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. O

    How to create a uniformly charged sphere?

    Can we create at least any one of the following in laboratory? How? 1. A uniformly charged spherical shell of finite thickness 2. A uniformly charged sphere 3. A radially symmetrically charged spherical shell of finite thickness 4. A radially symmetrically charged sphere
  2. F

    Reducing the charge Q on an isolated charged conducting sphere to Q/8

    It seems to me that one can obtain the required result by using just one neutral sphere and one ground wire. Let A be the charged sphere and B be the neutral one. Initially ##Q_A=Q## and ##Q_B=0##. put A and B in contact. As a result ##Q_A=Q/2## and ##Q_B=Q/2##. ground B, so that ##Q_B=0##...
  3. G

    Electrostatic field of a sphere

    What is the electrostatic field of a non-conductive sphere (it's radius is R) which has a density charge distribution inside? ρ0 and R are parameters. I started solving this with Gauss's law: then: Solving the integral: This means the electrostatic field of the sphere in r is: Can you...
  4. G

    I Visualizing the cotangent space to a sphere

    Is it correct to say that: the cotangent is given by the gradients (*) to all the curves passing through a point and it actually spans the same tangent space to a point of a sphere? If you visualize them as geometric planes (**), the cotangent and the tangent spaces are more than isomorphic...
  5. G

    Electric field around a sphere with an internal charge distribution

    I'm just going to skip some of the step since I only need help with understanding the last part. After rearranging the equation stated at "Relevant equation" (and skipping some steps) we will get: E * 4*pi*e0*R^2 = integral pv * 4*pi*R^2 dR E = 1/(4*pi*e0*R^2) * 4*pi * integral pv*R^2 dR E =...
  6. Edward Candle

    Induced Charged on a Grounded Sphere

    I've come to the result (using cylindrical coordinates) #\sigma (z) = (-2q) / (pi*sqrt(R_0*(10R_0-6z)^3) )# and i tried to get #Q# by integrating #2*pi*sqrt(R_0^2-z^2)*\sigma(z)dz# from #-R_0# to #R_0#. But i can't solve that integral. I tried solving it numerically with arbitrary values and it...
  7. Z

    Electric field direction on a grounded conducting sphere

    I am required to find the direction of the electric field on the surface of a grounded conducting sphere in the proximity of a point charge ##+q##. The distance between the center of the sphere and the point charge is ##d## and using the method of images we find that the charge of the sphere is...
  8. Mounice

    Solving Potential of a Charge Outside a Sphere with Green Functions

    I was wondering if there is a way to deduce the solution of the potential of a charge outside a sphere given by the image method, though Green functions. Because of a Dirichlet condition (GD(R,r')=0), I know that a solution can be written as GD=Go+L, where ∇2L=0. But in order to approach this...
  9. Leo Liu

    Understanding the Electric Field of a Charged Sphere

    This page claims that "[t]he electric field outside the sphere is given by: ##{E} = {{kQ} \over {r^2}}##, just like a point charge". I would like to know the reason we should treat the sphere as a point charge, even if the charges are uniformly distributed throughout the surface of the...
  10. F

    B Is εσT⁴ hemispherical or full sphere?

    Can someone advise on the εσT⁴ term of Stephan-Boltzmann law? Is this the power radiated from a point or flat surface into a hemisphere or a sphere fully enclosing a black body? thanks.
  11. Adrian555

    A Geodesics of the 2-sphere in terms of the arc length

    I'm trying to evaluate the arc length between two points on a 2-sphere. The geodesic equation of a 2-sphere is: $$\cot(\theta)=\sqrt{\frac{1-K^2}{K^2}}\cdot \sin(\phi-\phi_{0})$$ According to this article:http://vixra.org/pdf/1404.0016v1.pdfthe arc length parameterization of the 2-sphere...
  12. S

    Potential difference between the surface of a sphere and a point far away

    V at surface = k Q / r = 9 x 109 x (1 x 109 x (-1.6 x 10-19) / (1 x 10-2) = - 144 V V at a point far away = 0 V From the sentence "electric potential difference between the surface of this sphere and a point far away" means that the question asks about V at surface minus V at far away so the...
  13. S

