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.

View More On Wikipedia.org
Recent contents Top users
  1. B

    Comoving redshift and area of sphere

    Homework Statement Standard candles may be used to measure the "luminosity distance", using DL = (L/4F)1/2, where L is the source's intrinsic luminosity, and F is the observed flux. Inthis problem you will relate the luminosity distance to the previously discussed angular diameter...
  2. T

    How High Should the String Be Above the Floor to Keep the Sphere Sliding?

    Homework Statement A uniform sphere of radius r and weight W slides along the floor due to a constant horizontal force P applied by a string. Suppose the coefficient of friction is µ. Find the height of the string above the floor h. Homework Equations ƩT = 0 because there cannot be...
  3. S

    Fluid flow around a sphere, what's the net force?

    Homework Statement A sphere of radius ##a## is submerged in a fluid which is flowing in the z-hat direction. There is some associated viscosity in the fluid which will exert a force on the sphere. Use symmetry to argue that the net force will be in the z-direction. Show that it will have the...
  4. S

    Distance traveled winding around sphere

    Homework Statement I have a sphere that I want to travel from the north pole to the south pole. The route I take is winding around the sphere (instead of the obvious shortest path). My path is dictated by: ##\phi(t)=wt## ##z(t)=R-v_z t## in the negative z direction per N to S The form...
  5. V

    Electric flux leaving a sphere

    Homework Statement Within the spherical shell, 3 < r < 4 m, the electric flux density is given as D = 5(r − 3)3 ar C/m2 a) What is the volume charge density at r = 4? b) How much electric flux leaves the sphere r = 4? Homework Equations ρv=Div D Electric flux = ∫sD.ds=∫vρvdvThe Attempt at a...
  6. B

    Calculating Flux through a Sphere using Cylindrical Coordinates

    I was told it might be better to post this here. Homework Statement The trick to this problem is the E field is in cylindrical coordinates. ##E(\vec{r})=Cs^2\hat{s}## Homework Equations ##\int E \cdot dA## The Attempt at a Solution I tried converting the E field into spherical...
  7. B

    Flux Through Sphere: Cylindrical Coordinates

    Homework Statement The trick to this problem is the E field is in cylindrical coordinates. ##E(\vec{r})=Cs^2\hat{s}##Homework Equations ##\int E \cdot dA##The Attempt at a Solution I tried converting the E field into spherical coords and I can find the flux that way but it is a complicated...
  8. Philosophaie

    Surface Area and Volume of a Sphere

    I need to know the Surface Area and Volume of a spherical ball at the origin radius a. What I want is to evaluate the integrals at each integral. ##\oint_S dS =\int\int d? d? = 4 *\pi*r^2## ##\oint_V dV = \int_0^{\pi}\int_0^{2\pi}\int_0^a dr d\theta d\phi## = ##\frac{4}{3}*\pi*a^2##
  9. C

    Calculus of Variation - Shortest path on the surface of a sphere

    Refer to "2.jpg", it said that the shortest path on the surface of a sphere is Ay-Bx=z , which is a plane passing through the center of the sphere. I cannot really understand about this. Does it mean that the shortest path is a ring that connects two points with its center at the center of the...
  10. C

    E-field & hollow non-conducting sphere

    I know that the E-field around a hollow non-conducting sphere charged with Q charge comes immediately from Gauss' Law but I'm wondering what the situation is if we somehow go inside the material, we make a very small hole through the material of the sphere and go inside it. What would there be...
  11. U

    Minimum linear velocity attained by sphere

    Homework Statement A sphere of mass M and radius R is moving on a rough fixed surface, having co-efficient of friction μ, with a velocity v towards right and angular velocity ω clockwise. It will attain a minimum linear velocity at time (take v>ωR) The Attempt at a Solution Since v>ωR the...
  12. Q

    Calculating Change in Volume of a Shrinking Sphere

    Homework Statement http://i.minus.com/jbxIzu0P7sTqP0.png Homework Equations V(sphere) = 4/3(pi)(r^3) V = 36pi in^3 dr = -0.2 in dV = ? The Attempt at a Solution I basically solved for the radius, and took the derivative and plugged in the value of the radius and the...
  13. J

