Spherical Definition and 1000 Threads

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. Dvorak

    Are Di-Atomic Molecules spherical?

    It is usually said that molecules are spherical in shape, that is what we learn from our textbooks. May be what they are saying is true but only in the case of a monatomic molecule. If one considers a diatomic molecule there are two atoms that means two spheres and if it is polyatomic there is...
  2. L

    Electrical fields of spherical shells

    Hello, I am in a calc 1 general physics 2 summer session class and missed the lectures on this due to sickness. I'm really confused on applying coulomb's and gauss's laws to find the electrical field of a sphere or outside a sphere. This is of both variable and constant charge densities. I've...
  3. sergiokapone

    Field inside spherical hole inside dielectric

    Let we have a dielectric with field ##E## inside and with a little hole. I have problem. I get a two different answers on this problem, and I try to understand which one of them correct. As mentioned in http://www.feynmanlectures.caltech.edu/II_11.html#Ch11-S4 (11.25), the electric field in...
  4. anorlunda

    Planetoid formation, spherical or not.

    I am interested in the minimum size of a rocky planetoid needed to "crush" it into spherical shape. I'm also interested in its initial temperature because liquid or plastic masses obviously need much less crushing. The Wikipedia article "Giant Impact Hypothesis" says, In 2007, researchers...
  5. Quandemonitum

    Mathematical proof of the spherical electron?

    Why is/are an electron/electrons a sphere/spheres ? Need the mathematical proof...
  6. P

    Why is ##x = r \sin{\phi} \cos{\theta}## in spherical coordinates?

    My question is really about converting between spherical coordinates and cartesian coordinates. Suppose that ##\phi## and ##\theta## are defined as follows: ##\phi## is the angle between the position vector of a point and the ##z##-axis. ##\theta## is the angle between the projection of that...
  7. R

    Magnetic Field in Spherical Coordinate

    Homework Statement I am trying to use the equation ##B_{dip} (r) = \nabla \times A## to find the magnetic field due to a dipole at the origin pointing in the z direction (where A is the magnetic vector potential). The correct answer should be: ##B_{dip} (r) = \frac{\mu_0 m}{4 \pi r^3} \ (2...
  8. Dethrone

    MHB Confused by Spherical Wedge Graphing on z-y Plane

    Part (a) is easy to do by setting up a triple integral, but for part (b), I was a bit confused by the diagram provided by the solutions manual: Why is the spherical wedge (shaded) graphed on the z-y axis? In the most general case, shouldn't the two lines that form angle $\phi_1$ and $\phi_2$...
  9. E

    Time derivative of 3D Spherical Coordinate

    When we obtain the velocity vector for position vector (r, θ, φ) Why do we take the time derivative of the radial part in the 3D Spherical Coordinate system only? Don't we need to consider the polar angle and azimuthal angle part like (dr/dt, dθ/dt, dφ/dt)?
  10. w3dnesday

    Where is the Force on a Proton 0.79FR Relative to the Surface of a Charged Ball?

    Homework Statement Assume that a ball of charged particles has a uniformly distributed negative charge density except for a narrow radial tunnel through its center, from the surface on one side to the surface on the opposite side. Also assume that we can position a proton anywhere along the...
  11. PWiz

    Infinitesimal displacement in spherical coordinates

    I'm trying to derive what ##ds^2## equals to in spherical coordinates. In Euclidean space, $$ds^2= dx^2+dy^2+dz^2$$ Where ##x=r \ cos\theta \ sin\phi## , ##y=r \ sin\theta \ sin\phi## , ##z=r \ cos\phi## (I'm using ##\phi## for the polar angle) For simplicity, let ##cos...
  12. C

    Charge staggered over spherical volume.

    Homework Statement There is charge placed in a volume of a sphere, whose density changes by expression ρ(r)=ρ0a/r for 0<r≤a and ρ(0)=0. Where a and ρ0 are known variables , and r is a distance from the origin. Determine the potential of the point A(0,0,0) with regard to reference point at...
  13. Breo

    Why to change from momentum space integrals to spherical coordinate ones?

    So I was asked to compute loop contributions to the Higgs and compute the integrals in spherical coordinates, I gave a look to Halzen book but did not found anything. Why, when and how to make that change?
  14. H

    Triple integral in spherical coordinates

    Homework Statement Evaluate \int \int \int _R (x^2+y^2+z^2)dV where R is the cylinder 0\leq x^2+y^2\leq a^2, 0\leq z\leq h Homework Equations [/B] x = Rsin\phi cos\theta y = Rsin\phi sin\theta z = Rcos\phiThe Attempt at a Solution [/B] 2*\int_{0}^{\pi/2}d\phi \int_{0}^{2\pi}d\theta...
  15. Alettix

    What Is the Potential of a Spherical Shell with a Grounded Inner Sphere?

