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. RodolfoM

    Electric potential inside a hollow sphere with non-uniform charge

    I tried to find the charge distribution using the given potential but couldn't produce the correct result. Also, Gauss's Law doesn't help, as the electric flux is 0 but we don't have any symmetry. Can someone please shine a light on this? Thanks in advance..
  2. P

    Moment of Inertia of a sphere about an axis

    I = 2/5M R^2 + Md^2 This is analagous to Earth's movement about the Sun. Is the moment of inertia of Earth about the centre of mass of the Earth Sun system = 2/5MR^2 + Md^2, where: M = Mass of earth, R = Radius of Earth, d = distance from Earth to centre of mass of earth-sun system.
  3. E

    Motion of a sphere in a liquid

    in my head this is just a silly problem in which i need to determine the ↓ force (weight) and the ↑force (archimedes bouyant force) and then the difference must be the drag force ↑ (the force that involves velocity) but i can't get any sense out of this answer how is possible for the sphere to...
  4. J

    I Is the Surface of a Sphere Locally Flat?

    Given a certain manifold in ##R^3## I've been told that at every location ##p## it is possible to encounter a reference frame from which the metric is the euclidean at zero order from that point and its first correction is of second order. This, nevertheless does not match with the following...
  5. Eagle9

    So, can we see a cube instead of sphere?

    <mentor - Epistemology (how do we know or perceive) is not subject PF supports in the science forums - moved to General Discussion> Since my previous topic The arrangement of our visual system and the objective truth was closed I will open a new one, less philosophical. So, just imagine that...
  6. S

    I Exploring Life & Technology In A Red Giant Sun's Sphere

    I'm not sure this a strictly Astronomy question; perhaps it should go in Aerospace Engineering. I have always thought that Dyson's work was more of an Astronomy topic, although admittedly, from the POV of observing some other system's sphere. In any case, I was thinking about future...
  7. AN630078

    Rates of change: surface area and volume of a sphere

    The surface area of the sphere is 4πr^2. dr/dt is given as 3cm^-1. dS/dt=dS/dr*dr/dt Differentiating 4πr^2 is dS/dr= 8πr dS/dt=8πr*3 dS/dt=24πr Given that r=5 dS/dt=24π*5=120 π The volume of the sphere is 4/3πr^3, differentiating which is dV/dr=4πr^2 dV/dt=dV/dr*dr/dt dV/dt= 4πr^2*3...
  8. J

    Radiation - finding emissivity of a sphere

    So using the above equation, e=dQ/dt / (A*5.67E-8*303.8^4) The surface area of a sphere is 4(pi)r^2 and I get 136.8478 m^2. dQ/dt would be the net radiation (I think? Its in the correct units), 1074W. Plugging everything in I get 0.01625, but the answer is 0.0524. Now as I was writing this I...
  9. H

    Comparing the kinetic energies among a solid sphere, a cylinder and a hoop

    I did the question as attached, so I think A and D are correct but the given answer is E. Where am I wrong? Thanks.
  10. P

    Potential energy due to an external charge and a grounded sphere

    Let us attempt part C first, which is to find the total energy of the entire system. I can definitely find an expression for the force, as given by Coulomb's Law. However, why should I integrate this force from infinity to d, where d is the distance of the external charge to the centre of the...
  11. P

    Why is the pressure on a charged sphere only exerted on the contact area?

    The force per unit area (Pressure) on a part of the sphere is given by F = (E outside + E inside)/2 * Q = 0.5 (kQ/R^2) * (Q/ 4piR^2) = (Q^2/ 32pi^2 e0 R^4). I understand the above line. The solution then says this pressure is exerted on the contact area between the 2 spheres, as given by...
  12. P

    Why is static friction not considered in the work-energy theorem?

