Spherical Definition and 1000 Threads

In various other threads we have been kicking around various equations for a spherical shell and discussing the implications. In this thread I would like to present what (I think) I have worked out about how the shell metric relates to the vacuum metric inside and outside the shell. I hope to...

Hey. There's one thing I've always been wondering about when it comes to deriving the expression for the moment of inertia of a spherical shell. Namely, why is the length of the infinitesimal cylinder used in the derivations (like here ) equal to ##R d \theta##, instead of ##R d \theta...

Homework Statement Evaluate the iterated integral ∫ (from 0 to 1) ∫ [from -sqrt(1-x^2) to sqrt(1-x^2) ] ∫ (from 0 to 2-x^2-y^2) the function given as √(x^2 + y^2) dz dy dx The Attempt at a Solution I changed the coordinates and I got the new limits as ∫(from 0 to pi) ∫(from...

In chapter 5, magnetostatics, of Griffiths' Introduction to Electrodynamics (third edition), there's a problem in the back of the chapter that asks you to calculate the force of attraction between the northern and southern hemispheres of a spinning charged spherical shell. The problem in its...

I recently had to do an integral like the one in the thread below: http://math.stackexchange.com/questions/142235/three-dimensional-fourier-transform-of-radial-function-without-bessel-and-neuman The problem I had was also evaluating the product and I am quite sure that the answer in the thread...

I have been given the exercise above which relates to an article we are reading. I can calculate all the results but I can't interpret them physically. I think my major problem is that I don't see how self capacitance is a physically measureable quantity. For a recap the self capacitance is...

Hello People I need help with the following assignment: It states: Consider an ideal moderator with zero absorption cross section, Ʃa = 0, and a diffusion coefficient, D, which has a spherical shape with an extrapolated radius, R. If neutron sources emitting S neutrons/cm3sec are distributed...

I'm not sure whether this falls in the homework category, or the standard calculus section, so apologies in advance if this doesn't fall in the right category. Homework Statement Evaluate ∫∫∫e^[(x^2 + y^2 + z^2)^3/2]dV, where the region is the unit ball x^2 + y^2 + z^2 ≤ 1. (or any...

Homework Statement Double Integral Surface Area of Spherical Ball radius Homework Equations ##\int_S d\vec{S} = 4*\pi*a^2## The Attempt at a Solution ##\int\int_0^a f(r,?) dr d? = 4*\pi*a^2##

I want to find ##\Phi## and ##\vec{E}## for the general case of a Spherical Ball with uniform Charge Density centered at the origin radius d. ##\Phi = \frac{\rho}{4*\pi*\epsilon_0}\int\int\int\frac{r^2*sin\theta}{|r-r'|}dr d\theta d\phi## ##E =...

I am asked to show that when \(\hat{e_r}\), \(\hat{e_\theta}\), and \(\hat{e_\phi}\) are unit vectors in spherical coordinates, that the cartesian unit vectors $$\hat{i} = \sin{\phi}\cos{\theta}\hat{e_r} + \cos{\phi}\cos{\theta}\hat{e_\phi} - \sin{\theta}\hat{e_\theta}$$ $$\hat{j} =...

Homework Statement Spherical Ball centered at origin uniform ##\rho## with a radius a. Find E along x-axis. Homework Equations ##E = \frac{\rho}{4\pi\epsilon_0}\int\int\int\frac{r^2*sin\theta}{r_\rho^2} d\phi d\theta dr## The Attempt at a Solution Evaluate E spherically along the...

Homework Statement I have a problem that is the curl of jln(rsinθ) Since this is in spherical, why is there a bold j in the problem? Doesn't that indicate it's a unit vector in cartesian coordinates? Can I do the curl in spherical coordinates when I have a cartesian unit vector in the...

Firstly, I'm sorry if this is incorrect or if there is a specific place for such questions but as this is neither a problem posed to me, nor something that has been taught - I have little background with which to work with but it is something I need to do for my ERT and 2 maths teachers have...

Homework Statement z(x^2+y^2+z^2)^(-3/2) where x^2+y^2+z^2 ≤ 4 and z ≥ 1 The Attempt at a Solution So spherically this comes down to cos∅sin∅dpdθd∅ p goes from 0 to 2, theta goes from 0 to 2pi, but I don't know how to figure out what ∅ goes from? I'm trying use trig identities but...

Homework Statement Find the E produced by a spherical charge distribution with uniform charge density at a point inside the sphere, using triple integration. Homework Equations E = 1/4πε ∫f(x,y,z)/r^2 dV The Attempt at a Solution f(x,y,z) = p Radius of sphere = R Position of...

