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.

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  1. VVS2000

    I Spherical aberration in Biconvex and Plano Convex lenses

    I wanted to know about spherical aberration in a biconvex and plano convex lens as I was planning an experiment with them. I was reading about them and came upon the following passage. I don't know whether the given equation is an empirical one or a derived equation. Can anyone help me if you...
  2. A

    When to use the Jacobian in spherical coordinates?

    Greetings! here is the solution which I undertand very well: my question is: if we go the spherical coordinates shouldn't we use the jacobian r^2*sinv? thank you!
  3. V

    Potential on each of these concentric spherical shells

    Each spherical shell will contribute to potential on the surface of inner shell and the same will apply to outer shell. Due to inner shell ##V_1 = \frac {kQ} {{r_1}}## and due to outer shell ##V_1 = \frac {-kQ} {r_1}##. Therefore potential on inner surface is zero. But the answers are ##V_1...
  4. F

    Engineering Calculating the flux through the spherical surfaces at certain radius

    Relevant Equation: My attempt: Could someone please confirm my answer?
  5. P

    A Einstein Field Equations: Spherical Symmetry Solution

    [Moderator's Note: Thread spin off due to topic and level change.] For a spherically symmetric solution, if SET components were written in terms a single one of 4 coordinates, in a way plausible for a radial coordinate, the I believe solving the EFE would require spherical symmetry of the...
  6. C

    I Average field inside spherical shell of charge

    A known result is that the average field inside a sphere due to all the charges inside the sphere itself is proportional to the dipole momentum of the charge distribution (see, for example, here). I wonder whether the same result can be applied in the case of a spherical shell of non-uniform...
  7. Mayhem

    Deriving the Laplacian in spherical coordinates

    As a part of my self study, I am trying to derive the Laplacian in spherical coordinates to gain a deeper understanding of the mathematics of quantum mechanics. For reference, this the sphere I am using, where ##r## is constant and ##\theta = \theta (x,y, z), \phi = \phi(x,y)##. Given the...
  8. Danielle46

    I have to prove that vectors in spherical coordinates are clockwise

    I should use the cross product but I don´t know how. I tried to calculate it but it didn´t work out as expected. Please can you give me one example how to do it ?
  9. Shreya

    Variation of Electric Field at the centre of Spherical Shell

    My approach is thus: the shell will have induced charges if it's conducting resulting in E at the centre of shell(though flux at centre will be 0). For non conducting spheres there can be no induction only polarization of dipoles, therefore the E field at centre will remain 0. Is my approach...
  10. R

    Expanding potential in Legendre polynomials (or spherical harmonics)

    Using the generating function for the legendre polynomial: $$ \sum_{n=0}^{\infty} P_{n}(x) t^{n}=\frac{1}{\sqrt{1-2 x t+t^{2}}} $$ It's possible to expand the coulomb potential in a basis of legendre polynomials (or even spherical harmonic ) like this: $$ \begin{aligned} &\frac{1}{\left.\mid...
  11. redtree

    I Fourier transform of a function in spherical coordinates

    I am trying to understand the relationship between Fourier conjugates in the spherical basis. Thus for two functions ##f(\vec{x}_3)## and ##\hat{f}(\vec{k}_3)##, where \begin{equation} \begin{split} \hat{f}(\vec{k}_3) &= \int_{\mathbb{R}^3} e^{-2 \pi i \vec{k}_3 \cdot \vec{x}_3} f(\vec{x}_3...
  12. V

    Electric Field on the surface of charged conducting spherical shell

    When I look at the relevant equations, then there is no mention of field for a point on the surface of the shell, so it gets confusing. On the other hand, I feel the radial E will get stronger as we approach the surface of shell and magnitude of E will approach infinity.
  13. V

    Understanding Spherical Symmetry of Electron Clouds in External Fields

    The given diagram looks something like this: Electric force on nucleus from external field must balance attraction force from electron cloud and electric force from external field. $$e\vec{E}=\frac{k(\frac{L^3}{R^3}e)}{L^2}\hat{L}$$ where ##\vec{L}## is from center of electron cloud to...
  14. V

    Magnitude of electric field E on a concentric spherical shell

    The only explanation that I have seen in textbooks is that since the outer spherical shell is symmetrical relative to internal charged spherical shell so field every where on the outer shell is same in magnitude at every point on it. I can understand that electric field needs to be...
  15. jaumzaum

    B Why we use spherical mirrors instead of parabolic mirrors?

