Spheres Definition and 504 Threads

I believe there is an elementary way to solve this problem using some analogies with relevant models. First, I consider an electric model of polarization in uniform field. Here, there is a dielectric sphere oriented in uniform electric field ##E_0##. We can find out the electric fields inside...

This is about Euclidian geometry in n dimensions. On this subject it is often noted that the volume of the unit sphere goes to zero as n increases. However this has more to do with the unit cube getting larger than the sphere growing smaller. As n increases the diagonal of the unit cube...

There are two identical spheres with the same charge that are the vertices of an equilateral triangle. ##+3 \mu C## will exert an outward electric field, which is drawn in the FBD below (see the attached pic), Since the horizontal force components (1x and 2x) are equal and opposite at point P...

Hi, Here is the problem What is required to answer this question is two assumptions. Firstly, the component of the momentum normal to the centre line is the same before and after. Therefore, secondly, A must recoil entirely in the horizontal plane. This is the only way to answer this question...

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...

First assuming only one sphere at a potential of 1500 V, the charge would be q = 4πεrV = 4π(8.85×10 −12C2/N · m)(0.150 m)(1500 V) = 2.50×10−8C. The potential from the sphere at a distance of 10.0 m would be V =(1500V)(0.150m)/(10.0m) =22.5V. I don't understand the reasoning of the...

I think if we don't consider electron's/proton's mass then we can say that the amount of charge doesn't need to be equal according to Newton's 3rd Law. I mean having q on one ball and 2q on another ball , still makes the angles having the same size. Is it true ? What if we consider proton's...

For this part (b) of this problem, The solution is, However, would a better explanation be: As the spheres are conductors, there will be free electrons within and on the surface of the conductors that will be polarized by the external electric field between the conductor. This will decrease...

Prove that if there is a fiber bundle ##S^k \to S^m \to S^n##, then ##k = n-1## and ##m = 2n-1##.

Since the forces involved (gravity and electric force) are conservative we can use conservation of energy. The initial energy is ##E_i= k\frac{q_1q_2}{r_0}-G\frac{m^2}{r_0} ## and the final ##E_f=mv^2+k\frac{q_1q_2}{2r}-G\frac{m^2}{2r} ## so from ##E_i=E_f ## we get...

If I have a triangle on a sphere with two of its angles 90 degrees each, do I conclude that it's isosceles and that the shortest distance (on the sphere) beteeen the base and the vertix of the thid angle is 1/4 the circumference of a great circle on the sphere? This is the picture I have in...

Suppose I have two spheres in 3 dimensions of equal mass. In cartesian coordinates, sphere A is traveling with velocity uAi, and sphere B travels with vBi. They will collide elastically. I want to find the final velocities after the collision, ie uAf and vBf. Am I correct in saying that...

Let the radius of the small sphere = r 3r = 1 → r = 1/3 ##x=\sqrt{4r^2-r^2}=r\sqrt{3}## Volume of pyramid: $$=\frac{1}{3} \times (2r\sqrt{3})^2 \times r$$ $$=\frac{4}{27}$$ So m + n = 31, but the answer is 29. I guess my mistake is assuming line AB is tangent to the top sphere. How to do...

you can treat the center of two conducting sphere's like two point charges. Therefore it should be equal to ##k_e q^2/d^2##, but the answer is greater than ##k_e q^2/d^2##. Can someone explain how? Thank you

Hi, I have to found the number of microstates for a gas of N spheres of radius r and volume v in box taking into account the reduced volume after each sphere. V sphere << V box. I'm struggling to find the microstates in general. I don't see how to find the number of microstates without knowing...

I recently encountered this problem on a test where the solution for the above problem was given as follows: $$F= \frac{Gm_1m_2} {r^2} $$ (1) but $$ m=\frac{4}{3}\pi R^3 $$ substituting in equation (1) $$F= \frac{{G(\frac{4}{3}\pi R^3\rho})^2 }{2R^2} $$ where r=radii of the two spheres m=mass...

