Sphere Definition and 1000 Threads

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...

A sphere is rolling inclined wall (θ radian). and the momentum of that sphere is L = 1/2 mv^2 + 1/5 mv^2 + mgx sin(θ) = 7/10 m v^2 + mgx sin(θ) ∂L/∂v = p 7/5 m v = p but I can't understand why the factor of mv is 7/5. p is the linear momentum of sphere. which means the factor of mv must be...

Homework Statement A dielectric sphere of radius R with uniform dielectric constant ε has an azimuthally symmetric density charge σ = σ0 cos θ placed on the surface. Outside the sphere is vacuum. (a) Obtain the electrostatic potential inside the sphere, φin. (b) Obtain the electrostatic...

Homework Statement We consider a sphere of radius R which carries a uniform surface charge density \sigma and spins with angular velocity \omega around a diameter. We use spherical coordinates (r, \theta, \phi) with origin at the centre of the sphere and the z-axis along the rotation axis...

Hello! I'm Steven, and I'm currently working on the following problem: The Earth can be seen as a conducting sphere with an electric field: E= -(150V/m)r (on its surface) and where r is the unit vector . The Earth has a radius 6371 km. So, I am asked to calculate the electrostatic potential...

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...

Apologies if this question has already been asked, but is it not possible to derive the formula of a sphere by imagining a circle sliced in two, then rotating this semi-circle about the flat side (imagine the flat side is stuck to a skewer) by 2Pi, so as to sum up the semi-circles to "create"...

So we all know that it takes work to build up a sphere of charge by taking charge from infinity and piling it up into a sphere. Since the sphere wants to break apart under repulsion, its like a spring. It has intrinsic potential energy.However it doesn't seem the case with a sphere of mass, with...

$\tiny {205.23} $ $\text{ The volume }\displaystyle V=\frac{4}{3}\pi{r}^{3} \text{ of a spherical balloon changes with the radius.} $ $\text{a) at what rate } \displaystyle \frac{ft^3}{ft} \\$ $\text{does the volume change with respect to the radius when $r=2 ft.$} $...

Homework Statement Homework Equations spherical Jacobean The Attempt at a Solution I have (sorry, have to capture my work, too hard to type) then the integration of p3 ep2 = 1/2 ep2 (p2-3/2) ??

Homework Statement [/B] The sphere of radius R has the potential at the surface equal to $$ V_0 = \alpha sin^2(\theta) + \beta $$ where ## \alpha, \beta ## are some constants. Find the potential inside, and outside the sphere.Homework Equations $$V(r,\theta) = \sum_{l=0}^{\infty}(A_l r^l +...

Homework Statement Two solid spheres hung by thin threads from a horizontal support (Figure 1) are initially in contact with each other. Sphere 1 has inertia m1 = 0.040 kg , and sphere 2 has inertia m2 = 0.10 kg. When pulled to the left and released, sphere 1 collides elastically with sphere 2...

I just read through this thread* where a mathematician tries to shutdown another guys use of the "sphere to donut" analogy relative to ripping spacetime because a sphere is a "2-dimensional manifold". That is so completely and entirely irrelevant when both are still just 3 dimensional shapes...

Find E(r) inside and outside a uniformly charged spherical volume by superposing the electric fields produced by a collection of uniformly charged disks. a+b) Given equations, sketch of problem This is the equation in the handbook for a disk (but in the exercises the z becomes x, without loss of...

Homework Statement A spherical shell has inner and outer radii r_a and r_b, respectively, and the temperatures at the inner and outer surfaces are T_a and T_b. The thermal conductivity of he shell material is k. Derive an equation for the total heat current thought the shell in the steady...

Homework Statement A point charge q is located a distance d away from the centre of a grounded conducting sphere of radius R<d. I need to find the charge density on the sphere and the total induced charge on the sphere. This is very similar to example 2 here...

Using only a compass and a ruler, how can we measure the diameter of a hollow sphere?

I found this on the Internet . The formula is Surface Area = R^2 \displaystyle \int _0 ^ {2 \pi} \int _{0}^{\pi} \sin \theta d \theta d \phi I'm wondering why the limit of θ is from 0 to π only ? why not from 0 to 2π ? Because it's a perfect sphere...

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 Show that the equation represents a sphere, and find its center and radius. 3x2+3y2+3z2 = 10+ 6y+12z Homework EquationsThe Attempt at a Solution 3x2+3y2-6y +3z2 -12z =10 My equation is how the constants in-front of the squared terms affect the sphere formula? Besides that...

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...

The common explanation is this: If the conductor has a net charge, then the charges repel each other until they arrange themselves symmetrically around the outside of the sphere, and if you do the math the electric field will cancel out everywhere inside the conducting sphere. Alright, but what...

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 Hi everybody! I'm currently training at surface integrals of vector fields, and I'd like to check if my results are correct AND if there is any shortcut possible in the method I use. I'm preparing for an exam, and I found that it takes me way too much time to solve it. I...

NOTE: Other threads suggest solving it with Gauss' Law. I'd like to see an approach through direct integration, no full followthrough necessary.. 1. Homework Statement Consider a sphere with a uniform distribution of charge ρ (ro). Inside the sphere is a cavity (spherical). Calculate the...

