Download Homework # 5 Due date : Thursday, December 2nd JUST, Physics

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.
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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.
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