Spherical Definition and 1000 Threads

Wrote my question up in Latex Update: I have corrected the mistake when I stated the "textbook" version of the equation that the problem requires to be used. I have reposted the pdf with the same name.

Homework Statement Let V be the volume of the solid enclosed by the sphere x^2 + y^2 + z^2 - 2z = 0 , and the hemisphere x^2 + y^2 + z^2 = 9 , z ≥ 0. Find VHomework Equations Using spherical coordinates: x^2 + y^2 + z^2 = ρ^2 z = ρcos(ø) The Attempt at a Solution So I changed both of them to...

I've got a Green's function in which all the impulses are on the line from the north pole to the origin (polar angle θ=0) and terminating with a point impulse at the north pole. I've found its gradient at a field point, and I want to rotate everything to a new coordinate system with the source...

See image attached. (I've had a google but can't find anything). I am trying to understand the expression : Rdθ.2∏Rsinθ Here are my thoughts so far: Rdθ is the width of a strip, θ being the variable changing/to integrate over, giving arise to the elements. 2∏Rsinθ must then...

is it logical to ask this question in Spherical coordinates: Using the differential length dl , find the length where r=1 0<Θ<∏/4 ∏/2< θ <∏/4 where Θ is the azimuthal angle. What I mean by ∏/2< θ <∏/4 is that the line is a "diagonal" line which has an ascention of ∏/4 from the xy...

The question is: a conducting spherical shell of radius a rotates about the z axis with angular velocity ω, in a uniform magnetic field B= B_{0}\hat{z} . Find an expression for the EMF developed between: i) the north pole and the equator (2 marks); ii) the north pole and the south pole (1...

Hello i want to solve this problem via another approach question: Book Solution: my approach: Coulomb's law for surface charge: as we know the filed point is a fix point and i set the name of h instead of z (r is in spherical coordinate and haz in cartesian) so ar...

If the B-mode sky plots could be Fourier transformed what would be a plot of the lowest order B-mode harmonic plotted on a sphere look like? I guess we need two functions of spherical coordinates, one function for amplitude at points on a sphere and one function for the orientation at the...

Homework Statement A spherical capacitor consists of a spherical conducting shell charge -Q concentric with a smaller conducting sphere of radius 4.0 cm and charge Q. The larger conducting shell has inner and outer radii of 11.0 cm and 13.0 cm, respectively. What is the capacitance of the...

Homework Statement A conducting spherical shell of outer radius a and inner radius 3a/4 is cut in two pieces via a horizontal plane a distance a/2 above the center of the spherical shell, as shown in Figure 1. Let us label "A" the upper part of the shell and "B" the lower part of the shell...

Hello! Homework Statement I want to evaluate the derivative of spherical harmonics with respect to the azimuthal angle and express it in terms of spherical harmonics.2. Homework Equations and 3. The Attempt at a Solution I have calculated the derivative of the spherical harmonic with respect...

Homework Statement I want to show that <n',l',m'|\hat{z}|n,l,m> = 0 unless m=m', using the form of the spherical harmonics. Homework Equations Equations for spherical harmonics The Attempt at a Solution Not sure how to begin here since there aren't any simple eigenvalues for...

Homework Statement A solid insulating sphere of radius a = 5.6 cm is fixed at the origin of a co-ordinate system as shown. The sphere is uniformly charged with a charge density ρ = -494 μC/m3. Concentric with the sphere is an uncharged spherical conducting shell of inner radius b = 10.8 cm...

Homework Statement A metal sphere with radius a is supported on an insulating stand at the center of a hollow, metal spherical shell with radius b. There is charge +Q on the inner sphere and charge -Q on the outer shell. Take the potential V to be zero at infinite separation. Calculate...

Note: physics conventions, \theta is measured from z-axis We have a vector operator \vec{L} = -i \vec{r} \times \vec{\nabla} = -i\left(\hat{\phi} \frac{\partial}{\partial \theta} - \hat{\theta} \frac{1}{\sin\theta} \frac{\partial}{\partial \phi} \right) And apparently \vec{L}\cdot\vec{L}=...

Homework Statement A spherical capacitor with inner and outer radii a and b, contains a dielectric material with small conductivity \sigma between its spheres.Find the vector potential and magnetic field of this configuration. Homework Equations The Attempt at a Solution \nabla^2\phi=0...

Homework Statement In a spherical well in which.. V= \begin{cases} 0,\text{for }0 \le r < R \\ ∞, \text{for } r > R \end{cases} the s-wave eigenstates are \phi_n(r)=\frac{A}{r}\sin\left( \frac{n\pi r}{R} \right) where A is a normalization constant. If a particle is in the ground state and...

I have been trying to derive the Laplace in spherical co ordinates. I have attached a file which has basic equations. I am trying to get the following equation. d(phi)/dx= -sin(phi)/(r sin (theta)). I have also attached the materials I am referring to. Can someone please help me derive...

1-If we had a spherical capacitor and the voltage across it is 1000 V, I need to know the charge on every plate, I know 2 ways to solve this either using C=Q/V or V=Q/r * 1/4ε0∏. So, which one should I use, and why? 2-Another thing, according to hyperphysics the formula for the capacitance of...