Homework Statement So if you integrate over a spherical area, say a ball of radius 1, then 0≤p≤1, 0≤θ≤2∏, and 0≤∅≤∏. My question is why don't you integrate ∅ between 0 and 2∏? I mean if you are integrating over a sphere then you have to go around it vertically AND horizontally completely...

Homework Statement Consider the following electric field: \vec{E}=\frac{\rho }{3\varepsilon _{0}}\vec{r} where r\leq R and \vec{E}=\frac{\rho R^3 }{3\varepsilon _{0}r^2}\hat{e_{r}} where r>R (a) calculate the divergence of the electric field in the two regions (b)...

Homework Statement A hollow conducting spherical shell has radii of .80m and 1.20m. The sphere carries a net charge of -500 nC. A stationary point charge of +300 nC is present at the center origin. Calculate the electric field at points: a) 0.30m b) 1.00m c) 1.50m I have attached the image...

Homework Statement Suppose that in spherical coordinates the surface S is given by the equation rho * sin(phi)= 2 * cos(theta). Find an equation for the surface in cylindrical and rectangular coordinates. Describe the surface- what kind of surface is S? Homework Equations The...

say you have spherical metal conductor with a cavity with a positive charge inside, the field inside the cavity isn't zero and will induce an opposite charge/field on the surface of the cavity which will cancel the charge inside and lead to a zero Electric field inside the conductor. the...

Is it possible to express (cos(\theta)sin(\theta))^2 in terms of spherical harmonics?

Homework Statement Say I am given a spherically symmetric potential function V(r), written in terms of r and a bunch of other constants, and say it is just a polynomial of some type with r as the variable, \frac{q}{4\pi\varepsilon_o}P(r), and we are inside the sphere of radius R, so r<R…...

If there's a point charge Q at the center of a spherical surface(of radius a) made of conducting material that is connected to earth, why is the electric field past r>a zero ? Doesn't it imply that the spherical surface becomes charged with -Q ? And why is that? What would be the...

Homework Statement A point charge q is at the center of a spherical conducting shell of inner radius a and outer radius b. How much work would it take to remove the charge out to infinity? Homework Equations Potential, W = 1/2qV The Attempt at a Solution I am going at this in...

Q: Consider the solid that lies above the cone z=√(3x^2+3y^2) and below the sphere X^2+y^2+Z^2=36. It looks somewhat like an ice cream cone. Use spherical coordinates to write inequalities that describe this solid. What I tried to do: I started by graphing this on a 3D graph at...

Homework Statement (From Physics for Scientists and Engineers, 7E, Serway-Jewett Chapter 25 Q11) (i) A metallic sphere A of radius 1 cm is several centimeters away from a metallic spherical shell B of radius 2 cm. Charge 450 nC is placed on A, with no charge on B or anywhere nearby. Next...

Hello MHB, So when I change to space polar I Dont understand how facit got \frac{\pi}{4} \leq \theta \leq \frac{\pi}{2} Regards, |\pi\rangle \int\int\int_D(x^2y^2z)dxdydz where D is D={(x,y,z);0\leq z \leq \sqrt{x^2+y^2}, x^2+y^2+z^2 \leq 1}

Homework Statement Exercise 1.3 on uploaded Problem Sheet. Homework Equations Shown in Exercise 1.3 on Problem Sheet The Attempt at a Solution Uploaded working: I have found the inverse of the Transformation Matrix from Cartesian to Spherical Coordinates by transposing...

Homework Statement (a) Starting from a point on the equator of a sphere of radius R, a particle travels through an angle α eastward and then through an angle β along a great circle toward the north pole. If the initial position is taken to correspond to x = R, y = 0, z = 0, show that its...

I am trying to show that \[ Y_{\ell}^m(0,\varphi) = \delta_{m,0}\sqrt{\frac{2\ell + 1}{4\pi}}. \] When \(m = 0\), I obtain \(\sqrt{\frac{2\ell + 1}{4\pi}}\). However, I am not getting 0 for other \(m\). Plus, to show this is true, I can't methodically go through each \(m\). How can I do this?

Find the electric field for a non-conducting sphere of radius R = 1 meter that is surrounded by air in the region r > 1. The interior of the sphere has a charge density of ρ(r) = r. The answer is k(pi)/r^2, but I can't seem to get that. My problem is with finding the enclosed charge. I've tried...

I have calculated the Keplerian Elements for a particular Position and Velocity Vector for a Satellite around Earth. With a solution of the Geodetic Spherical Equation: u = \frac{1}{R} = \frac{μ}{h^2}(1+ e*Cos(\phi - \tilde\omega)) What does the "R" in this equation represent? where ω is the...

