Sphere Definition and 1000 Threads

A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of...

My idea was to use the continuity of parallel components of the magnetic field and the spherical coordinate system. Because the magnetic field in a superconducting material is 0 and the current is completely confined to the surface, there only is a ##\mathbf{B}_{above}## component. The equation...

Electric field inside conducting charged sphere is zero so the potential inside it will be constant, hence there will be no work to move a charge from the surface to the center. It means the work done is for moving the point charge 𝑞 to the surface of the conducting charged sphere. The ##d\vec...

Suppose there was a bijection ##\varphi## between the 2-sphere ##M## and the euclidean plane ##\mathbb R^2##. Then one could endow ##M## with the initial topology from ##\mathbb R^2## through ##\varphi## turning it into an homeomorphism (this topology on ##M## would be different from the subset...

We start with hydrostatic equilibrium: $$ \frac{dP(r)}{dr} = K \gamma \rho ^ {\gamma -1} \dfrac {d \rho}{dr} =-\frac{GM(r)\rho(r)}{r^2}$$ $$ K \gamma \rho ^ {\gamma -1} \dfrac {d \rho}{dr} = - \dfrac {G \rho (r) }{r^2} \int 4 \pi r^2 \rho (r) \, dr$$ $$ r^2 K \gamma \rho ^ {\gamma -2}...

Let the charge density be $\rho$, radius be $R$, total charge be $Q = \rho \frac{4}{3} \pi R^3$. We know from Gauss's law, $E (r) = \frac{Q r}{4 \pi \epsilon_0 R^3}$. We also know from Maxwell stress tensor $\sigma(r) = \half \epsilon_0 E^2$. We can compute the pressure due to the electric...

Alan Macdonald claims in "4 Appendix: The Equivalence Principle" of his text "Special and General Relativity based on the Physical Meaning of the Spacetime Interval", that his calculation regarding a 2D-surface of a sphere proves, that the equivalence principle is violated. He defines the...

Let's assume that we have a hollow sphere with holes at opposite ends of the diameter. What would be the field inside the hollow sphere? I know that we can look at this as the superposition of the hollow sphere without holes and 2 patches with opposite surface charge density. For some reason, in...

Hi, in the books I looked at, the 2-sphere manifold is introduced/defined using its embedding in Euclidean space ##\mathbb R^3##. On the other hand, Mobius strip and Klein bottle are defined "intrinsically" using quotient topologies and atlas charts. I believe the same view might also be...

My thinking would be to do a work integral Work = integral dWork = integral delta V dq = integral delta V 4(pi r^2) dr The problem is, is this possible with a single integral? Due to the offset cavity, the electric field E will not be constant at a given r.

If there is no field inside the conductor, how can there be electric potential? I think of potential very similar to gravity, as how much energy would be required to move a particle of mass/charge against the gravitational/electric field. If there is no field at all, how would there still be...

Four points lie on the surface of a sphere. Given the six distances between the points, calculate the radius of the sphere. This is (allegedly) an advanced high school level problem. However, it is a remembered problem, so it is possibly misremembered (i.e. there might have been some “bice...

Attempt : (Turns out, there is more mathematics in this problem than physics. The crucial part involves the use of vector calculus where one needs to find the volume of a region bounded at the top by a portion of a sphere. That is where am stuck.) The mass of water displaced by the ball...

Question 1: The sphere is at the electric potential of the top plate. As the sphere is small with respect to the capacitor, one can consider the bottom plate to be at infinity and therefore we can use the capacitance formula as C = 4 ∏ε0 R. The charge Q is therefore Q = C (V -0) = C V...

I don't understand where F comes from because in the problem there is only the tension of the cord. And I have another question the forces along y-axis always be equal to zero? And why T cos q - m g = 0 equal zero? if it is along the X-axis

Hello, this question may seem weird but I really need help on this. To bring the formula for the height h of the triangle above, I have to create a relation between potential and kinetic energies of the black ball with mass m (I can't find any other methods than this). For a sphere falling...