    My question is this: - Friction exists (for no slipping/pure rolling to occur) - Why is the work done against friction not accounted for in the conservation of energy equation? Thank you!
  13. A

    Capacitance of a Sphere: A Quick Calculation Method

    Could anyone tell me the capacitance of a one food diameter metal sphere. I know that this is a one terminal component but it still should have a capacitance. If a charge of Q is small and a 6 inch radius is drawn about the point then that 6 inch radius ( 12 in diameter ) should have a...
  14. A

    Point charges placed inside a charged sphere

    I traced a spherical X-ray Gaussian (green) where the negative charges were diametrically opposite. My question is this: I can transform the entire charge of the Gaussian sphere into a point charge placed in the center. So, can I analyze only the electrical forces of the two negative charges...
  15. H

    Potential difference between the center of a sphere and a point 4.0 cm away?

    I can use integral as attached to obtain the answer. However, I wonder why I can not use " V = KQ / r " the get the correct answer. Thanks.
  16. brotherbobby

    Intersection of a few surfaces

    Summary:: Describe what the intersection of the following surfaces - one on one - would look like? Cone, sphere and plane. My answers : (1) A cone intersects a sphere forming a circle. (2) A sphere intersects a plane forming a circle. (3) A plane intersects a cone forming (a pair of?)...
  17. A

    Electric Field from Non-Uniformly Polarized Sphere

    Solving for the volume and surface bound charge densities was easy using equations 1) and 2). The polarization only has an r component so ##ρ_b=-\frac 1 {r^2} \frac {d} {dr} (r^2 \vec P)=-α(n+2)r^{n-1}##, and ##\hat n=\hat r## so ##σ_b=αa^n##. To find ##\vec E## I intend to use equation 3)...
  18. .Scott

    I The Sub Photon Sphere Escape Game

    The Sub Photon Sphere Escape (SPSE) Game Game Board: Vast Empty Space Game Pieces: ##\space## 1) A large perfect Schwarzschild black hole ##\space## 2) A Carrier/Trigger. This is a massless device that sets the Player Device into a selected position and velocity and then triggers it...
  19. LCSphysicist

    Potential inside a concentric sphere

    I am rather confused how to answer this (Please focus on "find the potential at the center"): I thought that would be a good idea try to answer this with the Poisson equation. $$\nabla \phi = - \frac{\rho}{\epsilon}$$ So that, since the eletric field inside a hollow spherical shell is zero, the...
  20. ezfzx

    Ball rolling within a rolling cylinder

    Cylinders rolling down inclines are a common demo. But how do you model the movement of a sphere rolling within a rolling cylinder? I teaching a physics class and this question came up and my dynamics math is a little rusty. But I haven't found anything like this in any book or online. There's...
  21. .Scott

    I Investigating The Photon Sphere

    I've spent well over two hours searching the web for two functions of the radius of a Schwarzschild BH. One would give me the escape velocity of the BH assuming a perfectly vertical trajectory (so it isn't a normal escape velocity). The second relates to the trajectory of a photon that is...
  22. E

    I Riemann Curvature Tensor on 2D Sphere: Surprising Results

    I have worked out (and then verified against some sources) that ##R^\theta_{\phi\theta\phi} = sin^2(\theta)##. The rest of the components are either zero or the same as ##R^\theta_{\phi\theta\phi} ## some with the sign flipped. I was surprised at this, because it implies that the curvature...
  23. cwill53

    Integrating Mass of a Hollow Sphere: Multivariable Calculus Explained

    I know some multivariable calculus, I just want someone to walk me through the integration deriving the mass element dM and the integration of thin rings composing the hollow sphere. It would also be nice if you could show me doing it one way using the solid angle and one way without using the...
  24. B

    Understanding the electric field of a sphere with a hole

    Here's an image. O and O' are the respective centers, a is the distance between them, r is the distance from the center of the sphere to P, and r' = r - a, the distance from O' to P. The approach (which I don't understnad) given is to use Gauss' Law and superposition, so that we calculate the...
  25. Haorong Wu

    Stress-energy tensor for a rotating sphere

    The answer with no details is given by First, I considered a spherical shell because I thought the velocities at different radius ##r## will be different and hence the four-momentum will be different, as well. Then, I writed down the linear momenta by $$\epsilon^{ijk} r_i p_j = L_k$$ with...
  26. G