    Parabolas are the only geometrical shape in which we have a perfect focus (not an approximate one) and does not depend in the angle of incidence being small. So, why do we even build spherical mirrors and not only parabolic mirros?
  16. Eclair_de_XII

    Converting integration of rectangular integral to spherical.

    I'm going to type out my LaTeX solution later on. But in the meantime, can anyone check my work? I know it's sloppy, disorganized, and skips far more steps than I care to count, but I'd very much appreciate it. I'm not getting the answer as given in the book. I think I failed this time because I...
  17. D

    I Question about the vector cross product in spherical or cylindrical coordinates

    Hi If i calculate the vector product of a and b in cartesian coordinates i write it as a determinant with i , j , k in the top row. The 2nd row is the 3 components of a and the 3rd row is the components of b. Does this work for sphericals or cylindricals eg . can i put er , eθ , eφ in the top...
  18. E

    Scale factors in spherical coordinates

    how they got that value for the scale factors h?
  19. E

    I Precession of a spherical top in orbit around a rotating star

    Looking at L&L's solution to problem four of section §106. Lagrangian for a system of particles:\begin{align*} L = &\sum_a \frac{m_a' v_a^2}{2} \left( 1 + 3\sum_{b}' \frac{km_b}{c^2 r_{ab}} \right) + \sum_a \frac{m_a v_a^4}{8c^2} + \sum_a \sum_b' \frac{km_a m_b}{2r_{ab}} \\ &- \sum_a \sum_b'...
  20. A

    Divergence in Spherical Coordinate System by Metric Tensor

    The result equation doesn't fit with the familiar divergence form that are usually used in electrodynamics. I want to know the reason why I was wrong. My professor says about transformation of components. But I cannot close to answer by using this hint, because I don't have any idea about "x"...
  21. U

    Heat conduction from an isotherm spherical cap

    On the surface of a semi-infinite solid, a point heat source releases a power ##q##; apart from this, the surface of the solid is adiabatic. The heat melts the solid so that a molten pool forms and grows. Let's hypothesize that the pool temperature is homogeneously equal to the melting...
  22. Twigg

    Seismic ray-tracing, when does a spherical Earth matter in practice?

    I'm taking a geophysics class and the math makes sense but the context is lost on me. My understanding is that the primary use of seismic ray-tracing is to locate disturbances that cause waves to propagate radially. I also understand that 35km is the depth at which the Earth's spherical shape...
  23. J

    Potential Inside and Outside of a Charged Spherical Shell

    So here was my first go around at it: At first it made sense in my head but don't think my process is correct. Then i noticed the example in the book: I guess the reasoning isn't 100% there in my head and if i don't have an actual σ, how will i cancel out any legendre polynomials due to...
  24. G

    To calculate the centre of gravity of a spherical cap

    Could I please ask for help as to why I disagree with a book answer on the following question: Answer given is book is $$\frac{1}{2}(a+b)$$ Here's my proposed method: Prior to this question there is an example of a similar question: And here is the answer: So, to solve my question I...
  25. D

    Why smooth spherical waves with attenuation are only possible in 3-D

    Hi all, My question is about the attenuation and delay terms in part (1). what are attenuation and delay terms describing in physical phenomenon? thank you. What do "attenuation" and "delay" mean in terms of real-life physical phenomena? Consider the wave equation for spherical waves in...
  26. P

    Electrical energy stored by charged concentric spherical shells

    I thought up of this problem myself, so I do not have solutions. I would appreciate if you could correct my approach to solving this problem. Firstly, the charge induced on the inner surface of shell B is -q, and so the charge on the outer surface of shell B is Q+q. The energy stored can be...
  27. ScruffyNerf

