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. andyrk

    Earthing of a charged sphere

    When we Earth a positively charged sphere, the positive charge of the sphere vanishes. Why does this happen? When we connect two charged bodies doesn't charge redistribute till we get equipotential surface? Now when there is no charge on the sphere means the sphere has 0 potential. So that means...
  2. E

    Field of a grounded sphere - Scilab

    Homework Statement There is a grounded sphere of radius R in the origin of the coordinate system. In the distance L (L>R) from the sphere’s center there is a point charge Q. The electric field (both intensity and potential) should be computed in the area of radius rg = 5L (in the plane...
  3. O

    Average distance to surface of sphere

    Homework Statement First, the problem, quoted verbatim: "Neutrons are created (by a nuclear reaction) inside a hollow sphere of radius R. The newly created neutrons are uniformly distributed over the spherical volume. Assuming that all directions are equally probable (isotropy), what is...
  4. G

    Mie scattering for sphere with constant dipole moment

    Hi does anybody here know whether there already exists a theory that describe Mie scattering for spheres that have a constant dipole moment?
  5. T

    Electric field in a hollow charged sphere

    I am wondering about why the electric field in a hollow sphere with charges on its surface would be zero. I have thought about the gaussian law argument for it. It only guarantees that the net number of electric field lines that pass the encloved surface are zero. But it says nothing about...
  6. Ackbach

    MoI of a Sphere using Spherical Coordinates

    Homework Statement Calculate the moment of inertia of a uniformly distributed sphere about an axis through its center. Homework Equations I know that $$I= \frac{2}{5} M R^{2},$$ where ##M## is the mass and ##R## is the radius of the sphere. However, for some reason, when I do this...
  7. Astrum

    Electric Field of a Polarized Sphere with Nonuniform Polarization

    Homework Statement A sphere of radius R carries a polarization of \vec{P} = k \vec{r}, where k is a constant and \vec{r} is the vector from the center. Find the bopund charges, and the field inside and outside the sphere Homework Equations - \nabla \cdot \vec{P} \sigma _b = \vec{P} \cdot...
  8. G

    Greens function and the conducting sphere

    Can somebody explain to me, when equations 2.48 and 2.50 are applicable and what ##\Phi_s## and ##\Phi## actually are? The thing is, I want to find a general equations that determines the field produced by conducting spherical sphere in an external field and was wondering whether these are the...
  9. P

    Sphere has the minimum surface area?

    How do you prove that for a given volume, sphere has the minimum surface area?
  10. Crazymechanic

    Electric field , sphere , transformer.

    Hi , one of my electric field and how it works questions. My friend made a drawing that will help you visualize what i mean better. So we have a sphere made from wire or metal beams etc doesn't matter.Now we wrap wire around it and leave the ends separate, we attach a circuit of a resistor...
  11. R136a1

    Homotopic maps on a sphere

    If I take two arbitrary continuous maps ##f,g:S^n\rightarrow S^n## such that ##f(x) \neq -g(x)## for any ##x\in S^n##, then ##f## and ##g## are homotopic. How do I show this result? I really don't see how to use the condition that ##f## and ##g## never occupy two antipodal points. Any hint...
  12. mesa

    How to derive the surface area of a sphere?

    So, I have been trying to figure out how to derive the equation for the surface area of a sphere. All attempts have resulted in colossal failure and as such are not even worth posting on the forum. I know Archimedes was the first to come up with the formula but I have not been able to find...
  13. X

    How do you create a perfect sphere?

    the formula for the surface area of a sphere is SA = 4 (pi) r2, with pi = 22/7 and r = radius of the sphere. for example the SA for Earth with a radius of 6,378 km is 510,065,600 km2 what would the radius be in order that for you to lay a grid of perfect square on the surface of the sphere?
  14. Z

    Would a big enough manmade metal sphere have gravity?

    According to the theories that discuss gravity, if humans were to create a solid metal sphere with the same mass as Earth would it have the same gravity as earth? Would it have to be orbiting something and/or moving to have that same gravity? Or could it be stationary and have the same...
  15. 6

    Torque on a point on a sphere in a fluid/finding pressure?

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  16. Saitama

    Ray optics - Reflection from sphere

    Homework Statement A light ray parallel to the x-axis strikes the outer reflecting surface of a sphere at a point (2,2,0). Its center is at the point (0,0,-1). The unit vector along the direction of reflected ray is ##x\hat{i}+y\hat{j}+z\hat{k}##. Find the value of ##yz/x^2##. Homework...
  17. D

    What is the mass of the sphere?

