Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Homework # 5 Due date : Thursday, December 2nd JUST, Physics Department Mathematical Physics 701 Instructor Dr. Abdalla Obeidat 1. Find the geodesics on the cone z 2 = x2 + y 2 2. Write and solve the Euler equations to make the following integral stationary Z x2 ds x1 x 3. Fermat’s principle states that a ray of light in a medium with a variable index of refraction will follow the path which requires the shortest traveling time. For two-dimentional case, show that such a path is obtained by minimizing the integral Z x2 p 1 + y 02 dx n(x, y) x1 where n(x, y) is the index of refraction. For the particular case n = 1/y, show that the rays of light will follow semicircle paths. 2 2 4. You have an ellipse xa2 + yb2 = 1. Find the inscribed rectangle of maximum area. Show that the ratio of the area of the maximum area rectangle to the area of ellips is 2/π. (17.6.6) from ARFKEN. 5. For identical particles obeying the Pauli exclusion principle the probability of a given arrangement is WF D = Y i gi ! ni ! (gi − ni )! Show that maximizing WF D subject to a fixed number of particles and fixed total energy leads to ni = gi eλ1 +λ2 Ei +1 with λ1 = −E0 /kT and λ2 = 1/kT , this yields Fermi-Dirac statistics.(17.6.10) from ARFKEN. 1