Determine the equilibrium temperature distribution for a one-dimensional rod com- posed of two different materials in perfect thermal contact at x = 1. for 0 < x < 1, there is one material (crho = 1,k0 = 1) with a constant source (q =1), whereas for the other 1< x < 2, there are no sources (q = 0,crho = 2,k0 = 2) (see exercise 1.3.2) with u(0) = 0 and u(2)=0.

Consider a uniformly charged ring in the xy plane, centered at the origin. the ring has radius a and positive charge qdistributed evenly along its circumference. a)what is the direction of the electric field at any point on the z axis? . b)what is the magnitude of the electric field along the positive z axis? use k in your answer, where k=14πϵ0. d)the ball will oscillate along the z axis between z=d and z=−d in simple harmonic motion. what will be the angular frequency ω of these oscillations? use the approximation d≪a to simplify your calculation; that is, assume that d2+a2≈a2. express your answer in terms of given charges, dimensions, and constants