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. Delta2

    Potential energy of a sphere in the field of itself

    My attempt was to consider spherical shells of radius ##r## (##r\leq R##))and thickness ##dr## and then the potential energy of this shell would be in the field only of the "residual" sphere of radius ##r## (a result also known as shell theorem) $$U_{dr}=G\frac{\rho\frac{4}{3}\pi r^3 \rho 4\pi...
  2. Tesla In Person

    Flux of Electric field through sphere

    My attempt: We have 3 charges inside 2 +ve and 1 -ve so i just added them up. 4 + 5 +(-7) = 2q Then there is a -5q charge outside the sphere. I did 2q + (-5q)= -3q . The electric field flux formula is Flux= q/ E0 . So i got -3q/E0 which is obviously wrong : ) . After quick googling , I...
  3. Tesla In Person

    Electric Field Inside a Conducting Sphere: Is it Always Zero?

    Is the electric field inside a sphere always 0? Even if we have charges on the surface?
  4. Salmone

    I Solving a Particle on the Surface of a Sphere: Obtaining Eigenvalues

    The Hamiltonian of a particle of mass ##m## on the surface of a sphere of radius ##R## is ##H=\frac{L^2}{2mR^2}## where ##L## is the angular momentum operator. I want to solve the TISE ##\hat{H}\psi=E\psi## and in order to do that I rewrite ##L^2## in Schroedinger's representation in spherical...
  5. Salmone

    I Strange Hamiltonian of two particles on the surface of a sphere

    I have a problem with this Hamiltonian: two identical particles of mass ##m## and spin half are constrained to move on the surface of a sphere of radius ##R##. Their Hamiltonian is ##H=\frac{1}{2}mR^2(L_1^2+L_2^2+\frac{1}{2}L_1L_2+\frac{1}{2}S_1S_2)##. By introducing the two operators...
  6. Trysse

    Surface area of a sphere ##= \pi * (a^2+b^2+c^2+d^2)##

    I am not very good at proofs. The only thing I have come up with is the following regularity. However, I am not sure how this can be related to the above problem. Given a sphere ##S_a## with a center ##C## and a diameter of ##a##. I can now construct a line segment ##b## with the endpoints...
  7. alejo ortega

    Resistive force exerted on a sphere (air resistance)

    i don´t understand the question
  8. J

    Find magnetic field at center of rotating sphere

    if a sphere rotates, it's like multiple currents going around in a circle. I can find the magnetic field of each of those currents at the center point of the circle and add them together. We can integrate with respect to y and R. y ranges from 0 to 5 cm away from the center of the loop and the...
  9. G

    Potential difference metallic sphere

    a) We know that ##Q_1=1,2\, \textrm{nC}## and ##Q_2=6\, \textrm{nC}##. By the TOTAL influence theorem: $$-Q_1=Q_{2i}=-1,2\, \textrm{nC}$$ $$Q_2=Q_{2i}+Q_{2e}\rightarrow Q_{2e}=7,2\, \textrm{nC}$$ b) Electric potential difference crust: $$V_A-V_\infty=$$ How was this potential difference thing...
  10. M

    I Given three random numbers between 0 and 1, how to evenly populate a sphere?

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  11. A

    I Calculating the surface area of a sphere using dA

    Below is an image to calculate the surface area of a sphere using dA. I can see how ##rcos\theta d\phi## works, but I don't understand how that side can't just be ##rd\phi## with a slanted circle representing the arc length. The second part I don't understand is why it is integrated from...
  12. A

    Sphere and electric field of infinite plate

    The solution says that the tension in the string in the negative x direction is balanced by the force of the plate on the ball (red). Why is the repulsive force of the ball on the plate (in blue) not included in this calculation?
  13. F

    A Jackson Sec 2.6 on "general solution" of charge near sphere

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  14. mncyapntsi

    E-field of solid sphere with non-uniform charge density

    Hi! I've been trying to attempt this problem over here but the solutions state that the solution is this below? However, from integrating the density and then plugging it into Gauss's law, I get the exact same thing, except a 15 instead of a 5. Could any please help point out if there is an...
  15. Salmone

    I Hamiltonian of a particle moving on the surface of a sphere

    In a quantum mechanical exercise, I found the following Hamiltonian: Consider a particle of spin 1 constrained to move on the surface of a sphere of radius R with Hamiltonian ##H=\frac{\omega}{\hbar}L^2##. I knew that the Hamiltonian of a particle bound to move on the surface of a sphere was...
  16. J

