched...
  11. M

    Calculate No. of Spheres to Cover 50% of a 25m^2 Square

    If I have a square surface of 25 m^2 and I want to know how many sphere of radius r will cover 50% of the area, then how to proceed ? Should I calculate the area of circle which will form by projecting a sphere on a 2D surface, and divide 12.5 m^2 by this area to get the no. of spheres ?
  12. Alex_Neof

    Calculating the E-Field inside and outside a sphere.

    Homework Statement Consider a sphere of radius ##R##, with a charge density ##\rho(r)=\frac{\alpha} {r^2},## with ##\alpha## a constant. Use Gauss' law to calculate the electric field outside the sphere at a distance ##r## from the sphere's centre (ie. ##(r > R)## and inside the sphere (ie...
  13. Y

    Electric Potential Energy Inside a Charged Sphere

    Homework Statement A point charge q<0 lies just outside a uniformly and positively (non-conducting) charged ball. Assume the charge can pass through the ball freely. Describe the motion of the charge. Homework Equations Coulomb's force law, energy equation. The Attempt at a Solution Obviously...
  14. Quarlep

    Find average velocity of a sphere which expands and moves

    Homework Statement Find the average velocity of shell while its moves a velocity v' and expands a velocity v. Sphere radius is R Expansion velocity v Movement velocity v' Homework Equations I think there's no need an equation.[/B] The Attempt at a Solution I try to find average velocity...
  15. Quarlep

    Does a Moving Sphere's Kinetic Energy Affect Its Atomic Particles?

    Lets think we have a sphere and it moves a constant veloctity v.So it will have a kinetic energy.Is this kinetic energy efectts spheres particle energy.(sphere made up but atoms) Thanks
  16. M

    Verifying Stokes' Flow for Fluid Motion Around a Sphere

    Homework Statement Let a spherical object move through a fluid in R3. For slow velocities, assume Stokes’ equations apply. Take the point of view that the object is stationary and the fluid streams by. The setup for the boundary value problem is as follows: given U = (U, 0, 0), U constant, find...
  17. M

    Electric field on the surface of charged conducting sphere?

    just above the surface it's (kq/r^2) where r is the radius of the sphere and just below the surface it's zero, so is the electric field zero also exactly on the surface ? (as the q enclosed then will be zero since the flux is coming from the surface and not actually penetrating it) and...
  18. M

    Electric field inside charged conducting sphere?

    i know it s zero because of the electrostatic equilibrium, but in terms of point charges : from the charge distribution on the sphere surface if we consider 2 point charges opposite to each other in direction : it s logical that at the point in the mid distance between them the electric field...
  19. C

    Light Refraction on the Surface of a Sphere

    Hello All, Using Snell's Law, it is pretty obvious how to calculate the angle of refraction when both index of refractions are known. My question is how would I apply this to a 3 dimensional situation, such as light refraction in a sphere? Since there are two angles in relation to the normal...
  20. C

    Sphere of Uniform Density: Exact Solution to Einstein's Field Eqns?

    Is an exact solution to Einstein's Field Equations known for the interior of a sphere of uniform density (to approximate a star or planet, for example?)
  21. C

    Electric charge inside a uniformly distributed sphere

    1. The problem statement, all variables and given/known my book says inside of a uniformly distributed sphere is zero and it also says it is not it is increasing. I didnt understand any single thing it is like kidding me? Homework EquationsThe Attempt at a Solution
  22. Estefania_8

    Potential of a Charged Conducting Sphere

    Homework Statement A conducting sphere with radius R is charged to voltage V0 (relative to a point an infinite distance from the sphere where the potential is zero). What is the surface charge density σ? Express your answer in terms of the given quantities and ϵ0. Homework Equations Electric...
  23. Dethrone

    MHB Calculating Volume of a Hollow Sphere w/ Differentials

    This is probably an elementary question, but I stumbled upon it while thinking about total differentials. One of their many applications is calculating the error in a volume, for example, given uncertainties in its dimensions. I'm not in the mood to tackle a 3D problem, so let's revert to a 2D...
  24. nuuskur