Homework Statement I just want to know how to get from this: ∂ø^/∂ø = -x^cosø - y^sinø to this: = -(r^sinθ+θ^cosθ) Homework Equations All the equations found here in the Spherical Coordinates section: http://en.wikipedia.org/wiki/Unit_vector The Attempt at a Solution I've...

Homework Statement Find the surface area of the Earth (as a fraction of the total surface of the earth) that lies above 50 degrees latitude North. Homework Equations $$A = \int_R\sqrt{|\det(g)|}d\theta d\phi$$ The Attempt at a Solution Hence I get $$\int_0^{360}...

Homework Statement A conducting spherical shell is divided into upper and lower halves with a narrow insulating ring between them. The top half is at 10V and the bottom half is at -10V. Write down the appropriate expansion for Φ and use symmetry and the expected behavior at the origin to...

Homework Statement An object is placed 50 cm in front of a convex mirror and its image is found to be 20 cm behind the mirror. What is the focal length of the mirror? Homework Equations 1/d0 + 1/di = 1/f The Attempt at a Solution 1/50 + 1/20 = 1/-f 70/1000 = 1/-f f = 14.28...

1. Problem: I have a wave function ψ(r) = (x + y + z)*f(r) and want to find the expectation values of L2 and Lz. It is suggested that I first change the wave function to spherical coordinates, then put that in terms of spherical harmonics of the form Yl,m. 2. Homework Equations ...

Homework Statement A spherical metal shell has charge per unit area sigma and radius R. What is the magnitude of the electric field at a distance x from the surface of the sphere? You may include only these variables in your formula: sigma, R, epsilon_0, x Homework Equations...

My class hasn't delved into Gauss's Law much besides describing conductors at electrostatic equilibrium to have no net electric field or force within itself. For the picture, the question is: What are the magnitudes of the electric fields at: 1) r = a 2) r = 3/2 a 3) r = b...

Homework Statement Consider a spherical shell with inner radius r1=0.30 m and outer radius r2=1.00 m. The hollow inside the shell contains no charge; and charge is distributed on the inside surface of the shell and within the shell itself, such that the electrical field inside the shell itself...

Homework Statement A spherical stone of mass 0.500 kg and radius 10 cm is launched vertically from ground level with an initial speed of 20.0 m/2. As it moves upwards, it experiences drag from the air as approximated by Stokes drag, F=6η∏Rv, where the viscosity of air is 1.002 mPa*s...

When normalising the S.E. in spherical coordinates you split it up into 3 integrals, with respect to r, theta and phi. My question is, once you have found the constants for each, when writing out the normalised PSI do you simply place them as a product in the solution? i..e PSI...

Homework Statement Consider three concentric conducting shells, with potentials 0, ø_0, 0 and radius a, b ,c where a < b < c. (a)State conditions for Laplace to work and boundary conditions for E (b)Show ø is of the form: (c) Find ø and E everywhere. (d) Find the charge density and...

Hello, I came across the name spherical even even nuclei in an exercise about the hyperfine structure. What does it refer to? That the number of protons and the number of neutrons are both even? So that there is no nuclear magnetic moment?

Just a quick general question in applying Gauss's law. Not exactly homework, more a general question so I can understand my other homeworks better. I have a spherical shell with inner radius R_1 and outer radius R_2 and a point charge Q in its center. It is NOT a conducting sphere. In the...

I want to derive this equation: V(r) = \frac{1}{\epsilon_0} [\frac{1}{r} \int_0^r \! r'^2 \rho(r') \, d r' + \int_r^{\infty} \! r' \rho(r') \, d r' ] of a spherical charge distribution. I can do it with the general integral definition of the electrostatic potential (which is basically...

Homework Statement A spherical shell of radius R0 has a non-uniform surface charge density: η=η0*cos(2θ), where θ is the angle measured from the positive z axis and η0 is a constant. This shell is inserted into another spherical shell (this one has a volume) with its inner lip at radius R1...

I am considering the gravitational time dilation at the centre of a spherical, non-rotating body (such as the Earth). The usual formula for gravitational time dilation is √(1-r_s/r) where r_s is the Schwarzschild Radius and r is the radius of the clock compared to one at infinity, however, this...

I was looking at the Wikipedia entry on the Hamilton-Jacobi equation, and was confounded by the equation at the beginning of the section on spherical coordinates: http://en.wikipedia.org/wiki/Hamilton–Jacobi_equation#Spherical_coordinates Shouldn't the Hamiltonian simply be $$ H =...

1) If u(r,\theta,\phi)=\frac{1}{r}, is \frac{\partial{u}}{\partial {\theta}}=\frac{\partial{u}}{\partial {\phi}}=0 because u is independent of \theta and \;\phi? 2) If u(r,\theta,\phi)=\frac{1}{r}, is: \nabla^2u(r,\theta,\phi)=\frac{\partial^2{u}}{\partial...

When solving for the spherical harmonic equations of the orbital angular momentum this textbook I'm reading.. Does this mean that there must be a max value of Lz which is denoted by |ll>? Normally the ket would look like |lm>, and since m is maxed at m=l then |ll> is the ket consisting of the...

So, I was curious about this and found more or less what I was looking for here: http://electron9.phys.utk.edu/vectors/3dcoordinates.htm Except, something is bothering me about those equations. At the very bottom, the equation for Theta in a spherical coordinate system; shouldn't it be...

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} =...