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.
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?)...
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...
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...
Is it correct to say that:
the cotangent is given by the gradients (*) to all the curves passing through a point and it actually spans the same tangent space to a point of a sphere? If you visualize them as geometric planes (**), the cotangent and the tangent spaces are more than isomorphic...
I'm trying to evaluate the arc length between two points on a 2-sphere.
The geodesic equation of a 2-sphere is:
$$\cot(\theta)=\sqrt{\frac{1-K^2}{K^2}}\cdot \sin(\phi-\phi_{0})$$
According to this article:
http://vixra.org/pdf/1404.0016v1.pdf
the arc length parameterization of the...
I'm a bit confused about the notation used in the exercise statement, but if I'm not misunderstanding we have
$$\begin{align*}(\psi^+_1)^{-1}:\begin{array}{rcl}
\{\lambda^1,\lambda^2\in [a,b]\mid (\lambda^1)^2+(\lambda^2)^2<1\}&\longrightarrow& \{\pm x_1>0\}\subset \mathbb{S}^2\\...
I am working from Sean Carroll's Spacetime and Geometry : An Introduction to General Relativity and have got to the geodesic equation. I wanted to test it on the surface of a sphere where I know that great circles are geodesics and is about the simplest non-trivial case I can think of.
Carroll...
I apologise for the length of this question. It is probably possible to answer it by reading the first few lines. I fear I have made a childish error:
I am working on the geodesic equation for the surface of a sphere. While doing so I come across the partial derivative
\begin{align}...
Does the block move along the pink dotted lines as attached in the figure below?
I tried to draw the FBD of the small block ##m ## at the lowermost point which is also attached below.(The direction of ## v_0 ## is actually tangential)
Is the figure above correct? If not, why?
Seems the physics books agree that there is no difference in capacitance whether an isolated sphere is solid or hollow. And the reason mentioned for that always sounds something like the following:
"The reason that the capacitance C, and hence the charge Q, is not affected by whether or not the...
First off, I'm not a scientist or engineer and I apologize if I don't give a clear description of my question. I'm beginning a personal project and was hoping for some knowledge and assistance.
What I'm trying to achieve is to have a spherical object (it will be at least twice as wide as it is...
I have three points: A, B and C, which are all on the surface of the same sphere.
I need to find the xyz coordinates of C.
What I know:
- the radius of the sphere
- the origin of the sphere
- the xyz coordinates of A and B
- the arc distance from A to C and from B to C
- the angle between AB and...
I'm facing a problem with that rhyming title up there.
The design is thus: a downward-facing, vertical pipe with known constant flow and diameter has water flowing out of it, into a short (15cm-91cm) free fall. At the end of that fall is a bowl of indeterminate depth made of steel with holes...
Let there be a sphere whose inner surface consists of a perfectly reflecting surface.
It has a hole on it which allows a ray of light to enter.
Give the angle made with the normal of the hole when the ray of light enters such that the probability
that the ray comes out is the least?Assuming the...
Homework Statement
Homework Equations
gauss law
q=charge on sphere
Q=total charge enclosed by gaussian surface
Q=alpha/r x (4/3 pi r^3-4/3 pi R^3) + q
The Attempt at a Solution
EA=Q/ε[/B]
E=Q/(Aε)
now
for E to be independent of r,
alpha/r x 4/3 pi r^3 + q = 1/(4)(pi)(r^2)
alpha x 4/3...
A sphere of radius a carries a total charge q which is uniformly distributed over the volume of the sphere.
I'm trying to find the electric field distribution both inside and outside the sphere using Gauss Law.
We know that on the closed gaussian surface with spherically symmetric charge...
Homework Statement
What is the escape speed of an electron launched from the surface of a 1.0-cm-diameter glass sphere that has been charged to 10nC?
Homework Equations
Given:
d= 1.0cm
r= 0.05cm= 0.0005m
q1 = 10nC = 10 x 10-9 (sphere)
q2 = -1.6 x 10-19 (electron)
Equation:
U = (kq1q2)/r
KE=...