Hello everyone, I was trying to find volume of a sphere by doing some calculus, the area of a circle is ##{\pi}r^2## So I thought I would calculate the volume of one hemisphere and then multiply by two, but I got a different result than the standard formula, the standard formula is ##\frac 4 3...

is there an easier way to calculate the dipole moment? I described ## \vec r## in spherical coordinates. I thought at first that due to the symmetry I can assume that dipole-moment only points in the ##z##-direction, but the charge distribution is inhomogeneous, so I made the following...

The sum of the forces should be 0. Sin A'C'B' = px/b px = mg . sin alpha P should be px = - m.g. sin alpha and py = m.g.cos alpha Finally i fund as result F = -0.8 and R = -1.23 but for the second question i didn't fund the radius of the circle.

Think of a 3D rectilinear grid made of these rectangular cells, some of the cells will intersect with the sphere. I am trying to compute each intersecting area and the total sum. Ideally the total sum of the intersecting area should be close to ##4 \pi r^2##. I have not found any literature...

Ich wäre Ihnen sehr dankbar, wenn Sie sich meine Lösung der folgenden Übung ansehen: A sphere with radius ##R ## is spatially homogeneously loaded and rotates with constant angular velocity ##\vec{ \omega}## around the ##z ## axis running through the center of the sphere. Calculate the...

Hello everybody, I am thinking of the following problem: A sphere a radius r is in a much larger container of radius R. In this container, a fluid continuously flows with turbulent conditions from bottom to top. I would like to approximate the force pushing the sphere up. Some calculations I...

Consider a metal sphere connected to one end of the battery and the other end of the battery to be connected to the ground. Does the metal sphere become electrically charged with this method?

I am trying to understand the motion of the sphere in the image above, and I am a bit confused about the motion. How does the ball move down the cone? Will the rotation of the cone cause the ball to rotate with it, and which direction would the static friction be in? What does the path the ball...

Imagine a hollow sphere made of a material with high elasticity constant(e.g. steel). How much thickness should it have to prevent it from crushing when the air inside is pumped out? Is it valid to use Lame solution to quantify the answer? What about Finite Element Analysis?

I'm having trouble understanding how a charge distribution in a sphere can make this happen. My instinct is that the fact that it's radially directed is a big hint of something, but I don't know what that hint might be alluding to. If the net E-field is constant inside the sphere and is always...

I am sure that the radius of the two spheres changes equally. But in answer of the question it said that their change in volume is the same. Is it correct?! Link of website: https://www.toppr.com/ask/question/two-spheres-of-same-size-are-made-of-the-same-metal-but-one-is-hollow/ I think it is...

Suppose a sphere is rolling on horizontal surface. The point of contact is the instantaneous center of rotation. It's velocity momentarily is zero. So in a time dt it'll stay where it is. However during this time dt the point next to this contact point at its right side will move forward and...

I was thinking of using the sphere and point charge as an analog, but is quite diferent from what i have seen

Problem: Solution part a) where formula 6.14 is just M x n. We need to do part b without seperation of variables, I'm quite stuck. Will B just be the magnetic field inside a solenoid? How can I find the other fields.

The sphere floats on water so we should have: ##F_b=F_g## The buoyant force is equal to the weight of the displaced fluid, so : ##\rho _wV_wg=\rho _sV_sg## (w: water, s: sphere) From last equation we have : ##V_w=\frac {\rho _s}{\rho _w} V_s \rightarrow V_w=5 V_s ## The volume of displaced...

Zero does not have an inverse. And y=0 does not have an inverse. Does the wave form y=0 for all x have an inverse?

Hi there! I have a question about the Faraday's Nested Sphere Experiment, please see the attached pdf. I wonder why equation (1) and the electric field's equation ( coming after (1) ) consider only the charge Q. Why there aren't charge -Q in the equation? Ps. I'm thinking about point charges...

**Problem:** Find parametric equations for a simple closed curve of length 4π on the unit sphere which minimizes the mean spherical distance from the curve to the sphere; the solution must include proof of minimization. Can you solve this problem with arbitrary L > 2π instead of 4π? There...