    Rotate Points on Sphere by Theta and Phi

    Dear Forum, My goal is to rotate several points on a sphere by a theta and phi. For example, I have a sphere where the elevation is theta (90 to -90) and the azimuthal is phi (-180 to 180). I have the following points on the sphere: theta = [45 45 45 45] phi = [-180 90 90 180] This generate...
  17. K

    I Particle on a sphere problem in quantum mechanics and its solution

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  18. A

    Find the inertia of a sphere radius R with rotating axis through the center

    $$I = \int{r^2dm}$$ $$dm = \sigma dV$$ $$dV = 4\pi r^2dr$$ $$\sigma = \frac{M}{\frac{4}{3}\pi*R^3}$$ $$I = \sigma 4 \pi \int_0^R{r^4 dr} = \frac{3*MR^2}{5},$$ which is not the correct moment of inertia of a sphere
  19. guyvsdcsniper

    Finding Potential inside a hollow sphere

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  20. MatinSAR

    Is Electric Flux Through a Half Sphere Zero According to Gauss' Law?

    Picture : My answer : I guess net electric flux is 0. so electric flux passing through surface 1 = -(electric flux passing through surface 2) and electric flux passing through surface 1 is EA = E(pi)(r^2) Is it correct? Thank you ...
  21. samy4408

    I don't understand this problem -- Surface area of a section of a solid sphere

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  22. guyvsdcsniper

    Energy inside a sphere

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  23. rudransh verma

    Does Acceleration Affect Tension in a Sphere and Car System?

    If I draw the fbd then some force will accelerate the car in horizontal direction which I think does not effect the string in vertical direction. So same tension regardless of acceleration. But we know it will increase. So what will be the correct physics behind it?
  24. Ruda975

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  25. L

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  26. H

    Boundary conditions (E and D) for a dielectric sphere

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  27. wykk

    A uniformly charged rotating sphere does not radiate, why not?

    The problem says I have a spherically symmetric spinning constant charge distribution of charge Q and angular momentum w; I saw two possible explanations but none of them has made me realize why it is zero, one mentions thata constant w somehow implies a constant E which would mean there is no B...
  28. Daniel777

    Investigating the Charge Distribution on a Bulging Sphere

    In this question it is given that the sphere which is conducting is initially given a charge q then due to nonuniform mechanical strength and due to electrostatic force it creates a Small hemispherical bulge on its surface? okay my doubt is Let me define a term σ where σ is surface density...
  29. L

    Light incident on a sphere, focused at a distance ##2R##

    I used the equation for the refraction on a spherical surface: ##\frac{n_1}{p}+\frac{n_2}{q}=\frac{n_2-n_1}{R}##, where ##n_1=1## is the index of refraction of air, ##n_2## the index of refraction of the sphere, ##R## is the radius of the glass sphere, ##p## is the object distance which, since...
  30. Perchaddition

    B Can I use the circle circumference formula for a sphere?

    Trying to calculate a circumference of a sphere from a radius of 3.09 inches. Is 19.4 a correct answer? Just ran numbers in the first circumference calculator I found http://calcurator.org/circumference-calculator/. Can I use the same formula for a sphere? What can I say ...Geometry is not my...
  31. T

    Gauss's Law for a sphere with a cavity, solving for E(r)

    I am not sure how to solve for E(r) for R1<r<R2.
  32. GottfriedLenz

    Free body diagram for an inverted pendulum in the rolling sphere

    So, to obtain the motion equations I initially plotted the free-body diagram (see picture). Then I’ve tried to get equations, but I’m not sure, do I have done it rightl. I will be gratefull if someone could help me.
  33. Z

    Understanding the electric force felt by the charges on a sphere

    A thin shell in reality doesn't have zero thickness. Consider the image below, showing a cross-section of a small portion of the shell: Here we are considering a more general case in which we have electric fields of magnitude ##E_1## and ##E_2## on each side of the shell. Gauss's Law...
  34. Andrew1235

    Mass conservation in a sphere to find radial velocity of a flame

    I am not sure what form of mass conservation to use to solve the above problem from An Introduction to Combustion by Stephen Turns. Can anyone explain what form of mass conservation applies to a sphere in this context?
  35. D