    Three Charges inside a Metal Sphere

    I know that for a conductor the charge is uniformly distributed and the electric field is zero inside the shell. However, I am not sure how to calculate the charge inside the shell so I can know the electric field.
  14. karush

    MHB Apc.3.1.8 difference in sphere volume

    ?
  15. B

    Charging a magnetically levitating sphere to 1 gigavolt in a high vacuum

    What sort of limits would be encountered if you tried to charge a magnetically levitating sphere to as high a voltage as possible in an ultra high vacuum by using an electron beam aimed at the sphere? Assume the sphere is highly spherical and polished. If electrons have sufficient energy to...
  16. G

    Gauss-Theorem on a solid dielectric sphere

    The load system formed by the point load and the load distribution generates two regions in space corresponding to r<1m and r>1m, i.e. inside and outside the sphere. Given the symmetry of the distribution, by means of the Gaussian theorem we can find the modulus of the field at a distance r from...
  17. Z

    Find the electric field on the surface of a sphere using Coulomb's law

    Note that the solution is 5625 V/m in z direction which is found easier using Gauss' law, but I want to find the same result using Coulombs law for confirmation. Lets give the radius 0.04 the variable a = 0.04m. ##\rho## is the charge distribution distributed evenly on the surface of the...
  18. Saptarshi Sarkar

    Electric field inside a polarized dielectric sphere

    My attempt: I know from Gauss' law in dielectric ##\nabla .D = ρ_f## where ##D = ε_0E + P##, so as ##ρ_f = 0## (as there is no free charge in the sphere) => ##\nabla .D = 0## => ##ε_0\nabla .E = \nabla .P## from this I get ##E = \frac {-kr^2 \hat r} {ε_0}## But, I know that for a uniformly...
  19. fisher garry

    E field calculation for q sphere

    I have some questions about this answer. Why do they use absolute value when writing in the limits in the integral underlined with orange? And how do they get from this value where I have underlined with orange to the answer for E outside the sphere. Can someone do the rewriting? And last why is...
  20. B

    I Parallel transport of a vector on a sphere

    question1 : if you draw a small circle around the north pole (it should be the same at every points because of the symmetry of the sphere),then it is approximately a flat space ,then we can translate the vector on sphere just like what we have done in flat space(which translate the vector...
  21. Luke Tan

    On-Axis Field of a Uniformly Magnetized Sphere

    Since I am only required to find the on-axis field, I tried directly integrating the biot savart to find the field, rather than integrating to find the vector potential before taking the curl. However, on integration (by mathematica) it seems that the solution is an elliptic integral, very...
  22. E

    Finding the energy of a charged sphere

    In class we were taught that for spherical bodies we may use the formula below where the integral is done over the volume of the body. However, if we assume that the potential in infinity is 0, the potential inside the sphere is constant and equals KQ/R, where Q is the total charge of the...
  23. D

    Calculating the average linear speed of all points in a rotating sphere

    Is there a way to calculate the average linear speed of all points in the volume of a sphere rotating on a single axis? Since points closer to the axis of rotation and the poles move slower than points further out, would the average speed be a simple function of r/2 and pi? It would seem that...
  24. Heyxyz

    Moment of Inertia: Car vs Sphere

    Hello, I tried to put it in an equation, but it didn't really work out. In this situation, the car was about the size of a model, and, while not exact, the radius of each wheel couldn't have been more than like a centimeter. Conversely, the ball was like twice the size of the car and had a...
  25. M

    The sphere in general relativity

    I'm a bit confused about the notation used in the exercise statement, but if I'm not misunderstanding we have $$\begin{align*}(\psi^+_1)^{-1}:\begin{array}{rcl} \{\lambda^1,\lambda^2\in [a,b]\mid (\lambda^1)^2+(\lambda^2)^2<1\}&\longrightarrow& \{\pm x_1>0\}\subset \mathbb{S}^2\\...
  26. Arman777

    Electrostatic Energy of a solid sphere with a cavity

    I tried to use ##W = ε_0/2 \int E^2d\tau## for all space. So I find that ##E = \frac{(R^3 - b^3)\rho}{3ε_0r^2}## where ##\rho## is the charge denisty. So from here when I plug the equation I get something like $$W = \frac{(R^3 - b^3)^2\rho^2 4 \ pi}{18ε_0} \int_{?}^{\inf}1/r^2dr$$ Is this...
  27. E

    Line charge inside a conducting sphere?