    B Black Hole Observation: Outside Observer & Spherical Symmetry

    I know that for the infalling observer the horizon is a fake singularity that can be removed via the Eddington-Finkelstein co-ordinates but wouldn't the classic Swartsheild co-ordinates still apply for the outside observer? So, while for the infaller it takes a finite time, the outside...
  28. E

    Black body radiation -- Spherical shell surrounding a star

    I don't understand how this can be solved. The official solution was: F=\sigma*T^4 E=F*4\pi R^2*60*60 This doesn't make sense to me, as it seems to imply that the energy that the black body radiates depends on the radius of the shell. For a very large shell the body will reflect...
  29. D

    Find the wave function of a particle in a spherical cavity

    (a) Let the center of the concentric spheres be the origin at ##r=0##, where r is the radius defined in spherical coordinates. The potential is given by the piece-wise function $$V(r)=\infty, r<a$$ $$V(r)=0, a<r<R$$ $$V(r)=\infty, r<a$$ (b) we solve the Schrodinger equation and obtain...
  30. D

    Solve Spherical Harmonics: Y_{1,1} Eigenfunction of L^2 & L_z

    To show ##Y_{1,1}(\theta,\phi)## is an eigenfunction of ##\hat{L}^2## we operate on ##Y_{1,1}(\theta,\phi)## with ##\hat{L}^2## \begin{equation} \hat{L}^2Y_{1,1}(\theta,\phi)=\hat{L}^2\Big(-\sqrt{{\frac{3}{8\pi}}}sin\theta e^{i\phi}\Big) \end{equation} \begin{equation}...
  31. S

    Spherical components of a rotated operator

    The operator is the ##T_{xyz}## component of the rank 3 tensor ##T=\vec{r}\otimes\vec{r}\otimes\vec{r}## whose Cartesian components are ##T_{ijk}=r_ir_jr_k##. This tensor ##T## also has spherical components ##T_{q}^{(k)}## where ##k=0,1,2,3##, which in principle can be related to their Cartesian...
  32. T

    I On spherical geometry and its applications in physics

    Hello. Questions: do you know any applications of spherical geometry in physics? Are there any relations between spherical geometry and hyperbolic geometry? Why does Riemannian geometry use sphere theorems so much? Thank you.
  33. duchuy

    What is the formula for determining small angles in spherical diopters?

    Hi, I don't understand how the professor managed to determine the values of alpha, alpha' and omega. What is the formula tha´t is applied to determine alpha = SP / AS and so on... knowing that alpha is a really small angle. Cheers
  34. Ranku

    I Accelerating Universe and spherical distribution of matter

    If we take a spherical distribution of matter wherein gravitational force and cosmological-constant force are equal upon an object on its surface, then does the time that it took for that volume to grow to the size wherein the two forces are equal match the time it took for the universe to start...
  35. Mohomad

    Spherical charge distribution to generate this E-field

    A spherical volume charge (R<=1cm) with uniform density ρv0 is surrounded by a spherical surface charge ( R=2cm) with charge density 4 C/m2. If the electric field intensity at R=4cm is 5/Є0 ,deterime ρv0
  36. G

    Find the charge distribution from the given E-field (spherical)

    a) Static charge distribution should result in a static electric field? Legitimacy should be checked with curl of E = 0? b) Using the second equation should give is the answer?
  37. G

    Spherical shells (inner conducting and outer nonconducting)

    a) I think you find V by just integrating E in regards to R. Then we integrate from the point of interest, which is a, to the 0 potential which is (R = 2a)? b) If the same logic as a) applies here as well then we should integrate from the point of interest to the 0 potential. This should be...
  38. lmmoreira

    A Axisymmetric vs Spherical Metric Comparison

    Hello. I expect this question is not repeated. I look from it in the forum but I found nothing. I am confused on how an axisymmetric spacetime (generated by a rotating object) can manifest the spherically symmetric case. The axisymmetric spacetime should describe objects with any angular...
  39. F

    I Dot product in spherical coordinates

    I'm learing about antennas in a course, and we are using Jin's Electromagnetic text. This isn't a homework problem, I'm just trying to understand what I'm supposed to do in this situation. This part of the text discusses how to evaluate a radiation pattern. One of the steps to evaluate the...
  40. F

    What is happening to the sin(phi) factor in the spherical curl?