Homework Statement what will be Lagrange,s equation of motion for a particle confined to move on surface of sphere whose radius is expanding such that Homework Equations Euler-lagranges equation of motion d/dt(∂L/∂{dq/dt})-∂L/∂q=0 The Attempt at a Solution Z=(R+R0e^at)cosθ...

I have been given a problem with 4 equations, that need to be matched up to the corresponding image. I have worked the equations already and determined their center, but for the life of me I cannot seem to figure out which graph goes with which equation. The images are not that easy to read...

I was idly thinking of Dyson Spheres and solar energy when this idea came to me: say you were technologically advanced enough to build a perfect, Dyson sphere and were able to deal with the problems of drift--in other words you belong to a really, technologically advanced civilization! It...

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...

Is it possible to find the volume of a sphere(i know the formula) using definite integration ? And if possible how to proceed ?? Thanks in advance

Homework Statement A hollow sphere of inner radius 9.0 cm and outer radius 10.0 cm floats half submerged in a liquid of specific gravity 0.80. (a) Calculate the density of the material of which the sphere is made. (b) What would be the density of a liquid in which the hollow sphere would just...

Homework Statement Let be ##f : V \rightarrow \mathbb{R}## a ##C^{1}## function define on a neighbourhood V of the unit sphere ##S = S_{n-1}##(in ##\mathbb{R}^{n}## with its euclidian structure.). By compacity it exists u in S with ##f(u) = max_{x \in S}f(x) = m##. My goal is to show that ##u##...

I have posted this question earlier but I think it was ill-stated. I try to give it in a simpler fashion in this thread. In the problem it is stated that there a rotation around the x-axis of a (stereographic) sphere is given by $$\delta \phi = \cot \phi \cot \theta \delta \theta$$ where...

Homework Statement Show that the equations $$ \delta \phi = \cot \theta \cot \phi \delta \theta, \quad \delta \phi =- \cot \theta \tan \phi \delta \theta$$ represent rotations around the x and y axes respectively of a stereographic sphere. Both these two rotations map the sphere on itself and...

Greetings All, I have a rather odd question which has been bothering me. If you have a perfectly round sphere sitting on a perfectly flat plane, what is the area of surface contact between the two? Is there an actual value, or is it something which can't be calculated. I'm assuming the diameter...

Homework Statement We want to calculate the field of a uniformly polarized sphere of radius=R Homework Equations V(\vec{r}) = \frac{1}{4 \pi\epsilon_{0}} \oint_{S} \frac{\sigma_{b}}{r} da' + \int_{V} \frac{\rho_{b}}{r} d\tau' The Attempt at a Solution i)I know that \sigma_{b} = P...

please , I'm french , so i didn't quite get the meaning of this sentence.

Homework Statement A small sphere with charge 2.4 micro coulombs is suspended from a thread between 2 charged plates. The plates have a voltage of 62 and the distance between the plates is 3.1 cm. The sphere hangs at 18 degrees to the vertical. Homework Equations E = V/r FE = Eq Fnetx = FE -...

Qualitatively what is the nature of emitted light if we evenly pump a sphere of lasing material? Suppose there are no mirrors to favor one direction. Does the answer depend at all on the size of the sphere? Thanks!

I read in http://www-library.desy.de/preparch/books/vstatmp_engl.pdf page 29 (43 for pdf) that the mean volume is: <V> = \int_{-\infty}^\infty dr V(r) N(r| r_0,s) I have two questions. Q1: why do they take the radius to be from -infinity to +infinity and not from 0 to infinity? Q2: is there an...

The sphere is released at a height H above a fixed inclined plane, as shown in the attached figure. The coefficient of restitution at impact is e>0 (that is the sphere leaves the surface just after impact), the coefficient of friction between the sphere and the plane is \mu. I need a...

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...

Hello! When computing the density of states of electrons in a lattice, a material with dimensions L_x, L_y, L_z can be considered. The allowed \mathbf{k} vectors will have components k_x = \displaystyle \frac{\pi}{L_x}p k_y = \displaystyle \frac{\pi}{L_y}q k_z = \displaystyle \frac{\pi}{L_z}r...

I know the moment of inertia for both a solid sphere and a hollow sphere is , but my teacher has derived a moment of inertia of the sphere but am not sure about what axis she was deriving it , and she got this answer 3/5 MR^2

I read here, http://www.space.com/24781-big-bang-theory-alternatives-infographic.html , that, "What we call the "observable universe" (or the "Hubble Volume") is the spherical region, about 90 billion light-years in diameter, that is centered on any given observer. This is the only part of the...

Homework Statement Charged metal sphere hanging on an isolated thread of negligible mass is put in a homogeneous horizontal electric field so that the thread makes a 45 degree angle with the el. field. What angle does the thread with the sphere close with the el. field after we remove 40% of...

Homework Statement A spherical capacitor comprises two thin metal spheres of different radii but with a common centre. The following series of operations is completed: The spheres are mutually connected by an internal wire. The outer sphere is raised to potential +V with respect to ground. The...