    Hi! I have trouble with solving this problem and would be really thankful for some help. :) 1. Homework Statement Inside a thin, spherical metal-shell with a radius of 50 cm, a smaller homogenous metal-sphere with a radius of 20 cm is placed concentrically. The smal sphere is grounded through...
  16. S

    Capacitance of an isolated spherical conductor

    So it says here that a conducting sphere of radius R with a charge Q uniformly distributed over its surface has V = Q/4πεR , using infinity as the reference point having zero potential,,V (∞) = 0. This gives C = Q/|ΔV| = Q/(Q/4πεR)=4πεR. Does ,V (∞) mean that you are taking the potential of a...
  17. C

    CMB , Spherical Harmonics and Rotational Invariance

    In Dodelson's "Modern Cosmology" on p.241 he states that the ##a_{lm}##-s -- for a given ##l##-- corresponding to a spherical harmonic expansion of the photon-temperature fluctuations, are drawn from the same probability distribution regardless of the value of ##m##. Dodelson does not explain...
  18. gfd43tg

    Bound states in finite spherical well

    Homework Statement Homework EquationsThe Attempt at a Solution for ##r \le a## and ##l = 0##, the radial equation is $$- \frac {\hbar^{2}}{2m} \frac {d^{2}u}{dr^{2}} - V_{0} = Eu $$ $$- \frac {\hbar^{2}}{2m} \frac {d^{2}u}{dr^{2}} - [V_{0} + E]u = 0$$ call ##k^{2} = \frac...
  19. A

    Why are spherical instruments not more common?

    When it comes to waves, spherical harmonics are, like, da bomb. I'm no expert - probably obvious from the question - but it seem to me that an instrument which maximises the utilisation of harmonics/resonances would be spherical. And yet, I can think of no spherical instruments - the most...
  20. M

    MHB Integrating (triple) over spherical coordinates

    Hi, Set up the triple integral in spherical coordinates to find the volume bounded by z = \sqrt{4-x^2-y^2}, z=\sqrt{1-x^2-y^2}, where x \ge 0 and y \ge 0. \int_0^{2\pi} \int_0^2 \int_{-\sqrt{4-x^2-y^2}}^{\sqrt{4-x^2-y^2}} r\ dz\ dr\ d\theta
  21. brianeyes88677

    Charge on two concentric spherical shells

    There are two spherical shells in different sizes and they are concentric. Now if I connect a battery to the two spheres (connect the negative pole to the smaller sphere and connect the positive pole to the bigger sphere). Will this system become stable? Or is there any situation for the charges...
  22. Hardik Batra

    Electric flux linked with spherical shell having a hole?

    How much electric flux linked with the spherical shell having a hole?(consider the charge is outside the shell) I knew the flux linked with spherical shell is zero.(because it is closed loop.)
  23. YogiBear

    Show that w is solenoidal having spherical polar coordinates

    Homework Statement The gradient, divergence and curl in spherical polar coordinates r, ∅, Ψ are nablaΨ = ∂Φ/∂r * er + ∂Φ/∂∅ * e∅ 1/r + ∂Φ/∂Ψ * eΨ * 1/(r*sin(∅)) nabla * a = 1/r * ∂/∂r(r2*ar) + 1/(r*sin(∅)*∂/∂∅[sin(∅)a∅] + 1/r*sin(∅) * ∂aΨ/∂Ψ nabla x a = |er r*e∅ r*sin(∅)*eΨ | |∂/∂r ∂/∂∅...
  24. K

    Curl of a field in spherical polar coordinates

    Homework Statement I have a field w=wφ(r,θ)eφ^ (e^ is supposed to be 'e hat', a unit vector) Find wφ(r,θ) given the curl is zero and find a potential for w. Homework Equations I can't type the matrix for curl in curvilinear, don't even know where to start! I've been given it in the form...
  25. brianeyes88677

    Charge on two concentric spherical shells

    There are two spherical shells in different sizes and they are concentric. Now if I connect a battery to the two spheres (connect the negative pole to the smaller sphere and connect the positive pole to the bigger sphere). Will this system become stable? Is there any situation for the charges on...
  26. H

    Solving Laplace's equation in spherical coordinates

    The angular equation: ##\frac{d}{d\theta}(\sin\theta\,\frac{d\Theta}{d\theta})=-l(l+1)\sin\theta\,\Theta## Right now, ##l## can be any number. The solutions are Legendre polynomials in the variable ##\cos\theta##: ##\Theta(\theta)=P_l(\cos\theta)##, where ##l## is a non-negative integer...
  27. M

    Capacitance of a spherical capacitor?

