I am trying to find an equation or a series of equations that relates the current I carried by a solenoid of length L with turns N to the force F experienced by a magnetic dipole of strength M that rests at a distance d from the solenoid so that they are both aligned along their central axes. Specifically, I am trying to model this scenario inside of matlab, so it would be helpful if the answer could be expressed in such a way that differential equations are avoided. For example, I saw answers on other forums that simply stated that F = M dB/dx, but this isnt very helpful for me because the sh ..read more

I don't really believe in the Multiverse Theory, but hypothetically, if universes were like soap bubbles in a foam, what might that foam be like? I know this seems like an opinion-based question but what might the properties of true nothingness be based on what we know about the fabric of the universe? Would there be no gravity or some other repulsive force? Would there be high-energy states instead of zero energy? Complete nothingness without a single particle (true vacuum ..read more

Is it possible that photoelectric effect could remove all bounded electrons ..read more

Consider the surface of rotation shown in heavy lines in the accompanying sketch.It is made up of parts of the surfaces of two confocal ellipsoids of revolution,and that of a sphere.The inside surface is a perfect mirror.The foci are labled A and B.If two black blodies of equal temperature are initially placed at A and B,then all the radiation from A will reach B,but not vice versa,because radiation from B hitting the spherical surface will be reflected back.Therefore the temperature of B will increase spontaneously,while that of A will decrease spontaneously,and this would violate the second ..read more

I've seen this kind of simplification done very frequently in Gauss's law problems, assuming E is only radial and follows some "simple" geometry: $$\oint\vec{E}\cdot\vec{dA}=\frac{Q_{enc}}{\epsilon_0}$$ $$\oint |\vec{E}|\hat{r} \cdot \hat{r} dA = \frac{Q_{enc}}{\epsilon_0}$$ $$\int|\vec{E}| dA = \frac{Q_{enc}}{\epsilon_0}$$ I understand up to here. What I don't understand is the next step: $$\int|\vec{E}| dA = |\vec{E}| \int dA = \frac{Q_{enc}}{\epsilon_0}$$ Why can we just take $|\vec{E}|$ out of the integral? I've heard the argument that "it's because E doesn't depend on A", but then we coul ..read more

I have been practicing many questions regarding electrical field lines. However, I can't seem to understand when electrical field lines remain straight and when they start to curve. Does it depend on the shape of the charged object? Or does it have other factors which cause it to curve ..read more

The critical angle can be intuitively understood by Snell's law.If the incident medium has a bigger diffraction index than the refracted medium then according to Snell's law the refracted ray will be inside the incident medium which means it doesnt leave it in the first place as show in the picture: However why is the ray reflected only at the angle of incidence?I dont get it.Shouldnt part of the ray be reflected at the angle of incidence and part of the ray be "reflected" according to Snell's law?Thanks ..read more

So this is more of a clarifying question. A lot of online definitions state that momentum is a measure of how hard it is to stop or swerve an object, which makes sense. However, the formula for momentum is mass times velocity, and I'm just wondering how the individual components (mass and velocity, that is) make the object harder to stop. Also, are objects with more mass harder to stop because they have more inertia, or is there another reason? Thanks ..read more

The Magnus expansion and Dyson series are very similar to each other, in that they both give a way to approximate a time-evolution operator as a series expansion $$U(t) = \mathcal{T}\left(\exp\left[-i\int_0^tH(t')/\hbar\, dt'\right]\right)$$ where $\mathcal{T}$ is the time ordering operator*. The Dyson series directly expands the above exponential as a sum of integrals, while the Magnus expansion finds a series expansion for $\bar{H}$ such that $\exp(-i \bar{H} t) = U(t)$. The Magnus expansion expression for $U(t)$ is exactly unitary and has slightly better convergence properties, while the Dy ..read more

I'm confused about the difference between a Galilean transformation and boost with reference to their matrices. I was given four statements (listed below) but I'm not sure what I should be looking for to determine their validity. Apologies if this has been answered before I could only see boost vs translation but wasn't sure if this was the same. $ \begin{bmatrix} 1 & 0 & 0 & 0 \\ v & 1 & 0 & 1 \\ -v & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 \\ \end{bmatrix} $ is a Galilean transformation $ \begin{bmatrix} 1 & 0 & 0 & 0 \\ v & 0 & 0 ..read more