Spherical Definition and 1000 Threads

I know that $\arccos{(\cos{\phi_1}\cos{\phi_2}+\sin{\phi_1}\sin{\phi_2}\cos{(\theta_2-\theta_1)})}=\gamma$ But how can i answer the above question? If any member knows the proof of this formula may reply to this question with correct proof.

For the purpose of this thread the metric is ds2 = - (1-rs/r) c2 dt2 + dr2 / (1-rs/r) where rs = 2GM/c2. (I modified the above from https://jila.colorado.edu/~ajsh/bh/schwp.html .) I assume that the two spherical shells are stationary. Therefore dt = 0. The r coordinate for the radii of the...

When I try to do Gauss, the permeability is not always that of the free space, but it varies: up to a certain radius it is that of the void and then it is the relative one. How can I relate them? I'm trying to calculate the capacity of a spherical capacitor. The scheme looks like this: inside I...

In the contex of ##L^2## space, it is usually stated that any square-integrable function can be expanded as a linear combination of Spherical Harmonics: $$ f(\theta,\varphi)=\sum_{\ell=0}^\infty \sum_{m=-\ell}^\ell f_\ell^m \, Y_\ell^m(\theta,\varphi)\tag 2 $$ where ##Y_\ell^m( \theta , \varphi...

All are used to finding the image charge induced by a source charge outside a conducting sphere. The solution is supposed to also work for the case where the source charge is inside the conducting sphere, in which case the sphere is now a conducting cavity. But the solution suggests the image...

In order to compute de lagrangian in spherical coordinates, one usually writes the following expression for the kinetic energy: $$T = \dfrac{1}{2} m ( \dot{r}^2 + r^2 \dot{\theta}^2 + r^2 \sin^2 \theta \dot{\phi}^2 )\ ,$$ where ##\theta## is the colatitud or polar angle and ##\phi## is the...

I am trying to solve the following problem from my textbook: Formulate the vector field $$ \mathbf{\overrightarrow{a}} = x_{3}\mathbf{\hat{e_{1}}} + 2x_{1}\mathbf{\hat{e_{2}}} + x_{2}\mathbf{\hat{e_{3}}} $$ in spherical coordinates.My solution is the following: For the unit vectors I use the...

My teacher said me this commutator is zero because the spherical harmonics are eigenfunctions of L^2. Actually, he said that any operator must commute with its eigenfunctions. I tried an example: [L^2,Y_20] expressing L^2 on spherical coordinates and I determined this commutator is not zero...

Homework statement: Find the electric field a distance z from the center of a spherical shell of radius R that carries a uniform charge density σ. Relevant Equations: Gauss' Law $$\vec{E}=k\int\frac{\sigma}{r^2}\hat{r}da$$ My Attempt: By using the spherical symmetry, it is fairly obvious...

a. This solution is i can consider the charge Q as a point charge and the electric potential at a distance r is ## V = Q/(4πεοr)## b. This is where the confusion starts again when r2>r>r1, my answer ## V = ρ*4*π(r^3 - r_1^3)/(3*4πεοr) \\ V = ρ*(r^3-r_1^3)/3εοr; ## I know i am making some...

I know the divergence of any position vectors in spherical coordinates is just simply 3, which represents their dimension. But there's a little thing that confuses me. The vector field of A is written as follows, , and the divergence of a vector field A in spherical coordinates are written as...

Hi all, I can't find a single thing online that translates a cartesian velocity vector directly to spherical vector coordinate system. If I am given a cartesian point in space with a cartesian vector velocity and I want to convert it straight to spherical coordinates without the extra steps of...

a. For the question a the solution is If the uniform charge density is ρ then the charge of the sphere up to radius r is q = ρ * (4/3)*π * r3; Hence the electric field is E = (ρ *4π*r^3)/(3*εο*r^2); E = (ρ*r)/(3εο); b. I don't understand what is superposition? How to proceed? Please advise.

Summary:: If the conductor is having a cavity and is provided with some charge, with the cavity too having some charge then how the potential will be affected on the outer surface of the conductor. The center of cavity and the center of hollow sphere does not coincide. As if their centers do...