    Homework Statement Two large , horizontal charged plates are separated by 0.050m. A small plstic sphere is stationary and suspended between them and is experiencing an electric force of 4.5x10^-15 N. What is the mass of the sphere? F=ma My attempt. F=4.5x10^-15N a=9.8m/s^2...
  18. F

    Current induced in a charged hollow sphere

    Homework Statement A hollow spherical conducting shell is suspended in air by an insulating string, so that the sphere is electrically isolated. The total charge on the conductor is -6 μC. If an additional point charge of +2μC is placed in the hollow region inside the shell, what is the total...
  19. Feodalherren

    Volume of a Sphere: Solve with Calculus & Integration

    Homework Statement Show that the volume of a sphere of radius r is V = (4/3)πr^2 Homework Equations calculus, integration The Attempt at a Solution I have the solution in the book but it's confusing me, I'll attach a picture. So I get lost where it starts talking about...
  20. C

    Diffusion of a species through a sphere

    Hi guys, This is my first post here, this place looks like a great resource. Well, jumping straight in, I have a couple of questions on diffusion. At work, I did a couple of experiments with Ion Exchange Resins. Not getting the results we wanted, my boss asked me to do an analysis of...
  21. K

    The length of a path on a sphere (in spherical coordinates)

    So, I'm to show that in spherical coordinates, the length of a given path on a sphere of radius R is given by: L= R\int_{\theta_1}^{\theta_2} \sqrt{1+\sin^2(\theta) \phi'^2(\theta)}d\theta, where it is assumed \phi(\theta), and start coordinates are (\theta_1,\phi_1) and (\theta_2, \phi_2)...
  22. H

    Sphere particle dissolution, volume loss vs time

    Hi all, it might be a silly question, but my math is "a bit" rusty. I want to find the equation of the volume lost during the dissolution of a sphere particle as function of time: A=area of sphere V=volume of sphere t=time k=dissolution rate as volume dissolved over area time...
  23. M

    Effects on water level when a sphere is replaced by a new solid sphere

    A small solid sphere of mass M0, of radius R0, and of uniform density ρ0 is placed in a large bowl containing water. It floats and the level of the water in the dish is L. Given the information below, determine the possible effects on the water level L, (R-Rises, F-Falls, U-Unchanged), when that...
  24. A

    Potential of ring with sphere inside it

    Homework Statement Hi, I'm trying to find the potential of conducting grounded sphere with radius Rs which located in the center of charged ring with Rr (>Rs) with charge density λ, h meters up to the z axis (see the attached images) Rs=4.3[cm] Rr=6.6[cm] h=13.1[cm] λ=1.0[esu/cm]...
  25. D

    Slipping Sphere on Steep Incline

    I'm trying to wrap my head around how a rolling and slipping sphere would be behave on an incline (with an angle of θ) that is too steep for pure rolling. I believe I understand the behaviour up to that point but once we reach the position where the amount of friction required to maintain...
  26. B

    Collision of a sphere into a cluster of spheres (billiard balls)

    Hello all, I am attempting to simulate collisions of pool balls. For the moment i am doing so without taking rotational momentum or friction between balls into account. right now i have a system that successfully simulates a collision between 1 ball and another ball. Here's how it basically...
  27. andyrk

    Oblique Impact of a smooth sphere against a fixed plane

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  28. A

    Dielectric Sphere in Uniform Field

    Hi, One of the boundary conditions when solving for the potential, \Phi, outside a dielectric sphere placed within a uniform electric field is \lim_{r→0}\Phi(r,θ)<\infty Can anyone explain/prove why this so. Thanks,
  29. J

    Optimization: maximum curved surface area of a cylinder in sphere

    θHomework Statement The attached diagram depicts a sphere with several variables: the height of the cylinder, the radius of the cylinder and an angle. All that has been given to me is the hypotenuse of a triangle used. Homework Equations To my knowledge I was told to use 2sinθcosθ=sin2θ...
  30. S

    Finding the volume of a sphere with a double integral

    Homework Statement I know how to find the volume of a sphere just by adding the areas of circles, so I decided to do a double integral to find the same volume, just for fun. Here's what I've set up. I put 8 out front and designed the integrals to find an eighth of a sphere that has its center...
  31. S

    Type of curvature of gradient force from edge to center of a sphere

    I was doing some simple physics with a ball resting on a table and I made this curve (0,0) (25, 6.8) (50, 27.51) (75, 63.4) (100, 112.34) (125, 175.7) (150, 253.3) (175, 345.4) I was wondering if anyone could identify what sort of curve it is? I am just curious. This is not a homework...
  32. R

    Steady state conduction in a hollow sphere

    Hello, I'm having trouble with a conduction problem, I have access to the answer but not the solution. I did it on my own and my value is half of what the answer is. Now, my calculus is a little rusty, but I don't know where I am going wrong. So the dimensions and temperatures of the sphere...
  33. A

    Classical Limit formula for differential cross section for Hard Sphere

    I am looking for the derivation to an approximation formula for the differential cross section for hard sphere scattering in the limit of high energy. The paper that mentioned this had referred to Methods of Theoretical Physics, PM Morse and H. Feshbach page 1484 but I have no access to the...
  34. J

    Check volume of sphere

    Homework Statement Check that the volume of a sphere is 4/3(pi)r^3 (use disk method) So I don't understand it I got stuck so I looked. I still don't get their solution. Chapter 7, section 2, question 59. http://www.calcchat.com/book/Calculus-ETF-5e/ Homework Equations The...
  35. T