    Linear speed of sphere as it passes through lowest point

    Let ##m_s = 0.05, m_{s_1} = 0.02, m_r = 0.12, L = 0.8.## be the masses of the two spheres, mass of the rod, and length of the rod. Then the work done by gravity when the rod reaches the vertical position is ##(m_s(L/2) - m_{s_2}(L/2))g## and the kinetic energy equals ##\frac{1}2 (\frac{1}{12}...
  36. D

    B Name of projection to firmamentum (celestial sphere)

    Does the central projection of the position of a star to the firmamentum (celestial sphere) has a special name?
  37. Anonymous243

    Electrostatic potential energy of a non-uniformly charged sphere

    Hi, I'm new here, so I don't know how to write mathematical equations, and I may not be fully aware of the rules here, so I'm sorry if I made a mistake. I know how to calculate the electrostatic potential energy of a countable number of charged particles, but I don't know how to calculate the...
  38. H

    I Is the existence of a Dyson Sphere impossible?

    Specifically a monolithic Dyson Sphere; also, how would a Dyson Swarm work / be a better option?
  39. DeadInside

    Help! I'm Not Sure What I Did Wrong in Evaluating MOI of a Sphere

    I know I must have done something wrong somewhere here, but I cannot figure out exactly which one Answer is supposed to be (2/5)MR2 Whatever disaster I have in the last image does not evaluate closely to that at all. I'm not looking for another way to find the MOI of solid sphere, I would...
  40. E

    What happens when a conducting sphere rubs against metal?

    I know that metal is a "reservoir" of electrons, whereby electrons can flow out and in easily, so when conducting sphere is rubbed against metals, is there even a resulting charge on the conducting sphere?
  41. S

    I General Method for Mapping an Ellipsoid to Unit Sphere

    I have been working on a problem for a while and my progress has slowed enough I figured I'd try reaching out for some more experience. I am trying to map a point on an ellipsoid to its corresponding point on a sphere of arbitrary size centered at the origin. I would like to be able to shift any...
  42. vibha_ganji

    Alonso and Finn Volume 1 Chapter 4 -- A sphere is held against a wall by an inclined plane

    I’m pretty sure that the force on the sphere by the wall and plane has to equal mg so the sum of the normal force is steered by the wall and plane has to equal mg. I’m not sure where to go after this. Is mg the answer or is there something I’m missing?Here is Fig: 4-31:
  43. F

    Kinetic and potential energy of a particle attracted by charged sphere

    Hello, I have a particle at point A with charge ##q_A##, and an unmovable sphere of radius ##R_B## at point B with a volumic charge density ##\rho##. The distance from particle A to the centre of the sphere in B is ##r##. Both objects have opposed charges, so, the particle in A, initially at...
  44. J

    Understanding the Interaction Between a Charged Rod and a Metal Sphere

    (A) incorrect, because opposite signs attract, and the sphere would've been drawn to the charged rod. (B) correct, according to the answer key, but if the charge of the sphere and the charge of the rod are the same, then wouldn't they repel each other? I'm confused as to why this is the correct...
  45. LCSphysicist

    I'm not understanding how to calculate the magnetic field of a sphere

    I need to calculate the magnetic field generated by a static sphere at its center. On the surface of the sphere flows a constant current ##K \hat \phi##. Now, my guess was that the field produced would be equal to the field produced by a lot of rings, that is, i will split the sphere in a lot...
  46. yucheng

    How does metal sphere determine the potential of the plane by symmetry?

    The answer given states that: The entire x-y plane is obviously at the same potential since all the fields are strictly perpendicular to it (draw a diagram if youre confused). Since we choose the sphere to be at potential zero, the point on the sphere which cuts the x-y plane is also at zero...
  47. Russ Edmonds

    I The dramatic difference between wet and dry sphere splashes

    Caution: This video has music. Turn sound off before watching if this bothers you!
  48. eric11201120

    Expression for Electric Field Outside Sphere

    I'm not quite sure where to start. If someone could help me I would very much appreciate it.
  49. P

    Solving Sphere in Cone Problem w/Semivertical Angle

    Help please - okay so I have a question and struggling here. I need to know the radius of the sphere and how much water it displaces. One sphere inside an inverted cone One sphere for which the maximum possible amount of water is displaced. The problem is I’m only given the height of the...
  50. Drakkith

    B Mass and Surface Gravity of a Dyson Sphere

    I'm re-watching Star Trek TNG and I just started the episode where they encounter Scotty aboard a ship that's crashed into a Dyson sphere. That got me thinking. What would the mass and external surface gravity of a Dyson Sphere be? I've done the math myself, but I'd appreciate someone double...
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