For me is not to easy to understand volume element ##dV## in different coordinates. In Deckart coordinates ##dV=dxdydz##. In spherical coordinates it is ##dV=r^2drd\theta d\varphi##. If we have sphere ##V=\frac{4}{3}r^3 \pi## why then dV=4\pi r^2dr always?

Hello, I am trying to get a better understanding of optical lens's and came across a question I have not found an answer to. Say you have a lens made of either Polycarbonate or Trivex, and it is a spherical lens. A coating is applied to the side with the smaller radius to reflect certain light...

Ive found ##\delta x/\delta r## as ##sin\theta cos\phi## ##\delta r/\delta x## as ##csc\theta sec\phi## But unsure how to do the second part? Chain rule seems to give r/x not x/r?

Hi PF! When solving the Laplace equation in spherical coordinates, the spherical harmonics are functions of ##\phi,\theta## but not ##r##. Why don't they include the ##r## component? Thanks!

Hello! I came across spherical tensors, and I am a bit confused about the way they are applied. For example, Pauli matrices, can be grouped together to form a rank 1 (vector) spherical tensor as ##(\sigma_-, \sigma_z, \sigma_+)##, which are the raising operator, the z projection operator and the...

Hi. Since you can construct shells from a series of rings, why would there be an electric field inside a single ring but not inside a shell?

r,θ,ϕ For integration over the ##x y plane## the area element in polar coordinates is obviously ##r d \phi dr ## I can also easily see ,geometrically, how an area element on a sphere is ##r^2 sin\theta d\phi ## And I can verify these two cases with the Jacobian matrix. So that's where I'm at...

This should be a trivial question. I am trying to compute the spherical tensor ##T_0^{(0)} = \frac{(U_1 V_{-1} + U_{-1} V_1 - U_0 V_0)}{3}## using the general formula (Sakurai 3.11.27), but what I get is: $$ T_0^{(0)} = \sum_{q_1=-1}^1 \sum_{q_2=-1}^1 \langle 1,1;q_1,q_2|1,1;0,q\rangle...

Hi! I'm studying Shankar's Principle of quantum mechanics I didn't get the last conclusion, can someone help me understand it, please. Where did the l over rho come from?

Since the spherical wave equation is linear, the general solution is a summation of all normal modes. To find the particular solution for a given value of i, we can try using the method of separation of variables. $$ ψ(r,t)=R(r)T(t)ψ(r,t)=R(r)T(t) $$ Plug this separable solution into the...

. We have that energy in a infinitesimal Spherical layer with number of mols dn is: dU=Cv.T.dn (1) But by the ideal gas law: PV=nRT (2) Differentiation gives: PdV+VdP=RTdn (3) (3) in (1) and using CV=3R/2 (monoatomic) gives: dU=3/2.(PdV+VdP) (4) Integration of (4) over the whole gas will...

Homework Statement: Derive the formula for the moment of inertia of a thin spherical shell using spherical coordinates and multiple integrals. Homework Equations: Moment of Intertia is (2MR^2)/3 I = (2MR^2)/3

Hi, been a while since I last asked here something. I am restudying electrostatics right now, and I am facing difficulties in the following question: My attempt: I tried to use Gauss' law, what I got is the equation in the capture but that doesn't lead me anywhere as I am unable to find a...

The metric for 2-sphere is $$ds^2 = dr^2 + R^2sin(r/R)d\theta^2$$ Is there an equation to describe the area of an triangle by using metric. Note: I know the formulation by using the angles but I am asking for an equation by using only the metric.

I'm trying to solve the vector potential of a solid rotating sphere with a constant charge density. I'm at a point where I'm performing the final integral that looks like $$ -\left( \frac {\mu_0 i} {3} \right) \sqrt{\frac 3 {2\pi}} \frac {q\omega}{R^3} Y_{1,1} \int_0^R (r')^3 \frac {r_<}...

Hi! I need help with this problem. I tried to solve it by saying that it would be the same as the field of a the spherical shell alone plus the field of a point charge -q at A or B. For the field of the spherical shell I got ##E_1=\frac{q}{a\pi\epsilon_0 R^2}=\frac{\sigma}{\epsilon_0}## and for...