    Polarization of a solid sphere of nonconducting material

    a) Just using the equations gives us: surface charge density = ## \rho_{\rho s} = kR^2 ## volume charge density = ## \rho_\rho = -4kR ## b) I am not sure here but the Q on the shell is the same as within. If that's the case we can use gauss law to find Q which I guess is the total charge. ##...
  27. Chris Miller

    B Shape of Hubble sphere at relativistic velocity

    Consider the Hubble horizon as the proper distance over which Hubble expansion equals c, so that you are in the center of a Hubble sphere with a radius of about 13.5 billion light-years. As you approach light speed in any direction, does the Hubble horizon draw closer in that direction due to...
  28. TheGreatDeadOne

    Using the Divergence Theorem on the surface of a sphere

    The integral that I have to solve is as follows: \oint_{s} \frac{1}{|r-r'|}da', \quad\text{ integrating with respect to r '}, integrating with respect to r' Then I apply the divergence theorem, resulting in: \iiint \limits _{v} \nabla \cdot \frac{1}{|r-r'|}dv' =...
  29. S

    Deriving relations for a hard sphere phase diagram

    Ornstein-Zernike states that ##h(r_{12}) = c(r_{12}) + \rho \int d\mathbf{r}_3 c(r_{13})h(r_{32})## which after a Fourier transform becomes ##\hat{C} (\mathbf{k}) = \frac{\hat{H}(\mathbf{k})}{1+\rho \hat{H}(\mathbf{k})}## However, I don't see how to simplify this to the second equation he has...
  30. E

    Volume Of Intersection Between Square Pyramid And Sphere

    I'm assuming the way to go about it is to integrate in spherical coordinates, but I have no idea what the bounds would be since the bottom edges of the square pyramid are some function of r, theta, and phi.
  31. agnimusayoti

    Potential inside a uniformly charged solid sphere

    Well, in this problem, I try to use $$d \tau '= \mu ^2 \sin {\theta} {d\mu} {d\theta} {d\phi}$$ With these domain integration: $$0<\mu<r$$ $$0<\theta<\pi$$ $$0<\phi<2\pi$$ , I get $$V=\frac{1}{4\pi \epsilon_0} \frac{3Qr^2}{2R^3}$$ This result is wrong because doesn't match with Prob 2.21, which...
  32. B

    Electric Potential inside an insulating sphere

    I used the potential at the surface of the sphere for my reference point for computing the potential at a point r < R in the sphere. The potential at the surface of the sphere is ## V(R) = k \frac {Q} {R} ##. To find the potential inside the sphere, I used the Electric field inside of an...
  33. E

    Drag force on a descending sphere

    i tried a force balance around the sphere but the weight of the displaced fluid is greater than the weight of the sphere which gives a net acceleration upwards and no terminal velocity but the book says that the terminal velocity has a certain value from there the exercise is meaningless to me...
  34. A

    Image in a glass sphere (Ray Optics)

  35. Tony Hau

    The potential of a sphere with opposite hemisphere charge densities

    Here is what the solution says: As usual, quote the general potential formula: $$V(r,\theta)=\sum_{l=0}^{\infty}(A_lr^l+\frac{B_l}{r^{l+1}})P_l(cos\theta)$$ The potential outside the sphere is: $$V(r,{\theta})=\sum_{l=0}^{\infty}\frac{B_l}{r^{l+1}}P_l(cos\theta)$$, which makes sense to me...
  36. P

    What is the relationship between pressure and force inside a spherical object?

    So I already have a solution available to this problem and the link for the solution is: I have understood everything in the video except the part where they are equating the force dF=GM/r²*dm According to my reasoning the inner part of the sphere (the part below the dm element we have taken)...
  37. preachingpirate24

    Electric Field inside the material of a hollow conducting sphere

    Let's say I place a positive point charge inside a hollow conducting sphere. If we take a Gaussian surface through the material of the conductor, we know the field inside the material of the conductor is 0, which implies that there is a -ve charge on the inner wall to make the net enclosed...
  38. brotherbobby

