Write Poisson’s and Laplace’s equations
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A: Since you have posted multiple questions, we will provide the solution only to the first question as…
Write Poisson’s and Laplace’s equations
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- Determine the general solution of the 1-dimensional Laplace equation on the cylinder coordinates and ball coordinates!What’s Orthogonal Set in quantum mechanics? And Write its mathematical formulaAn 8-kg slender rod of length L is attached to collars that can slide freely along the guides shown. Knowing that the rod is in equilibrium and that β= 30°, determine (a) the angle θ that the rod forms with the vertical, (b) the reactions at A and B.
- Problem 2 The relativistic Lagrangian for a particle of rest mass m moving along the x-axis in a potential V(x) is given by 2 L = -mc² 1 V(x) c2 (a) Derive the Euler-Lagrange equation of motion. (b) Show that it reduces to Newton's equation in the limit |*| << c. (c) Compute the Hamiltonian H of the system. Eliminate ȧ from the Hamiltonian by using the equation ƏL p = ax and write H = H(p, x) as a function of x and p only.Class: Electromagnetics Consider a fixed hollow spherical shell with radius R and surface charge +? (sigma). A particle with mass m and charge -Q that is initially at rest falls in from infinity. Assume that a tiny hole has been cut in the shell to let the charge through. Determine the speed of the particle when it reaches the center of the shell. Your answer should be a function of ? (sigma), Q, and m.Suppose that we want to solve Laplace’s equation inside a hollow rectangular box, with sides of length a, b and c in the x, y and z directions, respectively. Let us set up the axes so that the origin is at one corner of the box, so that the faces are located at x = 0 and x = a; at y = 0 and y = b; and at z = 0 and z = c. Suppose that the faces are all held at zero potential, except for the face atz=c,onwhichthepotentialisspecifiedtobeV(x,y,c)=V0 =const. a) Find the electrostatic potential V at a generic point inside the box.b) Find the expression for the electrostatic potential evaluated at the center of the box, i.e. deter- mine V (a/2, b/2, c/2). Simplify your answer as much as you can! c) Suppose now that a = b = c, i.e. the box is a cube. Give a simple argument which gives theexact (and simple) expression for the potential at the center of the cube. (No calculations are asked here. Use physics, wave your hands, etc. and say “the answer is such and such because ...”)