    Proof: open sphere is an open set

    Homework Statement Prove that an open sphere in \mathbb{R}^m is an open set. Homework EquationsThe Attempt at a SolutionTo show that an open sphere is an open set, any point inside the sphere has to be an interior point: Let us have a sphere B(P_0, r), r > 0, where P_0 is the centerpoint and r...
  25. J

    Rotation operators on Bloch sphere

    Can anyone explain to me why the following operators are rotation operators: \begin{align*}R_x(\theta) &= e^{-i\theta X/2}=\cos(\frac{\theta}{2})I-i\sin(\frac{\theta}{2})X= \left(\!\begin{array}{cc}\cos(\frac{\theta}{2}) & -i\sin(\frac{\theta}{2}) \\ -i\sin(\frac{\theta}{2})&...
  26. P

    Newton's argument, gravitational force inside a sphere

    This is an excerpt from "Introduction to Mechanics" by Kleppner and Kolenkow: "The reason why gravitational force vanishes inside a spherical shell can be seen by a simple argument due to Newton. Consider the two small mass elements marked out by a conical surface with its apex at ##m##. The...
  27. LiHJ

    What is the percentage of water in a sphere given its volume and radius?

    Homework Statement Dear Mentors and PF helpers, I can do part (a) and (b) but don't really know how to do (c) and (d). Can somebody teach me how to go about solving it. Homework Equations Volume of cone: $$\frac{1}{3}πr^2h$$ Volume of cylinder: $$πr^2h$$ Volume of sphere...
  28. M

    Equation (with polar coordinates) of circle on a sphere

    hi, i'm a newbie... i have this problem: i have a sphere with known and constant R (obvious), i have two point with spherical coordinates P1=(R,p_1,t_1) and P0=(R, p_0, t_0) p_x = phi x = latitude x t_x = theta x =longitude x the distance between point is D=...
  29. M

    Find the weight of a sphere connected to a pulley and to wall

    Homework Statement The system of 2 spheres is in equilibrium. Figure: . If the weight of the second sphere P2=20N, ABC=60 degrees and BAC=30 degrees, find the weight of the first P1 and the force that is applied on the drum of the pulley. Homework Equations P1=m*a The Attempt at a Solution...
  30. R

    Working out a point / segment on a sphere

    Hi, I was hoping someone could help me figure out the problem below. It is a bit of a long winded questions so please bare with me! If you look at Fig 1 below, I have a sphere that spins about an axis in a clockwise direction. (the direction of the spin doesn't really matter) In this case the...
  31. cvex

    How to calculate velocities/forces from sphere collisions?

    Hi, Basically I have a point cloud that represents balls with different radii. They are all moving based on forces and sometimes they are intersecting with each other. Imagine the yellow ball is going in one direction while the blue balls goes in another direction. At one time they are...
  32. A

    Force on a point charge due to a sphere

    Homework Statement An insulated conducting sphere of radius ##R##, carrying a total charge of ##Q##, is in the field of a point charge ##q## of the same sign. Assume ##q\ll Q##. Calculate and plot the force exerted by the sphere on ##q## as a function of distance from the center. In particular...
  33. H

    Is this possible for hanging objects from spinning sphere?

    This youtube clip has a ride that is a spinning sphere with objects hanging by strings from the surface. (see at about 0:55) This is the only ride in the video I think is theoretically impossible (the others seem like they could be done but would be dangerous and/or costly). I think it's...
  34. Matejxx1

    Radius of insphere in a Tetrahedron

    Homework Statement What is the largest possible radius of a sphere which is inscribed in a regular tetrahedron a=10 ( this is the side of the tetrahedron) r=? r=5*√6/6 Homework EquationsThe Attempt at a Solution So first I calculated the Height of pyramid a2=(2/3*va)2+h2 h=√(a2-(2/3*a*√3/2)2)...
  35. G

    Electric potential of hollow conducting sphere

    Homework Statement A charge conducting hollow sphere and a point charge with radius of sphere and distancestors between their centers Homework Equations [/B] 3. The atempt at a solution I am unable to find the potential of Shere in presence of external point charge
  36. R

    Prove that a sphere is a conductor.