Homework Statement
A spherical shell of radius R has a surface charge distribution σ = k sinφ.
Calculate the dipole moment of the spherical shell.
Homework Equations
P[/B]' = ∫r' σ(r') da'
The Attempt at a Solution
So I believe my dipole will be directed along the y axis, as the function...
lrcarr
Thread
dipole
sphere
spherical coordinates
surface charge density
Homework Statement
A uniform charge density of 700 nC/m3 is distributed throughout a spherical volume of radius 6.00 cm. Consider a cubical Gaussian surface with its center at the center of the sphere.
[reference picture]
What is the electric flux through this cubical surface if its edge...
I am discussing physics with a friend and we need someone to confirm a thing that we're not agreeing on.
We are discussing incident light that is passing through different geometries, and I want to know how the light behaves when it reflects inside a half sphere (of glass for example). Maybe...
Homework Statement
A + q = 5 pC charge is uniformly distributed on a non-conducting sphere of radius a= 5 cm , which is placed in the center of a spherical conducting shell of inner radius b = 10 cm and outer radius c = 12 cm. The outer conducting shell is charged with a -q charge. Determine...
Homework Statement
Two conducting spheres having same charge density and with radius “R” & “2R” are brought in contact and separated by large distance. What are their final surface charge densities ?
Homework Equations
No equation in question.
The Attempt at a Solution
Tried using the fact...
I have done it by the parametric form of σ, but if I change σ to implicit form that is G(x,y,z)=x^2+y*2+z^2-R^2=0 I don't know how continue.
The theory is:
where Rxy is the projection of σ in plane xy so it's the circumference x^2+y^2=R^2
Hello,
I want to prove that a graph represent a manifold, for this i take the opposites edges of a vertex (edge connected between vertex connected to the current vertex) and this subgraph need to be homeomorphic for example to the 1-sphere if i want a 2 manifold. This criterion ensure that my...
Given a charged sphere, the electric field within it is zero at every point. Why is this? Why is not merely zero only at the center? If a sphere is conducting, then its charge is all across the surface. If electric field is inversely proportional to distance from charge squared, won't the field...
To find the surface area of a hemisphere of radius ##R##, we can do so by summing up rings of height ##Rd\theta## (arc length) and radius ##r=Rcos(\theta)##. So the surface area is then ##S=\int_0^{\frac{\pi}{2}}2\pi (Rcos(\theta))Rd\theta=2\pi R^2\int_0^{\frac{\pi}{2}}cos(\theta)d\theta=2\pi...
Homework Statement
In a lecture demonstration, a professor pulls apart two hemispherical steel shells (diameter D) with ease using their attached handles. She then places them together, pumps out the air to an absolute pressure of p, and hands them to a bodybuilder in the back row to pull...
Homework Statement
1. The field for an infinite charged sheet is found to be σ/2ε0. If we place 2 infinite sheets of opposite charge above one another, we say that the field in between the sheets is σ/ε0 due to the superposition of individual fields.
Why can't we say the same for a situation...
Homework Statement
My questions are just related to part a of this problem.
Homework Equations
The Attempt at a Solution
I know that potential inside a conductor is equivalent to potential on the surface of the conductor and potential at any point is an algebraic sum of potential...
Homework Statement
A thick, spherical shell of inner radius a and outer radius b carries a uniform volume charge density ρ.
Find an expression for the electric field strength in the region a<r<b.
Homework Equations
Gauss's Law ∫E⋅dA
Area of a sphere
The Attempt at a Solution
I know I am...
I am currently coding a small application that reproduces the transport of a vector along a geodesic on a 2D sphere.
Here's a capture of this application :
You can see as pink vectors the vectors of curvilinear coordinates and in cyan the transported vector.
The transport of vector along...
Homework Statement
Find the potential of an uncharged metal sphere provided that a point charge q is located at a
distance r from its center
2. The attempt at a solution
As far as the charges are concerned , some negative charges will build up at the side of the charge because of induction ...
