What is Spherical: 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.
Homework Statement
A solid sphere of radius R has a uniform charge density ρ and total charge Q. Derive an expression for its total electric potential energy. Suggestion: Imagine that the sphere is constructed by adding successive layers of concentric shells of charge dq = (4\pi r^{2} dr) ρ...
Homework Statement
What is the dot product of two unit vectors in spherical coordinates?Homework Equations
A∙B = ||A|| ||B|| cos(\theta) = cos(\theta)The Attempt at a Solution
The above equation is the only relevant form of the dot product in terms of the angle \theta that I can find. However...
Homework Statement
The potential inside a spherical shell is given by:
V_{-}(x,y,z)= \frac{V_0}{R^2}(6z^2-3x^2-3y^2)
P_n(\cos(\theta )) where \theta is the polar angle.
The potential on the surface carries a surface charge density \sigma. Besides this, ther's no other charges and no outher...
Hi Guys,
Suppose we have a spherical shell with charge density on the surface \sigma and radius R. The potential inside the shell is given by:
V_(x,y,z) = \frac{V0}{R^{2}}(6z^2+ax^2+by^2)
It is assumed, that the potential is rotational symmetric around the z-axis inside and outside the...
Hi,
I seem to have forgotten some of my math how-to, as I haven't done this in a while. Looking through my notes, Bird, Stewart and Lightfoot, Greenberg, etc. don't really help.
My equation is this, at steady state:
0 = 1/r^2 ∂/∂r (D*r^2 ∂C/∂r) + P
Where P is some production rate...
Hi Guys,
need some assistance. I am sure what I am asking is trivial but i still need help. How could i find the angle within a spherical triangle (triangle formed on a sphere). Now this triangle has equal lengths on all 3 sides.
Pleas help!
What, "Physically" is a Spherical Harmonic?
I'm trying to use spherical harmonics to get an equation to fit a set of data I have. I'm fine with that, I've found a derivation of what the general form is and I crunch that into MATLAB. My problem is derivations online really don't help me...
Homework Statement
A conducting spherical shell that has zero net charge has an inner radius R1 and an outer radius R2. A postive point charge q is placed at the center of the cell. The 1st part was to find the electric fields at the 3 diff places. The part I need help on is where we have to...
Homework Statement
A field is given in spherical coordinates as F=[cos(θ)/r2]∙ar+[sin(θ)/r]∙aθ. Express F in terms of x, y, z, ax, ay, azHomework Equations
ar∙ax=sin(θ)cos(∅)
ar∙ay=sin(θ)sin(∅)
ar∙az=cos(θ)
aθ∙ax=cos(θ)cos(∅)
aθ∙ay=cos(θ)sin(∅)
aθ∙az=-sin(θ)
x=r*sin(θ)*cos(∅)...
1. Homework Statement [/b]
A typical virus is a packet of protein and DNA (or RNA) and can be spherical in shape. The influenza A virus is a spherical virus that has a diameter of 85 nm. If the volume of saliva coughed onto you by your friend with the flu is 0.044 cm3 and 10−9 of that volume...
Homework Statement
Prove that the spherical harmonic wave function \frac{1}{r}e^{i(kr-{\omega}t)} is a solution of the three-dimensional wave equation, where r = (x^2+y^2+z^2)^{\frac{1}{2}} . The proof is easier if spherical coordinates are used.
Homework Equations
Wave function...
Hello,
in this diagram, the shaded regions are spherical conductors.
What's the potential at A=B?
Ignoring the outer sphere, it should be kQ/R.
When you add the outer sphere, potential at C=D=0 and electric field between B and C is kQ/x^2
so i integrated (kQ/x^2) dx with interval [2R, R]...
I want to study antenna patterns of different arrangements. I am looking for a very cheap software ( free is even better) to plot graph if I provide the \;R,\theta,\phi.
Even if 2D plot would be helpful like keeping either \;\theta\;\hbox { or }\; \phi\; constant and vary the other angle to...
Hello, I'm studyng relativistic quantum mechanics by the book Relativistic quantum mechanics. Wave equations - Greiner, W. and I'm trying to derive the energy eingenvalues for s1/2 states, so I have the equation that I uploaded with the name eq1.jpg. In the text the author says, "If we assume R0...
So there are two concentric conducting spherical shells one with radius R and another 2R with charge +Q and +2Q respectively... Now the two are connected by a conducting wire. Why does the entire charge flow to the outer shell?
Please clarify my doubts. I will be grateful.