    Depth to which the sphere is submerged

    I draw a sketch of the diagram to the right. The body has a radius of ##r_B = 0.2## m. The volume of the body is given by ##V_B = \frac{4}{3}\times \pi \times 0.2^3 = 0.034\;\text{m}^3##. The mass of the body : ##w_B = \rho_B V_B = 400\times 0.034 = 13.37\,\text{kg}##. Hence the mass of the...
  39. R

    Applying Stokes' Theorem to the part of a Sphere Above a Plane

    I've tried a few ways of solving this, both directly and by using Stokes' Theorem. I may be messing up what the surface is in the first place F= r x (i + j+ k) = (y-z, z-x, x-y) Idea 1: Solve directly. So ∇ x F = (-2,-2,-2). I was unsure on which surface I could use for the normal vector...
  40. M

    Integrating ##\sigma=\chi\int{dA/A}## for a sphere

    I am trying to integrate ##\sigma=\chi\int\frac{dA}{A}## for a sphere. The answer is supposed to be ##\sigma(R)=\chi(R^2/R_0^2-1)##. The answer I keep getting is ##\sigma(R)=2\chi ln\frac{R}{R_0}##. I also tried doing it in spherical coordinates, and all I get for the integration of...
  41. rxh140630

    Drilling a hole through the center of a solid sphere

    The volume of the sphere = \frac{32{\pi}r^{3}}{3} The answer given at the back of the book is (\frac {32}{3} - 4\sqrt{3}){\pi}r^3 To drill a hole completely through the sphere, the hole would have to have a length of 4r. To get the answer in the back of the book, it requires setting the...
  42. Amitkumarr

    To measure the diameter of a sphere using a screw gauge

    Least count of the screw gauge = Pitch÷No. of divisions on circular scale=1.5÷100 mm =0.015mm According to me,in this case the main scale reading should be taken as 2 mm because it is the one which is visible and circular scale reading should be 76. So, Diameter=2 mm + 0.015×76 mm = 2 mm + 1.14...
  43. omar558

    What is the Electric Field Equation for an Electrified Sphere?

    I don't know if i can use V=E · d. I've read about gradient but i never used it so i was asking if there is another way to get tje electric field equation
  44. Nexus99

    Period and Velocity of Oscillating Sphere Attached to Spring

    A homogeneous sphere of mass M and radius R is at rest on a rough horizontal plane with coefficient of static friction μ . A spring of elastic constant k, is connected to the rotation axis of the sphere illustrated in the figure. The center of mass of the sphere is positioned at rest so that...
  45. Z

    What happens if you put a sphere (ball) on the top of a pyramid?

    I was wondering what happens if you put a perfect sphere (a ball) on the top of a perfect pyramid. To which side will the ball fall and why? It is random? An if it is, does a pattern emerge after many attempts?
  46. G

    Modulus of the electric field created by a sphere

    I think the right solution is c). I'll pass on my reasoning to you: R=6\, \textrm{cm}=0'06\, \textrm{m} \sigma =\dfrac{10}{\pi} \, \textrm{nC/m}^2=\dfrac{1\cdot 10^{-8}}{\pi}\, \textrm{C/m}^2 P=0'03\, \textrm{m} P'=10\, \textrm{cm}=0,1\, \textrm{m} Point P: \left. \phi =\oint E\cdot...
  47. P

    Electric field at the center of a sphere

    My first impression was the electric field is 0 at the center of the sphere, but it turned out not the case. My understanding when problems refer surface charge density, is that the charge exists only on the surface and it is hollow inside the sphere. Am i correct? Using the electric field...
  48. A

    Rotational Mechanics -- A solid sphere is rolled on a rough surface

    I found out the time when rotation ceases to be 4 ##v_0## /5*mew*g, where mew=coefficent of friction of surface but I am unable to plot the graph post that time
  49. M

    Work done by pushing a proton into the sphere with non-uniform charge

    I have already calculated full charge inside the sphere: e = ∫ρ dV = 2πBr^2 And I know that electric potential on the edge of the sphere is: U = e/ 4πεr The idea is that I calculate work by the change of electric potential energy, but to do that, I have to calculate electric potential energy in...
  50. Z

    B Would a Dyson sphere need a "relief valve?"

    Would a Dyson sphere need a "relief valve?"
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