Homework Statement Compute the line integral of \vec{v} = (rcos^{2}\theta)\widehat{r} - (rcos\theta sin\theta)\widehat{\theta} + 3r\widehat{\phi} over the line from (0,1,0) to (0,1,2) (in Cartesian coordinates) The Attempt at a Solution Well, I expressed the path as a...

So i can see by symmetry arguments why The electric field inside a uniformly charged spherical shell would be zero inside. But what about a non uniformly charged spherical shell. Say most of the charge is located on one side, why is the electric field still zero? I can see that the flux...

Homework Statement For some work I am doing I wish to be able to define the potential distribution as a function of the radius (ρ) between two concentric electrodes. Homework Equations One solution (from reliable literature) defines the varying radial potential as: V(ρ)=2V0(ρ0/ρ...

If the solution to the electric part of the spherical wave equations is: E(r, t) = ( A/r)exp{i(k.r-ωt) What happens when t=0 and the waves originates at the origin, i.e. r=0 ... which I assume can't be right as you of course cannot divide by zero. Thanks!

If the solution to the electric part of the spherical wave equations is: E(r, t) = ( A/r)exp{i(k.r-ωt) What happens when t=0 and the waves originates at the origin, i.e. r=0 ... which I assume can't be right as you of course cannot divide by zero. Thanks!

Homework Statement I have a question about notation. My professor posted an older practice test with some different notation techniques than I am used to. "Sphere Charge Find the electric field 2.5 m from the center of a region of space with a charge density given by ro=5.5 E-15 R**(2.3)"...

Homework Statement OK, we've been asked to derive the equations of motion in spherical coordinates. According to the assignment, we should end up with this: $$ \bf \vec{v} \rm = \frac{d \bf \vec{r} \rm}{dt} = \dot{r} \bf \hat{r} \rm + r \dot{\theta}\hat{\boldsymbol \theta} \rm + r...

Problem: For the vector function \vec{F}(\vec{r})=\frac{r\hat{r}}{(r^2+{\epsilon}^2)^{3/2}} a. Calculate the divergence of ##\vec{F}(\vec{r})##, and sketch a plot of the divergence as a function ##r##, for ##\epsilon##<<1, ##\epsilon##≈1 , and ##\epsilon##>>1. b. Calculate the flux of...

Problem: Say we have a vector function ##\vec{F} (\vec{r})=\hat{\phi}##. a. Calculate ##\oint_C \vec{F} \cdot d\vec{\ell}##, where C is the circle of radius R in the xy plane centered at the origin b. Calculate ##\int_H \nabla \times \vec{F} \cdot d\vec{a}##, where H is the hemisphere...

Homework Statement The formula for divergence in the spherical coordinate system can be defined as follows: \nabla\bullet\vec{f} = \frac{1}{r^2} \frac{\partial}{\partial r} (r^2 f_r) + \frac{1}{r sinθ} \frac{\partial}{\partial θ} (f_θ sinθ) + \frac{1}{r sinθ}\frac{\partial f_\phi}{\partial...

Hi, can I say that a sphere is a plane, because in spherical coordinates, I can simply express it as <(r, \theta, \varphi)^T, (1, 0, 0)^T> = R? It does sound too easy to me. I'm asking because I'm thinking about whether it is valid to generalize results from the John-Radon transform (over...

Homework Statement A spherical conductor has a spherical cavity in its interior. The cavity is not centered on the center of the conductor. If a positive charge is placed on the conductor, the electric field in the cavity A. points generally toward the outer surface of the conductor. B...

Hello, I need some information about spherical mirrors that I can't find in internet or this forum. How to calculate the amount of light that is focused in the mirror's focus point depending on the mirror's area and the amount of light emited by the source? If that light is reflected by...

Is partial derivative of ##u(x,y,z)## equals to \frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}+\frac{\partial u}{\partial z} Is partial derivative of ##u(r,\theta,\phi)## in Spherical Coordinates equals to \frac{\partial u}{\partial r}+\frac{\partial u}{\partial...

Hi guys, This isn't really a homework problem but I just need a bit of help grasping rotations in spherical coordinates. My main question is, Is it possible to rotate a vector r about the y-axis by an angle β if r is expressed in spherical coordinates and you don't want to convert r...

A spherical shell and a conducting sphere each of radius R are charged to maximum potential. Which of the two has more charge? My attempt: Since in a conductor, no charge can reside inside the conductor so all charge is on the surface of the conductor just like the spherical shell. Now ...

We know that the Newton binomial formula is valid for numbers in elementary algebra. Is there an equivalent formula for commuting spherical tensors? If so, how is it? To be specific let us suppose that A and B are two spherical tensors of rank 1 and I want to calculate (A + B)4 and I want...