I took the divergence of the function 1/r2\widehat{r} in spherical coordinate system and immediately got the answer as zero, but when I do it in cartesian coordiantes I get the answer as 5/r3.
for \widehat{r} I used (xi+yj+zk)/(x2+y2+z2)1/2
what am i missing?
Hey, I've been stuck on this question for quite a while now:
Homework Statement
1a. Write down an expression for the position vector r in spherical polar coordinates.
1b. Show that for any function g(r) of r only, where r = |r|, the result \nabla x [g(r)r] = 0 is true. Why does this...
Homework Statement
Consider the study of the motion of a two bodies system interacting with only gravitational forces.
If the two bodies (or even one of them) has not spherical symmetry, how will you proceed? Indeed the Earth and the moon does not have spherical symmetry mass distributions...
Homework Statement
A concave spherical mirror has a radius of curvature of magnitude 27.1 cm. Determine the object position for which the resulting image is inverted and larger than the object by a factor of 4.00.
Homework Equations
Mirror equation in terms of focal length: 1/p + 1/q =...
Homework Statement
A dentist uses a spherical mirror to examine a tooth. The tooth is 1.13 cm in front of the mirror, and the image is formed 10.8 cm behind the mirror. Determine the mirror's radius of curvature.
Homework Equations
1/p+1/q=1/f
f=R/2
The Attempt at a Solution...
Hi
Here's the problem I am trying to do.
a) Is the state \psi (\theta ,\phi)=e^{-3\imath \;\phi} \cos \theta
an eigenfunction of \hat{A_{\phi}}=\partial / \partial \phi or of
\hat{B_{\theta}}=\partial / \partial \theta ?
b) Are \hat{A_{\phi}} \;\mbox{and} \;\hat{B_{\theta}}...
For my investigation regarding the aerodynamic forces on a spherical projectile, I really need to know the theoretical ratio of rotational kinetic energy to linear kinetic energy of a spherical projectile (assuming the only spin is forward spin and there is no Magnus effect).
Can someone please...
Homework Statement
f(x) is a differentiable function let
F(t)= \int\int\int_{x^2+y^2+z^2\leq t^2} f(x^2+y^2+z^2) dx dy dz
compute F^{'}(t)
Homework Equations
x=p sin \phi cos\theta
y= p sin \phi sin\theta
z= p cos \phi
spherical bounds 0<p<t 0<\phi<\Pi 0<\theta < 2\Pi
p^2...
I wanted to derive the volume of a sphere using triple integration with spherical coordinates, but instead of taking the limits of θ as (0° ≤ θ ≤ 180°), I chose to take (0° ≤ θ ≤ 360°), and therefore, for φ as (0° ≤ φ < 180°),
Now of course the integral of sin(θ) from 0° to 360° is zero, and...
Homework Statement
Using spherical coordinates, set up but DO NOT EVALUATE the triple integral of f(x,y,z) = x(x^2+y^2+z^2)^(-3/2) over the ball x^2 + y^2 + z^2 ≤ 16 where 2 ≤ z.
Homework Equations
x = ρ sin ϕ cos θ
y = ρ sin ϕ sin θ
z = ρ cos ϕ
ρ^2 = x^2 + y^2 + z^2
∫∫∫w...
Hello Everyone,
I was just wondering if there was a way to add two vectors that are determined by spherical coordinates (radius, theta, phi). For example, if I have v1 = (5, Pi/4, Pi/2) and v2 = (3, Pi, -Pi/2) is there a way to add these using their respective radii, thetas, and phis or do I...
So I've been trying to derive the moment of inertia equation for a thin spherical shell and I've slammed into a dead end algebraically. I was able to derive an equation for a hollow sphere:
I = (2/5) M (Ro^5 - Ri^5)/(Ro^3 - Ri^3)
where Ro is the distance to the very outside of the sphere...
Homework Statement
The polarizatiob charge on the surface of a spherical capacitor is -\sigma_e \cos(\theta), at a point whose radius vector from the centre makes an angle \theta witha given axis Oz. Prove that the field strength at the centre is \frac{\sigma_e}{3 \epsilon_0}, Homework...
(b]1. Homework Statement [/b]
A spherical weather balloon has a radius of 1m when it is 1500m high.
You observe that the radius increases at a rate of 2cm/min as it continues to rise.
At what rate is the surface area increasing when the radius is 4m?
Homework Equations
I thought...