  28. S

    Force on a sphere due to a conducting plate

    I tried to find the the Electric field due to the image charge. So the potential due to the image charge is V=-(pR^2)/√(4R^2-4rRcos(θ)+r^2). When I took the gradient of that in spherical coordinates, I got a mess that doesn't seem to be possible to integrate.
  29. A

    Work of gravity on an object sliding down a frictionless sphere

    Homework Statement: A perfect hemisphere of frictionless ice has radius R=6.5 meters. Sitting on the top of the ice, motionless, is a box of mass m=6 kg. The box starts to slide to the right, down the sloping surface of the ice. After it has moved by an angle 20 degrees from the top, how much...
  30. S

    [E&M] Question on the Image charge method of a grounding sphere

    So I was reading Jackson's discussion on Image charge method of a grounding sphere. He first assumed an image charge q inside Sphere with radius a, so the potential for real change and image charge is . The by set potential equal to 0 at x=a, he solved q' and y' Then he can get potential...
  31. sdefresco

    Electrodynamics: Conducting sphere of radius R cut in half

    Summary: Electrodynamics: Conducting Sphere cut in half to form a gap, and a charge q is placed on the first half-sphere. Find all four σ. A sphere of radius R is cut in half to form a gap of s << R (ignore edge effects) - the first hemisphere is charged with q, and the second hemisphere is left...
  32. snatchingthepi

    Quadrupole term for uniformly charged sphere (where did I go wrong?)

    So I got an assignment returned to me with fewer marks than I had expected. One part in particular is confusing to me. The professor is only available on Monday for a tutorial, but I'd like to see what is wrong before then. Can anyone spot why this is incorrect?
  33. bob012345

    A What is the solution for two electrons on a sphere?

    The Quantum Mechanical solution for a particle on a sphere is well known. I'm looking for a treatment of two particles on a sphere where both particles are electrons. I assume it's analytically solvable. Of course, I am not expecting someone to actually solve it from scratch (unless you want to)...
  34. Celso

    I Curve Inside a Sphere: Differentiating Alpha

    Honestly I don't know where to begin. I started differentiating alpha trying to show that its absolute value is constant, but the equation got complicated and didn't seem right.
  35. F

    Why is the magnitude of the electric field in a sphere the same?

    I was looking at a sphere that has a positive point charge at the center of a sphere with radius R. Now, I understand that the electric field is pointing outwards (in the direction of dA), so $$d\phi = EdA$$ However, I am told that since the magnitude electrical field is the same because the...
  36. Kaushik

    Moment of Inertia of a hollow sphere with Mass 'M' and radius 'R'.

    Homework Statement: Derive the formula for moment of inertia of a hollow sphere. Homework Equations: Required answer ##\frac{2MR^2}{3}## Consider a Hollow sphere. At an angle ##Θ## with the vertical, consider a circular ring whose moment of inertia is given by ##MR^2##. The most basic...
  37. K

    I Locally Cartesian Coordinates on the Sphere

    I was trying to construct locally Euclidean metrics. Consider the sphere with the usual coordinate system induced from spherical coordinates in ##\mathbb R^3##. Consider a point ##p## in the Equator having coordinates ##(\theta_0, \phi_0) = (\pi/2, 0)##. If you make the coordinate change ##\xi^1...
  38. George Keeling

    I Question about geodesics on a sphere

    I am working from Sean Carroll's Spacetime and Geometry : An Introduction to General Relativity and have got to the geodesic equation. I wanted to test it on the surface of a sphere where I know that great circles are geodesics and is about the simplest non-trivial case I can think of. Carroll...
  39. George Keeling