Homework Statement
A metal sphere of radius ##a=1cm## is charged with ##Q_a=1nC##. Around a sphere is placed a spherical shell of inner radius ##b=2cm## and outer radius ##c=3cm##. The electrical potential of the shell in refenrence to a point in the infinity is ##V=150V##. The spheres are in...
Homework Statement
A sphere of radius ##a## is non-uniformly charged on its surface with a charge whose surface density is ##ρ_s(φ)=ρ_{so}(cosφ)^2## where ##φ## is the angle measures from the z axis, (0≤φ≤π) and ##ρ_{s0}## is a constant. Determine the expression for the total charge distributed...
Hi.
I'd like to show that a conducting, charged spherical shell can shield the field of an inside opposite point charge even if this charge is not at the center. I was thinking about a Gaussian surface just outside the sphere, such that if there were electric field vectors they would be...
Homework Statement
A solid sphere of density ##ρ## and radius ##R## is centered at the origin. It has a spherical cavity in it that is of radius ##R/4## and which is centered at ##(R/2, 0, 0)##, i.e. a small sphere of material has been removed from the large sphere. What is the the center of...
Hi.
The capacitance of an ideal plate capacitor (coaxial cable) goes to zero as the plate distance (outer radius) goes to infinity. This doesn't happen with concentric spheres as we let the outer radius go to infinity, hence a single sphere has a nonzero capacitance.
What's the exact reason...
Homework Statement
Let P be a point on the sphere with center O, the origin, diameter AB, and radius r. Prove the triangle APB is a right triangle
Homework Equations
|AB|^2 = |AP|^2 + |PB|^2
|AB}^2 = 4r^2
The Attempt at a Solution
Not sure if showing the above equations are true is the...
Homework Statement
Suppose the nonconducting sphere of Example 22-4 has a spherical cavity of radius r1 centered at the sphere's center (see the figure). Assuming the charge Q is distributed uniformly in the "shell" (between r = r1 and r = r0), determine the electric field as a function of r...
Hey everyone,
I've been stuck on this one piece of HW for days and was hoping someone could help me.
It reads:
The surface area, A, of a sphere with radius R is given by
A=4πR^2
Re-derive this formula and write down the 3 essential steps. This formula is usually derived from a double...
Is there a potential on the inner surface of a charged spherical shell?
I know that there is no electric field on the inner surface, as shown by Gauss's Law, but that isn't enough information to say that the potential (V) there is zero since E = dV/dr, so V could be a nonzero constant.
If...
Homework Statement
In the figure a nonconducting spherical shell of inner radius a = 2.07 cm and outer radius b = 2.51 cm has (within its thickness) a positive volume charge density ρ = A/r, where A is a constant and r is the distance from the center of the shell. In addition, a small ball of...
Homework Statement
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...
Homework Statement
Given a sphere with radius R, centered at (0,0,0), it's dipole density given as ##P\left(\vec{r}\right)=\alpha\left(R-r\right)\hat{z}## where r is the distance from the center of the ball.
I'm required to find:
Bound charge density inside the sphere, bound charge density on...
I'm studing Gauss law for gravitational field flux for a mass that has spherical symmetry.
Maybe it is an obvious question but what are exactly the propreties of a spherical simmetric body?
Firstly does this imply that the body in question must be a sphere?
Secondly is it correct to...
Homework Statement
Hello everyone, I am new here and have a question regarding method of images in my electricity and magnetism class. I need help to even get the ball rolling. The question is as follows:
a) What is the image of a dipole, oriented toward the center of the conducting sphere, if...
Hi, I have an interesting problem.
I have three GPS coordinates, creating two lines across the surface of a sphere (assuming the Earth is spherical). I want to be able to create a new line (across the surface of a sphere) with a gradient that is in between the gradient of the two existing...
hi guys,
i have a question.
i saw this picture, and i don't really understand how they derived with the formula. The aim is basically to find the formula for the surface area of a spherical cap.
why do you differentiate the x=sqrt(rˆ2-yˆ2)? how does that help to find the surface?
and then...