    This is from my E&M textbook. I'm doing a problem where I need to take the Curl in spherical coordinates but I'm getting the wrong answer. I tried applying the matrix, but it doesn't seem like it make sense with the expansion that they show in the textbook (screenshot below). If I apply the...
  41. P

    I Second Matrices from Spherical Harmonics with Eigenvalue l+1

    See the first post in the previous thread ‘Matrices from Spherical Harmonics with Eigenvalue l+1’ first. Originally when I came across the Lxyz operator and the Rlm matrices I had a different question. If this had to do with something like the quantum Hydrogen atom then why did it appear that...
  42. Athenian

    I Transforming Cartesian Coordinates in terms of Spherical Harmonics

    As the subject title states, I am wondering how would one go about transforming Cartesian coordinates in terms of spherical harmonics. To my understanding, cartesian coordinates can be transformed into spherical coordinates as shown below. $$x=\rho \sin \phi \cos \theta$$ $$y= \rho \sin \phi...
  43. LCSphysicist

    Energy of a configuration of two concentric spherical charged shells

    I found the total work done is: ##\frac{q^2}{8\pi \varepsilon a} + \frac{q^2}{8\pi \varepsilon b} + \epsilon \int E_{1}.E_{2} dv## The third is a little troublesome i think, but i separated into threeregions, inside the "inside" shell, between both shell and outside both. Inside => ##E_{1}.E_{2}...
  44. P

    A Matrices from Spherical Harmonics with Eigenvalue l+1

    I’m New to the forum. I’m Interested if a certain set of matrices have any significance. To start out the unit vectors ##\vec i , \vec j, and ~\vec k ## are replaced with two dimensional matrices. ##\sigma r = \begin{pmatrix}1&0\\0&1\\\end{pmatrix}, ~\sigma z = \begin{pmatrix}1&0\\...
  45. agnimusayoti

    Electric field a distance z from the center of a spherical surface

    Well, I really don't understand what is the use of the hint. I try to solve this problem with Coulomb's Law and try to do in spherical coordinates and got very messy infinitesimal field due to the charge of infinitesimal surface element of the sphere. Here what I got: $$\vec{r}=\vec{r_P} +...
  46. Another

    Why Is Velocity in Spherical Coordinates Given by This Equation?

    Why the velocity in spherical coordinates equal to ## v^2 = v \dot{} v = \dot{r}^2 + \dot{r}^2\dot{\theta}^2## maybe ## v^2 = v \dot{} v = (\hat{ \theta } \dot{ \theta } r +\hat{r} \dot{r} + \hat{ \phi } \dot{\phi } r \sin{ \theta}) \dot{} (\hat{ \theta } \dot{ \theta } r +\hat{r} \dot{r} +...
  47. aspodkfpo

    Change in radius over time for a spherical ball formula

    Algebra in this answer does not seem to flow right. Firstly, the 16, secondly the n term. Can someone explain or show me the right answer?
  48. curiosissimo

    Electric field in a spherical shell

    So for the Gaussian theorem we know that $$ \frac{Q}{e} = \vec E \cdot \vec S $$ Q's value is known so we don't need to express it as $$Q=(4/3)\pi*(R_2 ^3-R_1 ^3)*d$$ where d is the density of the charge in the volume. I've expressed the surface $$S=4\pi*x^2$$ where x is the distance of a point...
  49. Taz

    A Normalization of the radial part of the spherical harmonics

    Im trying to solve the equation 62.7 of this numerical on mathematica. Whenever i try to normalized the function it shows function diverges. As the Bessel function contains trigonometry term so it diverges. I don't know how to solve the integral. Can i use the hydrogen atom wavefunction in exp...
  50. S

    I How do I create a sewing pattern for a spherical lune in 2D from an n-hosohedra?

    I ultimately want to make a sewing pattern of a ball. If I have an n-hosohedra, how do I figure out the equation of the curves that make up each lune in 2D?
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