    Electrostatics - hollow charged sphere

    So we have a cross sectoin of a sphere that is charged with Q (refer to attachment). electrostatics say the electric field within a charged conductor is 0, and the electric field is perpendicular to the surface. But for a hollow charged sphere (like in the attachment), does the hollow area...
  14. M

    Angular and CoM Velocities of a Solid Sphere

    Homework Statement A solid sphere of mass M and radius R is rolling,without slipping, down a curved rail. The sphere is initially at rest at a height of h1. Find the angular velocity ω2 and the center of mass velocity of the sphere vcm at the end of the rail of height h2. You may assume that...
  15. PsychonautQQ

    Integrating over a sphere

    Homework Statement Triple Integral: x^2+y^2+z^2dV over the ball x^2+y^2+z^2 ≤ 9 Homework Equations The Attempt at a Solution so With my integral I had Triple Integral: p^3sin∅dpd∅dθ 0≥p≥3 0≥∅≥∏ 0≥θ≤2∏ Does this look like the correct integral? I swear it is! Yet my answer...
  16. PsychonautQQ

    Finding area between sphere and parabloid

    Homework Statement Find the volume above the sphere x^2+y^2+z^2 = 6 and below the parabloid z = 4-x^2-y^2. Homework Equations The Attempt at a Solution I did a triple integral in cylindrical coordinates Triple Integral: dzdrdθ where z is between (6-r^2)^(1/2) to (4-r^2) and dr...
  17. PsychonautQQ

    Computing integral over a sphere

    Homework Statement Computer the integral of f(x,y,z) = x^2+y^2 over the sphere S of radius 4 centered at the origin. The Attempt at a Solution so if the parameters for a sphere are in terms of (p,θ,∅) , triple integral (p^2((psin∅cosθ)^2+(psin∅sinθ)^2))dpdθd∅) where the boundaries...
  18. ShayanJ

    Center of mass of a uniform sphere

    We know,by symmetry,that the center of mass of a uniform sphere is at its center.So we expect the formula r_{com}=\frac{\int r \rho d\tau}{\int \rho d\tau} to give us zero for this case.So let's see: r_{com}=\frac{\int_0^{R} \int_0^{\pi}\int_0^{2\pi} r^3 \sin{\theta} d\phi d\theta...
  19. L

    Geometric phase of a parallel transport over the surface of a sphere

    I have this question on the calculation of the geometric phase (Berry phase) of a parallel transporting vector over the surface of a sphere, illustrated by Prof. Berry for example in the attached file starting on page 2. The vector performing parallel transport is defined as ψ=(e+ie')/√2...
  20. B

    B Field Inside of Sphere using Sep. Variables

    Done editing I hope. Homework Statement If Jf = 0 everywhere, then (as we showed in class), one can express H as the gradient of a scalar potential, W. W satisfies Poisson’s equation with ∇⋅M as the source. Use this fact to find the field inside a uniformly magnetized sphere. (Griffiths has...
  21. B

    Geometry Question About A Sphere

    If I had a sphere with a radius of 100 meters, a diameter of 200 meters, a volume of 4,188,790.20 square meters, and I wanted to place within this sphere a single dot (one dimensional so it doesn't take up any extra space and there is no displacement --if you're thinking in terms of water--)...
  22. J

    Volume of Sphere: Find w/ Pappus Theorem

    Homework Statement Use the theorem of pappus to find the volume of the given solid A sphere of radius r Homework Equations V = 2∏xA The Attempt at a Solution V = 2∏(4r/3∏)(4∏r^2) = (16/3)∏r^3 So something is wrong I should end up with (4/3)∏r^3 no?
  23. M

    Work, energy stored in solid sphere

    Homework Statement Find the energy stored in a uniformly charged sphere of charge q, radius R Homework Equations The Attempt at a Solution Ein=\frac{qr}{4\pi\epsilon o R^3}, Eout=\frac{q}{4\pi\epsilon o r^2}... W=\int_{0}^ {R}\int_{0}^{2\pi}\int_{0}^{\pi}[\frac{qr}{4\pi\epsilon...
  24. B

    Potential inside sphere with empty cavity

    Homework Statement An insulating sphere of radius R , centered at point A, has uniform chagre density ρ. A spherical cavity of radius R / 2 , centered at point C, is then cut out and left empty, see Fig. (a) Find magnitude and direction of the electric field at points A and B. (b) Find...
  25. R

