A very thin ring of radius carries a positive charge uniformly distributed on it. The ring lies in the - plane with its center at the origin. A short uniformly charged rod of length with an identical positive charge is located along at and oriented in the -direction with one end at and the other end at . Both the ring and the rod are insulators, and the charges are fixed in place. For the system consisting of the rod and the ring, find the magnitude and direction of the electric field at the point . Express your symbolic answer in terms of , , , , hati for , hatj for , and hatk for as needed. [Recall that .]

Aparticle of mass m is projected with an initial velocity v0 in a direction making an angle α with the horizontal level ground as shown in the figure. the motion of the particle occurs under a uniform gravitational field g pointing downward. (a) write down the lagrangian of the system by using the cartesian coordinates (x, y). (b) is there any cyclic coordinate(s). if so, interpret it (them) physically. (c) find the euler-lagrange equations. find at least one constant of motion. (d) solve the differential equation in part (c) and obtain x and y coordinates of the projectile as a function of time. (e) construct the hamiltonian of the system, h, and write down the hamilton’s equations (canonical equations) of motion.

Holes drilled several kilometers into earth’s crust provide direct evidence about earth’s interior in the form of

Two 1.20-m nonconducting wires meet at a right angle. one segment carries + 2.50 µc of charge distributed uniformly along its length, and the other carries - 2.50 µc distributed uniformly along it, as shown in fig. 21.50. ( a. find the magnitude and direction of the electric field these wires produce at point p, which is 60.0 cm from each wire. ( b. if an electron is released at p, what are the magnitude and direction of the net force that these wires exert on it?