Homework Statement
Homework Equations
The Attempt at a Solution
I'm trying to find the steady state solution to the heat equation for a system of spherical shells (looks like http://correlatingcancer.com/wp-content/uploads/2009/01/nanoshell-thumb.jpg" ) where heat generation Q occurs in...
consider this following triple integral
1/(x^2+y^2+z^2)dxdydz
bounded above by sphere z=(9-x^2-y^2)^1/2 and below by the cone z=(x^2+y^2)^1/2
what i have done:
z=Pcospi
P^2=x^2+y^2+z^2
9=x^2+y^2+z^2
P=0 to 3
pi=0 to pi/4
theta=0 to 2pi
is this the correct range?
Homework Statement
A -5-nC point charge is located at the center of a conducting spherical shell. The shell has an inner radius of 2 m, an outer radius of 4 m, and a charge of +7 nC. (Let the radially outward direction be positive.)
(a) What is the electric field at r = 1 m? (Indicate the...
I am an ameteur physicist (i actually have my degree in meteorology), and i have some questions about the EM properties of liquid metals or ferrous liquids when in spherical form. I understand if you are too busy or if i sound off, but if you do have the time to answer a few questions, it would...
Homework Statement
I want to understand spherical harmonics. I want to really grok them deeply. I want to be able to visualize them and understand them.
I'm the sort who can't take anything on faith, especially where quantum mechanics is concerned. So I want to understand angular...
Homework Statement
The problem is on page 40 of this PDF:
http://web.mit.edu/8.02t/www/802TEAL3D/visualizations/coursenotes/modules/guide05.pdf
Find the capacitance of a spherical capacitor with 2 different homogeneous dielectrics arranged concentrically.The Attempt at a Solution
In that...
Homework Statement
Compute the electric field generated by a spherically symmetric charged sphere of radius R with charge density of \rho = kr^{2}
Homework Equations
\oint _S \vec{E} \cdot \vec{dA} = \frac{Q_{enclosed}}{\epsilon_0}
The Attempt at a Solution
I know that this question...
Does anyone remember the formula for the area of a spherical patch in terms of two angles?
Obviously you parametrize the surface and do the surface integral but I'm a bit too lazy/busy right now. So does anyone just remember the result?
By spherical patch I mean something like this:
I want...
Im trying to use spherical trig to solve a problem
In the standard method to compute the angles, I already have only the following four angles
A, a, B, b
but from these, how can I compute either C, or c?
Thanks
Hi folks,
I'm looking for a derivation of the following statement (formula 76)
http://img845.imageshack.us/img845/1550/screenshot4op.png
Do you know any reference, where I can find a bit more detailed description? I reckon, you can find it in Jackson's electrodynamic book, but I couldn't find...
On May 25, 2011, the journal Nature published an article stating that the electron was experimentally found to be extremely spherical. In Volume II, Chapter 5 of Feynman's Lectures on Physics, he states that the electric field of an electron has been experimentally determined to vary...
Homework Statement
Give a physical explanation of why a spherically symmetric Ylm cannot describe the state of a system with non-zero angular momentum.
Homework Equations
The Attempt at a Solution
I was thinking that if Ylm is spherically symmetric then the particle is equally...
hi everybody,
i would appreciate it if someone could clarify the concept of spherical aberrations in the context of high NA objectives in which use lenses are used that are not exclusively of the spherical type.
a common thing that you hear is that some objective is corrected for 0.17mm...
Homework Statement
Evaluate the following triple integral by switching it to spherical coordinates?
The integrand is r dzdrdθ
The limits for the inner integral are 0 to r
The limits for the middle integral are 0 to 3
The limits for the outer integral are 0 to 2π
Homework Equations...
1. The problem statement
Consider an infinite spherical charge distribution with constant charge density. According to symmetry of the problem, I expect the electric field at any point to be zero. But if you construct a Gaussian sphere and apply Gauss theorem, it will give you some finite field...
Hi
I was trying to do one problem from Griffiths's EM book. The problem says to find the
magnetic force of attraction between the northern and southern hemispheres of a spinning
charged spherical shell. There was an earlier problem in the book to find the electric
force of attraction...
Homework Statement
Using spherical coordinares, find the smaller volume bounded by the cone z=r and the sphere z^2+y^2+x^2=4
Homework Equations
x^2+y^2+z^2=4 ; rho=2, z=rhocosphi
The Attempt at a Solution
Shot in the dark:
Tried function integrating (rho squared - rhocosphi)...
atomic orbitals, what i just know about them is they are the regions around the nucleus where the probability of finding an electron is high and that is OK with s orbital because it is spherical.
but when an electron in p orbital spins the probability of finding an electron should spin...
I've read many times now that the electric field inside a spherical shell is zero and all explanations of that fact have been based on imo false interpretation of the gauss law. The law itself says no such thing, atleast not explicitly. It merely says that the flux is zero. The only thing i can...