##\require{physics}##I am trying to understand how the Ewald's sphere works in the context of X-ray diffraction (XRD). I am reading from Kittel's book, as well as a few lecture series. Let me first state what I have learnt in this context (please correct me if I am wrong). For any real lattice...

Hello, I am wondering if in an n-ball the number of lattice points is finite. First, we have a ball which is bounded by the radius. The distance between two lattice points is given by the successive minimum. Theoretically, one could now draw a ball* around each lattice point in the (big)...

neglect friction and motion (sliding) and G(sphere)=20N. In this question I reached two different result with two different solving method.But one of them is false according to answer key. My question is why first solving way is false? Because the first solution way makes sense to me. If we...

How do I use the ewald sphere to calculate h, k and l?

I am unsure wether or not all I can use the Ewald sphere for is to calculate d_hkl for the diffracted wave vector. For cubic lattices for example d= a/sqrt(h^2+k^2+l^2). To determine the lattice constant "a" you would then need to know exactly what your h,k and l are or you use lattice-dependent...

I have a flat planar part made of crystalline sapphire (about ~2k weight, and polished to a mirror finish) that rests on three ball bearings, and I want to minimize the static friction at these 3 interfaces. The ball bearings are fixed so they cannot roll, and the sapphire part can only slip...

I consider a disc of thickness ## R d\theta ## as shown in the figure. Then, $$ dV = \pi R^2 sin^2 \theta R d\theta $$ ( Area of the disc * its thickness) Hence, $$ V = \int^{\pi}_{0} \pi R^2 sin^2 \theta R d\theta $$ $$ V = \frac 1 {2} {\pi}^2 R^3 $$ ....(1) While $$ V =...

I - A point divides a line into two parts; II - A line divides a plane into two parts; III - Does a smaller sphere divide a larger sphere into two parts, like layers of an onion? Note that the first two statements, the question of infinity must be considered. For the third statement, is the...

I'm supposed to do the surface integral on A by using spherical coordinates. $$A = (rsin\theta cos\phi, rsin\theta sin\phi, rcos\theta)/r^{3/2}$$ $$dS = h_{\theta}h_{\phi} d_{\theta}d_{\phi} = r^2sin\theta d_{\theta}d_{\phi}$$ Now I'm trying to do $$\iint A dS = (rsin\theta cos\phi, rsin\theta...

Flux=$$\iint(xz, -yz, y^2)\cdot(x,y,z)/\sqrt{2} dA=\int_0^{2\pi}\int_0^1 r^2\cos^2\theta \sqrt{1-r^2/2} rdrd\theta$$. Integrating this doesn't give the correct answer of ##\pi/4##.

I drew a diagram using all of the information however, I am stuck and not to sure how to get one of the charges

I solved that the hollowed out mass is M/8, which is correct. I don't understand why it is incorrect to substitute the remaining mass (7M/8) back into the F = G*m1m2/r to produce the force. Why is the solution the force of the whole lead sphere minus the force of the “hole” lead sphere, which is...

I can calculate the electric field strength at any point above the plane with Gauss' Law (##E = \frac{\eta}{\varepsilon_0}##) and so the electric potential at any point a perpendicular distance ##z## above the conducting plane (##V=−\frac{\eta}{\varepsilon_0}z##). But I'm having trouble taking...

I agree with Doc. Al that, "For the simplified case of a uniform density spherical planet, the gravitational field varies linearly from 0 at the center to its full value at the surface." But, what is the effect, if any, on the shape and density distribution of such a sphere over time (e.g...

Hello and please know I am very greatful for your help! I am wanting to learn how to measure light. I have chosen a specific light for this to help me better understand. Lm- 7800 CD-620.7 So, I got that far, lol. I don't really know how to input the numbers for the Steradians equation, I have an...

Suppose you had a thin-walled sphere fully submerged in a liquid. The sphere is filled to the equator with a liquid of sufficient density to reach buoyant equilibrium. Will the lateral cross-sectional areas of the thin-walled sphere experience tensile stresses in the longitudinal axis? Why or...