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. L

    MHB Minimizing the surface of a sphere and cylinder

    279) A body is formed by a straight circular cylinder which ends up in a hemisphere. What are the dimensions that should have this body so the total surface area is minimal, if your volume is answer Cubic sqrt( 3V/5 pi) i tried to post an image of my notes and i couldnot i will type later Vt...
  2. L

    MHB Optimization of the sum of the surfaces of a sphere and cube

    If the sum of the surfaces of a cube and a sphere as is constant, deierminar the minion of the diameter of the sphere to the edge of the cube in cases in which: 272) The sum of the volumes is minimal 273) The sum of the volumes is maximum And the answer are 272 = 1 and 273 = infinit Ok Vs =...
  3. M

    Potential Difference problem with sphere and capacitor

    Homework Statement A thin spherical shell made of plastic carries a uniformly distributed negative charge -6e-10 coulombs (indicated as -Q1 in the diagram). Two large thin disks made of glass carry uniformly distributed positive and negative charges 1.5e-05 coulombs and -1.5e-05 coulombs...
  4. Z

    Quick question - Flux through a sphere

    Homework Statement Evaluate ∫∫r.ndS where r=(x,y,z) and n is a normal unit vector to the surface S, which is a sphere of radius a centred on the origin. 2. The attempt at a solution I decided to use polar coordinates. The radius of the sphere is clearly constant, a. So a surface...
  5. Duderonimous

    Gaussian sphere problem, non uniform charge

    Homework Statement A solid non conducting sphere of radius R=5.60 cm has a nonuniform charge ditribution ρ=(14.1 pC/m^3)r/R, where r is the radial distance from the spheres center. (a) What is the sphere's total charge? What is the magnitude E of the electric field at (b) r=0, (c) r=R/2.00...
  6. L

    Why is the electric field inside of a conducting sphere zero?

    Homework Statement Consider a solid conducting sphere with a radius a and charge Q1 on it. There is a conducting spherical shell concentric to the sphere. The shell has an inner radius b > a, outer radius c and a net charge Q2 on the shell. Denote the charge on the inner surface of the shell...
  7. H

    MHB Radius of Sphere Tangent to Two Lines

    I need help getting around this Calculus 3 problem. Any hints will be gladly appreciated: Find the radius of smallest sphere that is tangent to both the lines L1 : x=t+1 y=2t+4 z=−3t+5 L2 : x=4t−12 y=t+5 z=t+17
  8. L

    MHB Maximizing the weight of a cylinder cut from a sphere

    A sphere weighs P kg what is the weight of the higher straight circular cylinder that can cut from the sphere? Answer sqrt(3)P/3
  9. S

    MHB These two problems are based on Vectors, dot product and distance for sphere.

    Problem 1: Let S1 be a sphere centered at(0, 1, -3) with radius 1 and let S2 be a sphere centered at (3, 5, -9) with radius 2. Find the distance between the two spheres. problem 2: Given three non-zero vectors v1, v2, v3 we say that they are mutually orthogonal when v1 dot v2= 0, v1 dot v3=0 ...
  10. M

    Light refracting through a plastic sphere

    Homework Statement A small light bulb is placed 10.0 cm from the center of a plastic sphere of radius 1.0 cm and refractive index 1.40. Where is the image of the bulb? Homework Equations 1) Thin lens equation 2) Thick lens equation The Attempt at a Solution I realize that I'm...
  11. C

    Gradient of sphere level fxn with 2 parameters inside the parametric e

    Level function [L(x,y,z)] = (1/r^2) (x^2 + y^2 + z^2) = 1 Vector [N([x(h,g)], [y(h,g)], [z(g)])] = parametric equation to sphere Level function [L(x,y,z)] The parametric equations have 2 parameters, h and g [x(h,g)] = (r [sin (a + gv)]) [cos (b + hw)] [y(h,g)] = (r [sin (a + gv)]) [sin (b +...
  12. R

    Image Charges and a hollow conducting sphere

    Homework Statement Hi! Bear in mind, before shooting me down, that I'm very new to electrostatics and extremely (i.e. today) new to the method of image charges, and all my learning is dodgy book-learning and not learning from asking the learned questions. This said, I just want my...
  13. F

    Fluid Inside an Ideal Sphere: What Happens?