    Eletric potential inside charged sphere with hole inside

    Homework Statement Consider a charge density of ρ=k/r , k>0 , located between a sphere surface of r=a and another sphere surface of r=b, b>a. I'm supposed to find the electric field on all space, which I did. Now I have to find the electric potential in all space, which I also did for r>b...
  26. N

    Sphere Volume Calculation: Bore Hole of Radius r in Sphere of Radius R

    Homework Statement A hole of radius r is bored through the center of a sphere of radius R. Find the volume V of the remaining portion of the sphere. Homework Equations The Attempt at a Solution Wouldn't is be (4/3)∏(R^3-r^3)?
  27. M

    Find index of refraction of a sphere given the beam path

    Homework Statement A beam of parallel light rays from a laser is incident on a solid transparent sphere of index of refraction n1 (see figure). (a) If a point image is produced at the back of the sphere, what is the index of refraction of the sphere? (b)What index of refraction, if any, will...
  28. J

    Accelerated charge inside sphere (again)

    Sorry to go on about this scenario again but I think something is going on here. Imagine a stationary charge ##q##, with mass ##m##, at the center of a stationary hollow spherical dielectric shell with radius ##R##, mass ##M## and total charge ##-Q##. I apply a force ##\mathbf{F}## to charge...
  29. C

    Charged Metallic Sphere Touching Spherical Shell From Inside

    Homework Statement (From Physics for Scientists and Engineers, 7E, Serway-Jewett Chapter 25 Q11) (i) A metallic sphere A of radius 1 cm is several centimeters away from a metallic spherical shell B of radius 2 cm. Charge 450 nC is placed on A, with no charge on B or anywhere nearby. Next...
  30. B

    Work required to increase the radius of a sphere by dr

    Homework Statement To assemble a uniformly charged sphere, assemble it like a snowball, layer by layer, each time bringing in an infinitesimal charge dq from far away and smearing it uniformly over the surface, thereby increasing the radius. How much work dW does it take to build up the radius...
  31. C

    Help with magnetic field in sphere problem

    Homework Statement Consider a polycrystalline sample of Subscript[CaSO, 4]\[CenterDot]2 Subscript[H, 2]O in an external magnetic field Overscript[B, \[RightVector]] in the z direction. The internal magnetic field (in the z direction) produced at the position of a given proton in the...
  32. Z

    Magnetic sphere moving through iron dust; find velocity & other things

    Homework Statement A small magnetic sphere of initial mass Mo and initial radius Ro is moving through a space filled with iron dust. During its motion, 5% of displaced dust is deposited uniformly onto the surface of sphere. Given the density of dust to be ρ, find: 1. relation rate of increase...
  33. T

    Why integrate at these points? (Electric potential of a sphere)

    Homework Statement Find the potential inside and outside a uniformly charged solid sphere of radius R and total charge q. Homework Equations V(r) = -∫E dl The Attempt at a Solution I just have a question about finding the potential inside the sphere. Why integrate from infinity...
  34. O

    Work on charge from outside to inside of sphere

    Homework Statement A solid non-conducting sphere of radius R = 1.12m. The sphere posses a total charge Qtot spread uniformally throughout its volume. a) derive equations for electric field for 1) 0<r<R 2) r>R result in terms of r R and Q b) Derive an equation that gives...
  35. M

    Point Charge in the presence of charged, insulated, conducting sphere

    So here we are talking about solving this problem by method of images. The approach taken by most of electrodynamics textbooks is as follows: "If we wish to consider the problem of an insulated conducting sphere with total charge Q in the presence of a point charge q, we can build up the...
  36. P

    Electrodynamics Potential from charged sphere. I am lost :/

    ~Electrodynamics~ Potential from charged sphere. I am lost :/ Homework Statement A sphere of radius R, centered at the origin, carries a charge density ρ(r,θ)=κ/r^2(R-2r)sin^2(θ). κ is constant. Find exact potential. Homework Equations 1/4∏ε∫ρ∂t/r The Attempt at a Solution Question and...
  37. J

    How much induced voltage on sphere?