    1-capacitance of sphere with radius r : 4πεr 2-capacitance of 2 concentric spheres with inner radius a and outer radius b : 4πε(a.b/b-a) 3-when the outer the outer sphere is earthed it gives the same capacitance as in number 2 4-when the inner sphere is earthed it gives the sum of capacitances...
  28. J

    How Do Particles Leak from a Container and Reach a Spherical Cap?

    Homework Statement An ideal gas satisfying the Maxwell-Boltzmann distribution is leaking from a container of the volume V through a circular hole of area A'. The gas is kept in the container under pressure P and temperature T. The initial number density (concentration) is given by n0=N/V...
  29. U

    Finding limits of integral in spherical coordinates

    Homework Statement The question asks me to convert the following integral to spherical coordinates and to solve it Homework EquationsThe Attempt at a Solution just the notations θ = theta and ∅= phi dx dy dz = r2 sinθ dr dθ d∅ r2 sinθ being the jacobian and eventually solving gets me ∫ ∫ ∫...
  30. S

    Physics principle(s) to explain how to stack spherical items

    I need to think of all the physics principles to explain how one can stack spherical items (ex. baseballs) on top of each other. So far I've thought of one. 1. Newton's third law In this case, the reaction is the normal force in each baseball that is stacked and the action is the force of...
  31. H

    Derive grad T in spherical coordinates

    Homework Statement ##x=r\sin\theta\cos\phi,\,\,\,\,\,y=r\sin\theta\sin\phi,\,\,\,\,\,z=r\cos\theta## ##\hat{x}=\sin\theta\cos\phi\,\hat{r}+\cos\theta\cos\phi\,\hat{\theta}-\sin\phi\,\hat{\phi}## ##\hat{y}=\sin\theta\sin\phi\,\hat{r}+\cos\theta\sin\phi\,\hat{\theta}+\cos\phi\,\hat{\phi}##...
  32. S

    Can I Find an Analytical Solution for Spherical Trilateration?

    Hi I am looking for an equation of intersection of 3 circles or 3 spheres, on the surface of the fourth (central) sphere, in a spherical coordinate circle. This should really be just a simple trilateration problem. I know this is usually done by transforming the spherical coordinate system to...
  33. beer

    Refesher Part Duex - Rectangular -> Cylindrical -> Spherical

    I'm at it again today. Helping the same friend (plus another) study for their final calculus exams; it's a good refresher for me as well. (I'm a senior industrial engineering major so I'm "done" with calculus, and it isn't used much in our upper level classes nor professionally to the best of...
  34. Y

    Ball roll down from the top of a rough spherical dome

    Homework Statement A ball is rolling down from the top of a rough spherical dome with negligible initial velocity and angular velocity. Show that the ball must slide before losing the contact with the dome. Homework Equations ΣF=ma Στ = Fr = Iα fs = μsN vcm = rω Δmgh = 0.5mvcm2 + 0.5Icmω2 I =...
  35. carllacan

    How to Find Electrostatic Energy in a Spherical Capacitor?

    Homework Statement The space between two spherical shells kept at potentials V1 and V2, respectively, is filled with a dielectric medium. Find the electrostatic energy on the medium. Homework EquationsThe Attempt at a Solution I know how to get the energy if I am given the electric field or...
  36. Rumo

    Lorentz transf. of a spherical wave in Euclidean space

    This thread is not about the lorentz invariance of the wave equation: \frac{1}{c^2}\frac{\partial^2\Phi}{\partial t^2}-\Delta \Phi = 0 It is about an interesting feature of a standing spherical wave: A\frac{\sin(kr)}{r}\cos(wt) It still solves the wave equation above, when it is boosted in...
  37. Estefania_8

    Potential of a Charged Spherical Shell

    Homework Statement A hollow spherical conductor, carrying a net charge +Q = 47 pC, has inner radius r1 = 5.9 cm and outer radius r2 = 11.9 cm. At the center of the sphere is a point charge +Q/2. a. Find the potential at r = 18.0 cm. b. Find the potential at r = 10.0 cm. c. Find the potential...
  38. Raman Choudhary