    Hi all! Imagine you have an ideal sphere (made of steel or plastics). Then fill it completely with some fluid, e.g. water. Then rotate it very fast. What happens to a fluid inside? I suppose that it will try to change its shape (from spherical to flat). But there is no space for any size...
  14. L

    MHB (2) Find the height of the cone's lateral surface minimum confined to a sphere of RADIUS R.

    (2) Find the height of the cone's lateral surface minimum confined to a sphere of RADIUS R. the answer is (2 + sqrt(2)) R
  15. B

    Mass of a sphere with a non-uniform density.

    Homework Statement A sphere is given by x^2+y^2+z^2 ≤ 1. The density is given by ρ(x,y,z) = x+y+z. Show that the mass is 3π/2. Homework Equations m = ∫ρ ∂V ∂V=ρ^2sinϕ∂ρ∂ϕ∂θ The Attempt at a Solution I have converted the x, y and z in the density function to spherical...
  16. M

    Can someone explain thoroughly the celestial sphere

    Hello, I am trying to get into amateur astronomy but first I have to understand the sky. Because of the tilt of the earth, there is more than half the surface area of the Earth radiated by the sun's rays and therefore those areas which are on the hemisphere of more sunlight have longer days...
  17. C

    One Way mirror solar panel sphere.

    I saw a previous thread on this forum that asked about the practical usage of a one way sphere ball and after reading through it I've come to terms with an idea that uses such an object in a very efficient manor. If you take your one way mirror sphere and wire solar panels on the interior of...
  18. G

    Compute voltage inside sphere of uniform charge

    Homework Statement Problem 2.21 from Introduction to Electrodynamics, David J. Griffiths, Third Edition. Find the potential inside and outside a uniformly charged solid sphere who's radius is R and whose total charge is q. Use infinity as your reference point. Homework Equations...
  19. A

    Voltage on a electrostatic charged sphere

    I understand that voltage depends on how far the electrons are, so for a sphere it would be q*k / r. But what if a sphere with 100kv , then we put with contact to it a metal box , becoming on body, will the volt remain 100kv
  20. H

    Moment of Inertia of Hollow Sphere about Center Axis x-y-z method

    Homework Statement Find the moment of inertia of a hollow sphere about a vertical axis through its center in terms of its mass M and radius R. Homework Equations I=\int r^{2} dm The Attempt at a Solution I've been curious about different methods for finding moments of inertia...
  21. J

    Relativity - Sphere flattening due to relativistic speed

    Relativity -- Sphere flattening due to relativistic speed Homework Statement Gum balls are spherical, and about 1.5 cm in diameter. Smarties are circular in one cross section, with the same diameter, but perpendicular to this circular cross section, they are flattened, with the smallest...
  22. H

    Charged sphere hanging from a string-Find the charge?

    Charged sphere hanging from a string--Find the charge? A small, plastic sphere of mass m = 126 g is attached to a string as shown in the figure. There is an electric field of 151 N/C directed along the + x axis. If the string makes an angle 30 degrees with the y-axis when the sphere is in...
  23. I

    Array of magnets around a sphere

    An array of N magnets which can turn freely about their centers in any direction in 3d space is distributed uniformly around a spherical surface (their centers). What is the configuration of equilibrium of the system after some time? (minimum energy) Do you know of any work showing that...
  24. J

    What Is the Angle of Incidence for Light to Emerge Parallel from a Sphere?

    [Mentor's note: this thread does not use the normal homework forum template because it was originally posted in a non-homework forum, then moved here.] A light beam is incident on a sphere with refractive index n=√3 at an angle i from air and emerges parallel to the horizontal axis passing...
  25. G

    Partial Derivative of Sphere in Terms of x and y

    Hi everyone! I'm not sure if this is the right forum to post my question. If I'm wrong, let me know it. The question: Let us consider the functions \theta=\theta(x,y), and M=M(\theta), where M is a operator, but i doesn't relevant to the problem. I need to know the derivative \frac{\partial...
  26. H

    Integral of magnetic field over the sphere

    If all the currents were inside a sphere with the radius R, then we would have \int B \,dV= 2/3\mu_0 M where M is magnetic moment of all the currents and B is magnetic field. If all the current were outside the sphere, then we would have\int B \,dV= 4/3 \pi R^3 B(0) where B(0)is magnetic field...
  27. H

    Electric field inside a charged sphere

    If we have a uniformly charged spherical shell, supposing that the shell is non-conducting, could we have any electric field inside the sphere? Why?
  28. Z

    Which scatters light more: a sphere or a hemisphere?

