Fall06_solutions

Fall06_solutions - Qualifying Exam Fall 2006 Solutions...

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Qualifying Exam Fall 2006 Solutions Solution 1 -The AD-axis of the cube has threefold symmetry, i.e. the cube is invariant under rotations by about that axis. 0 120 ± -Hence, the corners , and are equivalent and have the same potential. 1 B 2 B 3 B -Also, the corners , and are equivalent and have the same potential. 1 C 2 C 3 3 C If a current I enters A, then: - A current 3 I circulates through each of the resistors . i AB - A current 6 I circulates through each of the six resistors . j i C B - A current 3 I passes through each of the three resistors . D C j - The potential drop between A and is i B 3 rI . - The potential drop between and is i B j C 6 rI . - The potential drop between and D is j C 6 rI . The potential difference between A and D is: rI rI 6 5 3 1 6 1 3 1 = + + Hence r R 6 5 =
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Solution 2 a.) [] z y x S i i i i i i i i i i i i i i i i i i S S h h h h h h = = = = 1 0 0 0 0 0 0 0 1 0 0 0 0 0 2 0 0 0 0 0 2 0 1 0 1 0 1 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 0 1 0 1 0 1 0 2 , 2 2 2 2 2 b.) A measurement along the x-axis has as an outcome, one of the eigenvalues of . z S 0 2 0 2 2 0 2 det = λ h h h h h h ± = = + , 0 0 2 2 2 3 λλ The possible values are: h h , 0 , . c.) The largest possible value along the x-axis is ; obtain eigenvector in the basis of quantization along the z-axis: h 0 2 0 2 2 0 2 = c b a h h h h h h h c b a = = 2 The normalized eigenvector: ( ) 1 0 2 1 2 1 + + + Probabilities: 4 1 ) 1 ( = + P 2 1 ) 0 ( = P 4 1 ) 1 ( = P
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Solution 3 Conservation of energy given by the sum of potential energy due to gravity and kinetic energy can be used to determine escape velocity. In the case of Earth along the potential is given by: r m M G r E = ) ( φ where m is the mass of the book. The book will escape if initial kinetic energy is high enough to overcome the potential at E R r = . Thus E E E R m M G mv = 2 2 thus s km R GM v E E E / 11 2 = = In the Earth-Moon case the potential is r R m M G r m M G r EM M E = ) ( where MM = ME. The potential is a symmetric double-well and in order to leave the surface of the earth the kinetic energy must be high enough to overcome a saddle point right in the middle between earth and moon. Thus the condition for escape velocity is ME E E EM E E E R m GM R R R m GM mv 4 1 1 2 2 = + This equation solved for escape velocity gives . / 7 . 7 s km v E = Solution 4 a) Calculate the capacitance of the capacitor.
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Fall06_solutions - Qualifying Exam Fall 2006 Solutions...

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