    Homework Statement How do I prove that a sphere is a conductor? Homework Equations E = kQ/rThe Attempt at a Solution In my mind, if a sphere is a conductor, the charges formed during induction will move to the surface of the sphere as they can move freely in the conductor, and the same...
  37. CARNiVORE

    Velocity of Center of Mass for a Downwardly-Rotating Sphere

    Homework Statement mass = M radius = r rot. inertia = i height = h Sphere of mass M is released from rest at the top of an inclined plane. The speed of the center of mass at the bottom of the incline, without friction, is sqrt(2gh). I need to find the velocity of the center of mass assuming...
  38. W

    Electric Potential V inside nonconducting sphere with cavity

    Homework Statement A nonconducting sphere of radius r_2 contains a concentric spherical cavity of radius r_1. The material between r_1 and r_2 carries a uniform charge density rho_E(C/m^3). Determine the electric potential V, relative to V=0 at r= infinity, as a function of the distance r from...
  39. Aristotle

    Solid Insulator Sphere Inside Hollow Sphere Conductor

    Homework Statement I was looking for some practice problems in my textbook and found this problem that I was just a little stuck on. I drew the diagram from my textbook with the givens of the problem. Homework Equations ∲E*dA = Q (inside) / ɛ0 The Attempt at a Solution For r less...
  40. Dennydont

    Viscosity of liquid for falling sphere viscometer

    Homework Statement A falling sphere viscometer measures the viscosity of a liquid from the terminal velocity of a tiny, falling sphere. One such device determines that a tiny sphere of radius 46 μm falls through a liquid with a terminal velocity of 2.5 mm/s. If the density of the sphere is 4171...
  41. H

    Moment of inertia of a solid sphere

    Homework Statement Find the moment of inertia of a solid sphere of uniform mass density (like a billiard ball) about an axis through its center Homework Equations I = ∫rρdV The Attempt at a Solution I =ρ ∫r4πr2dr = ρ4π∫r4 Then I integrate this from 0 (the center) to R, so I = (ρ4π)*(R5/5) And...
  42. D

    Determining the oscillations of an electron within a sphere

    Homework Statement Using an electron as a point particle of charge −e inside a positively charged sphere of radius R ≈ 10^(−10) m and total charge +e, find the density ρ(r) of the positive charge for which the electron oscillates harmonically about the center of the sphere assuming that the...
  43. P

    Electric flux through a sphere

    Homework Statement An uncharged nonconductive hollow sphere of radius 12.0 cm surrounds a 11.0 µC charge located at the origin of a cartesian coordinate system. A drill with a radius of 1.00 mm is aligned along the z axis, and a hole is drilled in the sphere. Calculate the electric flux through...
  44. S

    Point Charge and Charged Sphere

    Homework Statement A point charge q1 = -6.1 μC is located at the center of a thick conducting shell of inner radius a = 2.8 cm and outer radius b = 4.8 cm, The conducting shell has a net charge of q2 = 2.6 μC. Homework Equations E = (kQ)/r2 F = (kq1q2)/r2 The Attempt at a Solution I honestly...
  45. samjohnny

    Charges inside conducting sphere

    Homework Statement Three fixed point charges of +2 nC, −3 nC and +4 nC are located inside a thin uncharged metal spherical shell of radius R = 2 cm, as shown in the picture attached. Calculate the strength and direction of the electric field at position P, being 10 cm from the centre of the...
  46. T

    A non-conducting sphere, e-field and potential.

    Homework Statement A non-conducting sphere of radius R has volume charge density ρ = B/r. for r<R and ρ = - for r>R. B is a constant. a) Calculate E-field for r>R. b) Calculate E-field for r<R. c) Calculate potential for r>R. d) Calculate potential for r=R. e) Calculate potential for r<R...
  47. 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...
  48. 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...
  49. 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...
  50. 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...
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