ALLAMA IQBAL OPEN UNIVERSITY Level: = Ph.D Physics Semester: Spring 2015 Paper: Advanced Mathematical Physics (9701) Maximum Marks: 100 Time Allowed: 03 Hours Pass Marks: 50 Note: ATTEMPT FIVE QUESTIONS. ALL CARRY EQUAL MARKS. Q-NO 1. Assume that the Euler-Lagrange equation gives a minimizing curve. Show that the lowest value of the integral B ia Â£ Gy" ae â€˜A x where A is (â€”I, 1) and Bis (1, 1), is2 In(1+Â¥2). [20] Q-NO 2. The Lagrangian for a z-meson is given by Lx, )= V2.9" =| Vo? â€” 19%), where 11 is the meson mass and g(x, Â£) is its wave function. Assuming Hamiltonâ€™s principle â€œSs wave equation satisfied by g. [20] Q-NO 3 A general triangle has an@ls a, B and y and corresponding opposite sides a, b and c. Express length of each side in terms of the lengths of the other two sides an relevant cosines, writing the relationships in matrix merc form, using the vectors having components a, b, c and cas & cos B, cos y. Invert the matrix and hence deduce the cosine-law pr involving a, B and y. [20] Q-NO 4 A surface of revolution, equation in cylindrical polar coordinates is p = p(z), is Bounded by the circles p=a,z=te (a >c). Show that the function thatamakes the surface integral I= J pâ„¢'? ds stationary with respect to | variations is given by p(z)=K + Z/ (4k), whergRF [at (a â€”?)!7] /2. [20] Q-NO 5 A rectangular parallelepi, as all eight vertices on the ellipsoid X?43Y743Z7=1, Using the symmetry of tage licleniped about each of the planes Â¥= 0, Y = 0, Z= 0, write doâ€™ e surface area of the parallelepiped in terms of the coordinates o: ertex that lies in the octant X,Y,Z>0. Hence find the maximum ie of the surface area of such a parallelepiped. [20] Q-NO 6 Find the power series soluffos about z = 0 of dzy" + 2y' + y=0 (20) i i 3 Initially, one isatOÂ°C) Two identical copper bars ach of length a. and the other at 100 Â°C; they are then joined together end to end and \ 9 thermally isolated. Obtain in the form of a Fourier series an expression u(x, t) for the temperature at any point a distance x from the join at . later time Â¢. Bear in mind the heat flow conditions at the free ends o: the bars. Takin a= 0.5m estimate the time it takes for one of the free ends to attain a temperature of 55 Â°C. The thermal conductivity of copper is 3.8x107 Im! K-! 1, and its specific heat capacity is 3.4 x 10Â° Jm = KY The wave equation describing the transverse vibrations ofa stretched membrane under tension T and having a uniform surface density p is @u eu eu Pe Pe Gx* ay QNO7 QNO8 â€œhe i i stretched on a Find a separable solution appropriate to a membrane st frame of length aand width b, showing that the natural angular frequencies of such a membrane are given by eT frm ea ( = *) where n and m are any positive integers. [20]

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- Allama Iqbal Open University
- PhD Physics
- Advanced Mathematical Physics
- Spring 2015
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