Philosophy 2074F/G Study Guide - Justice As Fairness, Distributive Justice

The energy of a 2-dimensional oscillator is E =p^2x/2m+p^2y/2m+ax^2/2+by^2/2+ε, where px and py are its momentumcomponents, x and y are the Cartesian coordinates, m is the mass, aand b are constants, and ε is its internal energy which does notdepend on x, y, px and py. From Gibbs’s distribution finddistributions dwx and dwy of its coordinates x and y at temperatureT. Find average potential energy U. Sketch the distributionfunction f(x) for large and small a/kT.
[Additional problem of choice (you solve it only if you want to !):derive the probability distribution dwr of the distance r from theorigin = the probability that the oscillator is within the distancerange from r to r+dr (clue: change from (x, y) to the polarcoordinates (r, ϕ))].

An electric field can be created by a single charge or adistribution of charges. The electric field a distance from a pointcharge has magnitude.
The electric field points away from positive charges and towardnegative charges. A distribution of charges creates an electricfield that can be found by taking the vector sum of the fieldscreated by individual point charges. Note that if a charge isplaced in an electric field created by , will not significantlyaffect the electric field if it is small compared to .
Imagine an isolated positive point charge with a charge (many timeslarger than the charge on a single electron).

If the total positive charge is = 1.62×10-6 , what is the magnitudeof the electric field caused by this charge at point P, a distance= 1.53 from the charge?

Equations 20.43 and 20.44 show that for a collection of gas particles, which turns out to be true whenever the particles have a distribution of speeds. Let us explore this inequality for a two-particles gas. Let the speed of one particle be and the other particle has a speed . 

(a) Show that the average of these two speeds is .

(b) Show that .  

(c) Argue that the equation in part (b) proves that, in general, .

(d) Under what special condition will for the two-particle gas?