    In case the context, which this question is based needs to be provided, the context follows. A sub-project was being done on using sphere or hemisphere "scatterers" on TFSC (Thin-Film Solar Cells). These "scatterers" can be applied on the top of a TFSC in order so that the solar cells can...
  29. U

    Average and variance of a unit sphere

    Homework Statement Consider the sphere x2 + y2 + z2 = 1 Find the mean and variance. Homework Equations The Attempt at a Solution Mean = 0 (Symmetry) Variance Probability = \frac {dV}{\frac{4}{3} \pi R^3} = \frac {4 \pi r^2 dr}{\frac{4}{3} \pi R^3} = 3 \frac {r^2}{R^3} dr Variance =...
  30. S

    Vector field flow over upper surface of sphere

    Homework Statement Calculate the flow over the upper surface of sphere ##x^2+y^2+z^2=1## with normal vector pointed away from origin. Vector field is given as ##\vec{F}=(z^2x,\frac{1}{3}y^3+tan(z),x^2z+y^2)##Homework Equations Gaussian law: ##\int \int _{\partial \Sigma }\vec{F}d\vec{S}=\int...
  31. G

    Is \(\frac{\partial}{\partial \phi}\) a Killing Vector on the Unit Sphere?

    \bar{}Homework Statement Hi, I want to show that \frac{\partial}{\partial \phi} is a Killing vector on the unit sphere with metric ds^2 = d\theta^2 + \sin^2 \theta d \phi^2 Homework Equations I compute the Christoffel symbols to be \Gamma^\theta_{\phi \phi} = -\sin \theta \cos...
  32. T

    Sphere Slipping Against Blocks: Solving for the Speed of the Sphere's Center

    Sphere slipping against blocks Homework Statement In the given arrangement ,a sphere of radius R is placed on the two blocks A and B where A is fixed and B is moving at a constant speed ‘v’ towards left .Find the speed of the sphere’s center when the distance between the blocks is √R and...
  33. U

    Permeable sphere placed in external magnetic field

    Homework Statement A permeable sphere with μ_r is placed in a magnetic field of H. Show that it satisfies Laplace's equation and the potentials are of the form: Homework Equations The Attempt at a Solution Part (a) Starting from maxwell's equations, I showed that ø satisfies laplace's...
  34. Spinnor

    5 positivly charged particles on sphere, min energy configuration, rel

    Assume 5 charged particles (charge 1) constrained to live on the surface of a sphere are in a configuration that minimizes electrostatic potential energy. Are there configurations that are stable but that are not the minimum energy configuration? A simple computer program could quickly(?)...
  35. D

    Potential outside charged metal sphere.

    Homework Statement Find the potential outside a charged metal sphere (charge Q, radius R) placed in an otherwise uniform electric field E0. Explain clearly where you are setting the zero of potential. Homework Equations The Attempt at a Solution My only problem here is that I...
  36. T

    Hypothetical Hollow Steel Sphere: collapse from outside pressure

    Hello physics forums, Say you had a hollow steel sphere of thickness 1 mm and diameter of 1 meter (from outside to outside)? Inside the sphere is gas at 1 atm pressure. Outside is 1 atm of pressure. How much gas would I have to remove from the inside until the sphere collapsed from...
  37. 5

    Intensity of sound behind a sphere

    If I have a sphere with radius r, a distance d away from a sound source of intensity I0. What will the intensity I of the sound wave be on the point of the sphere directly opposite the source? Preferably I would like to find the intensity of the wave at any point on the sphere.
  38. S

    Method of Image grounded conducting sphere.