Suppose you have a sphere of radius a of positive charge, and a concentric shell from a to b of negative charge. The positive charge is equal to the negative charge. (non-conducting, uniform density) Is there an outward pressure at a of kqq/a2/(4πa2) - with pressure decreasing with radius...

Question My attempt for the 1st part, when r = 1.5 m Could someone confirm my answer?

I attempt to solve the problem in the picture below.

I can understand what happens with the conductor... (induction effects). But how can induction happen in insulators ? Is it due to the the induced dipole moment?

Before grounding (left picture), X will be positively charged and Y will be negatively charge. After grounding, I think electrons from Earth will flow to sphere Y and then move to sphere X so X will be neutral and Y will be negatively charged (answer C). But the answer key is D. Why? Thanks

Here are the answer choices: a) F1 = 2F2 b) F1 = 8F2 c) 2F1 = F2 d) F1 = 4F2 e) F1 = F2 I figured that Coulomb's law would tell us the magnitude of the forces are identical, so I answered E, but that was incorrect. (Screenshot of question attached)

I am not quite sure how to present my answer in the form of a function with relation to the distance from the centre. What I got so far is the E1 and E2, for the internal and external sphere respectively. For internal sphere, the charge is volume * 𝜌, so it is $$ \frac{4\pi r^{3}}{3} * 𝜌$$...

I seriously doubt that any of these things exist. For one thing there’s something better. Assuming the civilization has the technology to build a Dyson sphere or ring, would they? With that technology and resources it seems to me it would be much simpler to strip the rocky parts of a large...

The two situations are shown in the figures alongside. The hollow sphere has a thick heavy rim that compensates for the air inside it - both spheres have the same mass ##m_B## and radius ##r_B##. Since the bodies have the same mass ##m_B##, the mass of liquid displaced is the same ...

The following is from the 2018 AQA AS Further Maths/Mechanics Paper 2 exam: 'Two smooth spheres A and B of equal radius are free to move on a smooth horizontal surface. The masses of A and B are m and 4m respectively. The coefficient of restitution between the spheres is e. The spheres are...

Hi, I think this problem is solved in exactly as a similar problem where the two spheres are very far apart and connected by a very long thin conducting wire. I'm trying to explain this in words, since LaTeX does not seem to work any more (for some reason LaTeX syntax is not replaced by maths in...

I am trying to calculate the interaction energy of two interpenetrating spheres of uniform charge density. Here is my work: First I want to calculate the electric potential of one sphere as following; $$\Phi(\mathbf{r})=\frac{1}{4 \pi \epsilon_{0}} \int...

What would the fact that the fifth surface is connected to the ground imply: that V(r=R_5)=0 or that \sigma _5=0?

I literally don't know where to start with this, i drew a free body diagram to try and understand where the cylinder was affecting them, but it didn't get me anywhere

My answer was +Q/3. I was assuming that the charges would distributed themselves completely. But, apparently, I'm wrong? For example, if there were 12##e^-##s on Sphere C, then, in the first step in the system: the ##e^-##s would balance out until each sphere has 4 ##e^-##s each? What am I...

If you were to positively contact charge a small ~1 mm diameter sphere using a Van de Graaff generator, and were to charge it sufficiently high enough that field evaporation began to occur, what would happen? Would the rate of evaporation increase exponentially as the field strength would...

Can anyone tell me if gravity is greatest for someone standing on the surface of a hollow small sphere or a hollow larger sphere when the spheres are of equivalent mass, and the thickness of the shell is greater on the smaller sphere than the larger sphere (in order to maintain equivalent mass)...

Let ##(r,\phi, \theta)## be the radial, polar and azimuthal coordinates respectively. As ##\vec{B}## is confined to ##xz## plane such that ##\theta = \alpha## I assumed ##\vec{B}## on the surface of shell to be ##\vec{B} = a\sin(\alpha) \hat x + \cos(\alpha) \hat z \tag{1}## Surface area...