    Superposition of spherical harmonics

    i am a beginner and was going through (Donald Mcquarie's "quantum chemistry" ) some discussion regarding orbitals of H-atom but i didn't get the logic behind writing px and py orbitals as linear combinations of spherical harmonics? according to what i understood, a given spherical harmonic in...
  39. RJLiberator

    Spherical Coordinates - Help me find my bounds

    Homework Statement A vase is filled to the top with water of uniform density f = 1. The side profile of the barrel is given by the surface of revolution obtained by revolving the graph of g(z) = 2 + cos(z) over the z-axis, and bounded by 0 ≤ z ≤ π. Find the mass of the vase. Homework...
  40. Calpalned

    Spherical coordinates - phi vs theta

    Homework Statement My textbook states that when ##\theta = c ## where c is the constant angle with respect to the x-axis, the graph is a "half-plane". However, when ##\phi = c ## it is a half-cone. The only difference I see is that ##\phi## is the angle with respect to the z-axis, rather than...
  41. Calpalned

    Spherical vs cylindrical notation

    Homework Statement Plotting a point in spherical coordinates means using the format ##(\rho, \theta, \phi)## in place of ##(x, y, z)##. Taking a triple integral means replacing ##dV## with ##\rho ^2 sin(\phi) d\rho d\theta d\phi ## As you can see, ##\rho, \theta, \phi ## are all in the same...
  42. H

    Spherical Harmonics: Degree l & Order m Structure & Variation

    I am studying the Earths main magnetic field (internal, specifically the stuff at the Core-Mantle boundary) which has led me to spherical harmonics. I am curious... how is the structure of a spherical harmonic determined by its degree l and order m? What role do the first three coefficients...
  43. gfd43tg

    What do the color maps in spherical harmonics represent?

    Hello, I am watching a video about spherical harmonics, and I am at the point where the color map is being shown for various values of ##l## and ##m## My question is, what am I supposed to make of these plots? Pretty colors yes, but what do these things mean?
  44. S

    Spherical balloon related rates problem

    Homework Statement You are blowing air into a balloon at a rate of 4*pi/3 cubic inches per second. (The reason for this strange-looking rate is that it will simplify your algebra a little bit.) Assume the radius of your balloon is zero at time zero. Let r(t), A(t) and V(t) denote the radius...
  45. F

    Electric field in a spherical capacitor.

    Homework Statement The potential difference Δϕ between the plates of a spherical capacitor is kept constant. Show that then the electric field at the surface of the inner sphere will be a minimum if a=(1/2)b, find that minimum. Where: a=radius of the inner sphere, b=radius of the outside...
  46. M

    MHB Spherical coordinates - Orthonormal system

    Hey! :o Using spherical coordinates and the orthonormal system of vectors $\overrightarrow{e}_{\rho}, \overrightarrow{e}_{\theta}, \overrightarrow{e}_{\phi}$ describe each of the $\overrightarrow{e}_{\rho}$, $\overrightarrow{e}_{\theta}$ and $\overrightarrow{e}_{\phi}$ as a function of...
  47. Destroxia

    Capacitance of a Spherical Capacitor?

    Homework Statement The plates of a spherical capacitor have radii 51.8 mm and 55.0 mm. (a) Calculate the capacitance. (b) What must be the plate area of a parallel-plate capacitor with the same plate separation and capacitance? (givens) a = 51.8 mm b = 55.0 mm d = 3.2 mm ε0 = 8.85E-12...
  48. T

    Charge on spherical conducting shells

    Homework Statement [/B] There are two spherical conducting shells one inside the other. If the total charge on the inside shell is -3q and the total charge on the outside shell is +5q what's is the charge on the inner and outer surface of each shell? Attempt at solution My teacher said i...
  49. samjohnny

    Surface integral in Spherical Polar

    Homework Statement Attached. Homework Equations The Attempt at a Solution Hi, Ok, so for the first part of this question it asks to evaluate the integral of the dot product between A and dS. The magnitude of dS is as shown above, and it is in the radial direction in spherical polar...
  50. streeters

    Height of a spherical cap with known volume?

    Homework Statement What equation is needed to calculate the height of a spherical cap with a fixed volume and radius? Homework Equations V=πh/6 (3a^2 + h^2) Where V = volume, h is cap height, a is cap radius The Attempt at a Solution I have tried to separate the h out and got as far as...
Back
Top