    Homework Statement A point charge of 10 coulombs is placed at a distance d= 20cm from the centre of an earthed conducting sphere of radius a= 5cm.Find ① The maximum surface density of charge induced on the sphere. ② The force of attraction on the point charge ③ The Sphere is now Insulated...
  39. B

    A Question On Dividing a Sphere

    I would like to know how to divide a sphere's volume equally into 3 parts, by using two "slices" that are parallel planes. A good example would be cutting a round fruit into 3 equal parts by two slices with a knife. I would like to know the distance (fraction of the diameter) along the diameter...
  40. KiNGGeexD

    Components of torque in a solid sphere

    A uniform sphere of mass M and radius R has a point on its surface fixed at the origin. Its centre lies along a line in the direction of the position vector r = i + 2k + 3k at length R. Find the components of the torque acting on it due to gravity if the z-direction is upwards and gravity acts...
  41. M

    [ElectroStatics]Question Regarding Charge within a Metal Sphere

    Homework Statement I have this question on a practice exam in preparation fora final exam, and I am questioning my solution here: A hollow spherical metal shell has an outer radius equal to 1.5 m and a shell thickness of 0.5 m. A +100 nC point charge is located in the hollow 0.5 m from...
  42. Y

    Electric potential at the center of a sphere

    Homework Statement The electric field at the surface of a charged, solid, copper sphere with radius 0.19m is 2800 N/C , directed toward the center of the sphere. What is the potential at the center of the sphere, if we take the potential to be zero infinitely far from the sphere...
  43. B

    Find the magnetic field of a uniformly magnetized sphere.

    Homework Statement Find the magnetic field of a uniformly magnetized sphere. (this is all that was given in the problem) The Attempt at a Solution I chose the z axis in the same direction as M. J_b=\nabla \times M=0 and K_b=M \times \hat{n}=Msin\theta \hat{\phi} Apparently, I can...
  44. S

    Finding center of mass of surface of sphere contained within cone.

    Homework Statement Problem (also attached as TheProblem.jpg): Find the center of mass of the surface of the sphere x^2 + y^2 + z^2 = a^2 contained within the cone z tanγ = sqrt(x^2 + y^2), 0 < γ < π/2 a constant, if the density is proportional to the distance from the z axis. Hint: R_cm =...
  45. M

    Velocity at any point on a rotating sphere

    Homework Statement "A uniformly charged solid sphere, of radius R and total charge Q, is centered at the origin and spinning at a constant angular velocity ω about the z axis. Find the current density \vec{J} at any point (r,θ,\phi) within the sphere." Problem 5.6(b), p.223, from...
  46. T

    Volume of a sphere without cap

    Homework Statement A solid sphere of radius R has a spherical cap, defined by the cone theta = alpha, removed from its "north pole". Determine the volume of the sphere without cap. Homework Equations The Attempt at a Solution Well obviously, the volume would be volume of sphere -...
  47. N

    What Charge Must a Wooden Sphere Have to Float Above the Earth?

    Hello, everyone :-) We have a wooden sphere at a height of h = 1 m above the surface of the Earth which has a perimeter of RZ = 6 378 km and a weight of MZ = 5.97 · 10^24 kg. The sphere has a perimeter of r = 1 cm and is made of a wood which has the density of ρ = 550 kg·m − 3. Assume that...
  48. S

    Taking small element for integration purpose in SOLID sphere?

    Homework Statement The Question originally is to find the m of a solid uniformly charged solid sphere which is rotating uniformly with ω Now Homework Equations Now my question to you is how to take the small element? The Attempt at a Solution i take a small disc with...
  49. Saitama

    Understanding the Sphere Rolling Problem: Simplifying the Equations

    Homework Statement Homework Equations The Attempt at a Solution I have some difficulty understanding the problem statement. The problem says the paper moves horizontally but then in the very next line, it states the velocity is perpendicular to the initial velocity of sphere...
  50. K

    Circle Expansion on Expanding Sphere

    Homework Statement How would a circle on a sphere expand as a function of the sphere's radius as the sphere expands? Homework Equations none were provided The Attempt at a Solution \S=4\,\pi \,{R}^{2} A=\pi \,{r}^{2} {\it dA}=2\,\pi \,r{\it dr} {\it dS}=8\,\pi \,R{\it dR}...
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