3 Charged Closed Wire Consider a thin closed wire with the radius R localized at the origin on the xy-plane, which is charged uniformly with the total charge Q. (i) What is the charge density p in this problem? (ii) Calculate the electric field along the z axis? (Hint: First, consider an infinitessimal piece of the charge.)
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- A particle moves along line segments from the origin to the points (2, 0, 0), (2, 5, 1), (0, 5, 1), and back to the origin under the influence of the force field F(x, y, z) = z2i+ 3xyj + 3y2k. Use Stokes' Theorem to find the work done.5. Consider a coaxial cable which is an (infinitely) long wire of radius a, with given current flowing uniformly up it (so J(t) = I(t)/лa² goes uniformly through the wire). At the outside edge (r=a), there is an infinitesimally thin insulator, and just outside that, an infinitesimally thin sheath that returns all the current, basically a surface current going down the opposite direction carrying a total -I(t) back down. Figure out the B field everywhere in space (direction and magnitude) in terms of givens, and then find the self-inductance per length of this cable. Assume I(t) changes slowly in time, so we can use our usual quasi-static ideas. There are several ways to proceed. You might start from =LI, but this turns out to be tricky here, since the current is not confined to a single path! It is best to think about the total magnetic energy, W, stored by the magnetic field, and use our usual relation between energy and current: WL I²/2.Please show all work ง 5. Evaluate the vector surface integral JJ (+2², x + 2y + 32², 5x − 2y + 2³) . Ndo. S Here, S is the sphere Sy²+2 1. = Since S is a closed surface, the outside of S is considered as the positive side on S. Hint: use the divergence theorem.
- Suppose there is now friction and the equation of motion becomes x+2yx+w²x = 0, where 0 is some constant of friction. γ i. Find the general solution for x(t), under the assumption that w. You do not need to impose initial conditions here. ii. What is the range of for which the above solution is (1) heavily damped; (2) lightly damped. What happens to the above solution and the physical system when y = w?The two-dimensional wave equation describing the vibrations of an infi- nite string is Θη Ot2 2021 მ2 where n = n(x,t), — -x 0 is a constant. Its general solution can be written as n(x,t) = F(x-ct) + G(x + ct), with F and G arbitrary smooth functions. If at time t = 0 the shape of the string is n(x, t = 0) at 9 = 5+x4 and the string is released with velocity (x, t = 0) = 0, determine n(x,t) for t > 0.A system is described by the action S[r, r'] = [ dt (x² + 4rª + mr³4), r(t) where m is a constant. Find the Euler-Lagrange equation for r = which extremises S. Interpret each of the terms that appear in the Euler- Lagrange equations in terms of Newton's second law. Find a conserved quantity for this system.
- I need typing clear urjent no chatgpt used i will give 5 upvotes pls full explainEvaluate the integral. Check your answer by differentiating. (Remember the constant of integration.) (x4 − 9x2 + 2) dxWhat is the Riemannian metric? A) A function that defines the inner product of tangent vectors at each point on a manifold B) A function that measures the curvature of a surface C) A function that defines the volume of a manifold D) A function that determines the Euler characteristic of a manifold
- Sketch the region bounded by the curves x=y34,x=2, and y=0 then find the volume of the solid generated by revolving this region about the y-axis. I need answer typing clear urjent no chatgpt used i will give upvotes6) Circle the atom which is the most likely site to which a Lewis base would be attracted. Explain your answer. || H-C-N-H HI need typing clear urjent no chatgpt use i will give 5 upvotes