Hi! I need help with this problem. When the outer shell is grouded, its potential goes to zero, ##V_2=0## and so does it charge, right? ##-Q=0##. So the field would be the one produced by the inner shell ##E=\frac{Q}{4\pi\epsilon_0 R_1^2}##. When the inner shell is grounded, I think that...

In a recent test we were asked to calculate the electric field outside a concentric spherical metal shell, in which a point dipole of magnitude p was placed in the center. Given values are the outer radius of the shell, R, The thickness of the shell, ##\Delta R## and the magnitude of the dipole...

The textbook says ' A conducting sphere shell with radius R is charged until the magnitude of the electric field just outside its surface is E. Then the surface charge density is σ = ϵ0 * E. ' The textbook does show why. Can anybody explain for me?

Hi! I need help with this problem. I tried to do it the way you can see in the picture. I then has this: ##dE_z=dE\cdot \cos\theta## thus ##dE_z=\frac{\sigma dA}{4\pi\epsilon_0}\cos\theta=\frac{\sigma 2\pi L^2\sin\theta d\theta}{4\pi\epsilon_0 L^2}\cos\theta##. Then I integrated and ended up...

Stokes' Law gives us the value fo viscous force when a spherical body is under motion inside a fluid. ##F_{viscous} = 6\pi\eta av## (where ##a## is the radius of the spherical body and ##v## is the velocity with which it is moving) What is the reason for the Viscous drag to depend upon the...

Given that L is the luminosity of a single star and there are n stars evenly distributed throughout this thin spherical shell of radius r with thickness dr, what is the total intensity from this shell of stars? My calculations were as follows: Intensity is the power per unit area per steradian...

My apologies for not detailing my attempts at a solution; I'm not sure how to to digitally illustrate or describe the various setups I attempted before looking at the solution to this problem. I am also ONLY asking about the setup, though I included the full question for context. The solution to...

I'm trying to solve the diffusion equation in spherical co-ordinates with spherical symmetry. I have included the PDE in question and the scheme I'm using and although it works, it diverges which I don't understand as Crank-Nicholson should be unconditionally stable for the diffusion. The code...

Summary: A 1963 paper by Michael Wertheim uses a Laplace transform in spherical coordinates. How is the resulting equation obtained? In 1963, Michael Wertheim published a paper (relevant page attached here), where he presented the following equation (Eq. 1): $$ y(\bar{r}) = 1 + n...

Can an electron in a spherically symmetrical potential energy function have non-zero spin angular momentum?

I was just reading an intro text about GR, which considers the circumference of a circle on a sphere of radius R as an example of intrinsic curvature - the thought being that you know you're on a 2D curved surface because the circle's circumference will be less than ##2\pi r##. They draw a...

Hi, i am new to simulation and for my thesis i have to make a simple simulation by using mcnp4c2. Is anybody familiar with this version of MCNP? I need to calculate the fission reactions per second in a geometry of a spherical sub critical reactor of Uranium with low percentage of U 235 with...

If Jupiter is made of gas, how can it maintain its spherical shape without being contained in a spherical shaped container?

Hello everyone, There is an electrical field inside and outside (at the same time) the spherical hollow conductor when we place positive or negative charge inside, isn't it? I know this is because of the induced charges on the inner and outer surfaces of the conductor. There is no field inside...

Publication: Rafael G. González-Acuña and Héctor A. Chaparro-Romo, "General formula for bi-aspheric singlet lens design free of spherical aberration," Appl. Opt. 57, 9341-9345 (2018) Open access preprint: arXiv Given one surface of a lens, how does the other surface has to look like to avoid...

Case (a) is the textbook of a planar incident wavefront and below it in the figure is the known simple formula for the central spot and fringes, or minima and maxima, angular distribution with respect to the optical axis. So, the question here is regarding case (b). The position (usually...

I'm sure the average is going to be an integral, but \displaystyle\frac{1}{4\pi R^2}\oint\mathbf{E}\cdot d\mathbf{a} gives me a scalar, not a vector.