Consider a two-state paramagnet with 1023 elementary dipoles, with the total energy fixed at zero so that exactly half the dipoles point up and half point down.(a) How many microstates are "accessible" to this system?(b) Suppose that the microstate of this system changes a billion times per second. How many microstates will it explore in ten billion years (the age of the universe)?(c) Is it correct to say that, if you wait long enough, a system will eventually be found in every "accessible" microstate? Explain your answer, and discuss the meaning of the word "accessible."

A cylindrical shell of length L, radius R, and mass Mis uniformly charged on its surface with total charge Q, and is rotating with an angular speed of about its axis which is oriented along the z-axis, as shown. a) Consider a horizontal thin ring-like region on the surface of shell with thickness dz, as shown and calculate the infinitesimal current di due to the rotating charge on this ring, in terms of dz. b) Calculate the infinitesimal magnetic moment dụ due to this current. c) Calculate the infinitesimal magnetic field dB, on the z-axis of cylinder at the point P at the distance of d above the center of cylinder (d>L/2). d) Integrate this dB over dz, to find the magnetic field B at point P. Also find B at very far away. e) Integrate du over dz, to find total magnetic momentu. f) Show that is related to the angular momentum of the rotating sphere. Show the relation. g) Use this to calculate the B filed at point P when this point is very far away (d>> L & R). X @0 di P Z dz y

(a) What is charge and what does it mean that charge is quantized? (b) How does a conductor differ from an insulator? (c) What is electric flux? Include a sketch that indicates the angle used in calculating the electric flux.

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