    I Question about a partial derivative

    I apologise for the length of this question. It is probably possible to answer it by reading the first few lines. I fear I have made a childish error: I am working on the geodesic equation for the surface of a sphere. While doing so I come across the partial derivative \begin{align}...
  40. Kaushik

    Does the block move along the pink dotted lines

    Does the block move along the pink dotted lines as attached in the figure below? I tried to draw the FBD of the small block ##m ## at the lowermost point which is also attached below.(The direction of ## v_0 ## is actually tangential) Is the figure above correct? If not, why?
  41. elisagroup

    Capacitance of an isolated sphere - solid vs hollow

    Seems the physics books agree that there is no difference in capacitance whether an isolated sphere is solid or hollow. And the reason mentioned for that always sounds something like the following: "The reason that the capacitance C, and hence the charge Q, is not affected by whether or not the...
  42. G

    On the motion of a particle on the inner surface of a sphere

    Could anyone please help me out with the second part of this question: I've got the first part, u = √(5ga) Here's my diagram for the second part: Distance traveled is from bottom of sphere to peg is 2πa/3 which means angle traveled is 2π/3. So the particlee is going to travel 2π/3 radians...
  43. N

    Current density inside a sphere

    Since sphere is made of l.i.h material, $$\vec{J_f}= \sigma \vec{E}$$ We compute electric field E using $$\vec{E} = -\nabla V$$ $$= -\nabla \left(V_0cos\theta\right)$$ $$= -\frac{\hat\theta}{r}\frac{{\partial}}{{\partial\theta}}\left(V_0cos\theta\right)$$ $$\vec{E}=...
  44. confusedhome

    I Amount of energy inside a sphere of radius 1au around the Sun

    Suppose a sphere were to be placed around our Sun with a radius equal to the radius of the orbit of Earth, it would have a volume of 6.266x1022 m3. At any instant in time there would be 3.846x1027 Watts of energy within this volume in the form of photons from the Sun. Since...
  45. Lone Wolf

    What Is the Effect of Friction on the Height of a Sphere Rolling Down a Ramp?

    Figure: a) The mechanical energy of the sphere is conserved because the weight is the only force which does work. My problem with this question is mostly because the original picture (which I tried to recreate here) is kind of ambiguous, as in I don't know if H already accounts for the radius...
  46. J

    Understanding the physics behind an elastic sphere

    First off, I'm not a scientist or engineer and I apologize if I don't give a clear description of my question. I'm beginning a personal project and was hoping for some knowledge and assistance. What I'm trying to achieve is to have a spherical object (it will be at least twice as wide as it is...
  47. JD_PM

    Why a charged sphere whose radius oscillates in and out won't radiate?

    In chapter 10 (Radiation; just after example 2 from 'Radiation from an arbitrary source') of Introduction to Electrodynamics by G. Griffiths he asserts that a charged sphere with oscillating radius doesn't radiate because, by Gauss law, ##E## stays the same no matter where the charges are...
  48. J

    I Finding a linear combination to enter a sphere

    Let's say we have n vectors in ℝ3. And say we have defined a subspace inside ℝ3 in the form of a sphere with radius r, and the center of the spheare is at P, where P is a vector in ℝ3. What methods exists to find any linear combination of the n vectors, so that the sum of all of them, lies...
  49. migueldbg

    Potential from point charge at distance ##l## from conducting sphere

    After looking around a bit, I found that, considering the polar axis to be along the direction of the point charge as suggested by the exercise, the following Legendre polynomial expansion is true: $$\begin{equation}\frac{1}{|\mathbf{r} - \mathbf{r'}|} = \sum_{n=0}^\infty...
  50. brotherbobby

    The Indeterminacy of Problems with Multiple Friction Interfaces

    For convenience, I have marked the C.G. of the sphere as ##G_S## and that of the bar only as ##G_B##. The C.G. of the combination is halfway between them, shown by the green ##G_C##. My answer is that the combination will rotate about ##G_C## (centre of the bar) as it falls. I wonder if I am...
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