    Work Done by an Insulating Sphere on a test charge

    Homework Statement An insulating sphere of radius 0.240 has uniform charge density 6.50×10−9 . A small object that can be treated as a point charge is released from rest just outside the surface of the sphere. The small object has positive charge 4.10×10−6 How much work does the electric...
  36. E

    Mass conservation for pulsating sphere

    Hi, I'm trying to understand the mass conservation equation for a pulsating sphere which has thickness dr. Please refer to the attached solution. \rho = \rho_{0} (ambient density) + \rho' (small deviation) There are two things I don't follow. First, is that to obtain the mass, the area of...
  37. Y

    Intersection of a sphere and plane

    Homework Statement Show that the circle that is the intersection of the plane x + y + z = 0 and the sphere x2 + y2 + z2 = 1 can be expressed as: x(t) = [cos(t)-sqrt(3)sin(t)]/sqrt(6) y(t) = [cos(t)+sqrt(3)sin(t)]/sqrt(6) z(t) = -[2cos(t)]/sqrt(6) Homework Equations The...
  38. A

    How to uniformly charge an insulating sphere?

    In my Physics book there was this problem of finding electric field produced by the sphere, such that electric charge is distributed uniformly throughout the volume of an insulating sphere. I know that excess charge tends to distribute itself on the surfaces, but since this sphere is made...
  39. E

    Rotating disk/ sphere and moving mass along it

    Coriolis effect - In a non-friction system, f I roll something along the surface of the planet from on of the poles to the equator, it will appear to move to the west, it will essentially stay behind the planets rotation and actually rotate it in the opposite direction. Now, if we add friction...
  40. Saitama

    Conducting sphere connected to the ground

    Homework Statement Homework Equations The Attempt at a Solution The ground is at zero potential. Hence, the sphere should be also at zero potential. The net charge should be zero, so a charge of -Q should flow from the ground to sphere. But this is wrong. :confused:
  41. B

    Why does everything in the Universe form a ball or Sphere?

    I want to know why everything in the universe forms a Ball or Sphere? Is gravity the cause of this? If so why? For example, If we were to pick the sphere apart can we see the source of gravity? Is gravity also a ball or sphere and can we grasp and see it? Why do we not ever see square planets or...
  42. P

    Volume of two pieces of a sphere cut by a plane

    Homework Statement Consider the unit sphere x^{2} + y^{2} + z^{2} = 1 Find the volume of the two pieces of the sphere when the sphere is cut by a plane at z=a.The Attempt at a Solution My interpretation is that a is a point on the z-axis that the plane cuts at. So the height of the segment...
  43. J

    Any one interesed in tasking on a formula for the volume of a sphere.

    An alternate one by the by. My approach will be working with a base ten logarithm function(s). I'm going to graph the sphere on a 3d graph. I'm going to give the value of the radius, one unit. I will first look at the ten elevations up and down from the index, establish the area at each...
  44. P

    Finding Radius of Curvature of a Sphere Using Angle Excess

    On the surface of a sphere, we can find the radius of cuvature of the sphere by: angle excess / area = 1/ r_s^2 http://en.wikipedia.org/w/index.php?title=Angle_excess&oldid=543583039 If we use triangles, for instance, the angle excess is the sum of the angles of the triangle minus 180...
  45. skate_nerd

    MHB Implicit surface area, sphere cut by two plane

    Got a little problem here, just want to make sure what I'm doing is right because it's a little different from anything I've done that is using the same formula initially. So there's a sphere denoted as \(\Omega\): $$x^2+y^2+z^2=R^2$$ and its cut by two planes \(z=a\) & \(z=b\) where...
  46. I

    Charge surrounded by charged conducting sphere

    A spherical conducting shell with charge total q surrounds a charge –2q at the center of the shell. The charges on the inner and outer surfaces of the shell are respectively I know that if the conducting sphere has no charge the inner and outer charges would be +2q, and -2q respectively but...
  47. K

    Calculating Sphere Velocity on a Frictionless Ramp

    Homework Statement As of the past few hours I've been trying to make sense one how to calculate the velocity of a sphere that moves down a frictionless ramp. My biggest problem with this seems to be that I'm confusing myself with the linear velocity and the angular velocity. Note that the...
  48. P

    Sphere surface area and radius

    I am curious about the relationship between an ever expanding sphere's radius and its surface area. how would I relate the rate of change in radius to the rate of change in the surface area.
  49. J

    How would shell theorem act in a hollow sphere?

    What I understand is this: Shell theorem states that the force of gravity is focused at the center of an object. But, say that there is a large planet with a gravitational force equal to that of earth's. It is perfectly round… and hollow. Since it is hollow, how large would it be to have Earth's...
  50. _maxim_

    Small sphere or cylinder filled by a solvent

    how can I create a spheric glass ball or a closed cylinder containing a solvent as acetone or methanol? the ideal would be without air bubble inside and diameter in the range of 2-4 mm thanx max
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