    Hi, Assume that one has a pair of metal spheres, A and B, some distance apart. A is connected to, say, a small van der Graaf generator and B is connected to a voltmeter which is then connected to ground. I expect charge to be induced on sphere B. How would one calculate the voltage one...
  38. atyy

    Lattice Simulations on a Sphere in Condensed Matter

    Most lattices I've come across in condensed matter, like the Kitaev model, are regular lattices and don't fit on a sphere. Are lattice simulations ever put on a sphere in condensed matter, and if so what sort of lattice is used?
  39. K

    A Charged Sphere with a Cavity

    Homework Statement An insulating sphere of radius a, centered at the origin, has a uniform volume charge density ρ. A spherical cavity is excised from the inside of the sphere. The cavity has radius a/4 and is centered at position h(vector) , where |h(vector) |<(3/4)a, so that the entire...
  40. M

    Fluid flowing in a sphere and leaving

    Homework Statement Homework Equations The Attempt at a Solution Just checking if this is correct, and if the equation in part c implies that the flow rate out changes with time, or is just based off the initial height..I think it changes and therefore requires this integration. Also, I looked...
  41. T

    Find Electric Field Around a Non-Conducting Sphere

    Homework Statement I. A non-conducting sphere of radius a has a spherically symmetric, but non-uniform charge distribution is placed on it, given by the volume density function: p(r) = C·r, where C is a positive constant, and 0 < r < a. a. Find an algebraic expression for the total charge...
  42. H

    Differential equation for changing mass of a sphere

    The mass of a sphere with density as a function of radius is M=4\pi \int_0^r\rho(r) r^2dr Lets say the radius increases and decreases as a function of time t. So: M(t)=4\pi \int_{0}^{r(t)}\rho (r) r(t)^2dr I want to know the basic equation describing the mass added or removed...
  43. D

    Finding Electric Field from an insulating sphere

    An insulating sphere with a radius of (3.3E-2) m has a uniform charge density of (6.74E-6) C/m^3 throughout its volume. (1)Find the magnitude of the electric field at a point (5.7E-2) m from the center of the sphere. (2) Find the magnitude of the electric field at the surface of the...
  44. B

    Why do we need to pick a very small A for curved surfaces in electrostatics?

    Homework Statement Standard E field problem where I'm to find the field at 3 positions of a hollow sphere that has a charge density k/r^2 r ≤ a a ≤ r < b b ≤ r Homework Equations ∫Eda=Q/ε The Attempt at a Solution I guess the thing that is tripping me up are the limits. I know...
  45. Z

    Flux Through a Sphere Containing a Dipole

    So I am probably understanding something wrong but would gauss's law \frac{Qinside}{\epsilon0} for a sphere surrounding an electric dipole, with a point charge just outside of the sphere. If you imagine this scenario without the outside charge, the field lines through the surface all come...
  46. D

    Integral on the surface of a sphere - course notes

    Hello, In my electrodynamics course, there's a "maths" introduction and there's something i don't get ! Homework Statement It says that : the integral on the surface of a sphere is ∫1/r da = 4πr'/3 with r=|r'-R|, R the vector from the element da to the center. r'=r'*z^ The Attempt...
  47. H

    Puck on a sphere, energy & Newton's 2nd

    Homework Statement Consider a small frictionless puck perched at the top of a fixed sphere of radius R. If the puck is given a tiny nudge so that it begins to slide down, through what vertical height will it descend before it leaves the surface of the sphere? [Hint: Use conservation of...
  48. Y

    Electric Field of a Conducting Sphere

    Homework Statement At a distance of 0.206cm from the center of a charged conducting sphere with radius 0.100cm, the electric field is 485N/C . What is the electric field 0.612cm from the center of the sphere? Homework Equations E(r)=1/4∏ε_0 * qr/R^3 where r is radius of the Gaussian...
  49. M

    How can the surface area of a sphere be derived using integration of circles?

    Homework Statement Derive the formula for surface area of a sphere using integration of circlesHomework Equations Need to get : S = 4πr2The Attempt at a Solution Consider a sphere of radius r centred on the origin of a 3D space. Let y be an axis thru the origin. The sphere can be sliced into...
  50. andyrk

    Charged Spherical Shell and Solid Sphere

    A spherical shell and a conducting sphere each of radius R are charged to maximum potential. Which of the two has more charge? My attempt: Since in a conductor, no charge can reside inside the conductor so all charge is on the surface of the conductor just like the spherical shell. Now ...
Back
Top