I begin by drawing the problem. Let the center of the bigger sphere on the left by the origin for the system. Calculating ##x_C = \frac{10 \times 1.1 + 20 \times 2.15}{50+20+10} = \frac{54}{80} = 0.675\; \text{m} \;= \boxed{67.5 \; \text{cm}}\;##. Problem is, doesn't match the answer in the...

There have been some other threads on similar problems but none address one specific point I'm confused about. The change in GPE of a body is the negative of the work done on that body by a gravitational field between two points; by this logic, since the same (but opposite) gravitational forces...

Two identical conducting spheres A and B carry equal charge. They are separated by a distance much larger than their diameters. A third identical conducting sphere C is uncharged. Sphere C isfirst touched to A, then to B, and finally removed. As a result, the electrostatic force between A and B...

The force of the sphere increases because according to Coulomb's force increases as the distance decreases. I'm not sure if this is correct.

Homework Statement The solution to this problem is B, and I was able to get the answer by calculating the total potential at ##r = 2a##, however, what I don't seem to understand is why must the voltage be calculated at ##r=2a## but not ##r=3a##. Homework Equations ##V(r) = - \int_a^b E(r)...

Homework Statement Show that the force resisting change of the minimum distance h between the surfaces of two rigid spheres of radii a and b which are nearly touching is: $$6\pi\frac{\mu}{h({a^{-1} + b^{-1}})^2}\frac{dh}{dt}$$ provided $$\frac{\rho h}{\mu}\frac{dh}{dt}$$ Homework Equations...

I'm considering two identical spherical conductor each of radius ##a## and separated by a distance ##d##, and trying to figure out the capacitance of this configuration. My thoughts are that since capacitance is $$C=\frac {Q}{V}$$ and that the spherical conductors are equipotential surfaces...

Homework Statement Homework EquationsThe Attempt at a Solution I am having trouble figuring out why the answer is A) the electric field points radially between A and B. I think it is because since the point between A and B is mostly negative, the electric field would point outwords more...

Homework Statement Homework Equations V=kQ/R The Attempt at a Solution The answer is B)kQ/R. It is because V= k(2Q)/R. I don't understand why Q=2Q in this case. Isn't the point on the inside of the outer shell, so the Q for the equation is just Q?

Homework Statement Two thin conducting spherical shells have radii R1 and R2.Outer shell is charged to q and inner shell earthed.Find charge appearing on both the shells. Homework Equations The Attempt at a Solution Isnt the charge on inner shell 0 and charge on outer shell remains Q as it...

Homework Statement Sphere 1 has net positive charge Sphere 2 has net negative charge Sphere 3 has net positive charge The ranking of net charge magnitudes are SPHERE 3 > SPHERE 2 > SPHERE 1 All spheres are conductors Sphere 2 is moved away from Sphere 1 and toward Sphere 3 so that 2 and 3...

Homework Statement If two balls, being identical in volume, but different in density (one ball is made of iron, the other of aluminum) roll down from an inclined plane, which will reach the bottom first and which will cover a larger distance after having reached the bottom? IMPORTANT NOTE...

Homework Statement Homework Equations None (conceptual) The Attempt at a Solution My logic here is this, Sphere 3 has a net positive charge so it is repelling the positives in sphere 2 and attracting the negatives in sphere 2. This means that D has negative charge and C has positive charge...

Lets say I have two spheres of equal dimensions, one charged and one uncharged. Now I connect them with a conducting wire. They will now very quickly reach equal potential. Can it be said that the total charge on each sphere remains almost unchanged?

If I consider a tetrahedron of four densely packed spheres of unit radius, what it the radius of the largest sized sphere that can fit in the space in between?

Homework Statement Two identical uniform metal spheres of radius 47 cm are in free space with their centers exactly 1 meter apart. Each has a mass of 5000 kg. Without integrating, show that gravity will cause them to collide in less than 425 seconds. [/B] Source: Classical Mechanics, R...