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helpfulhomework20 - Physics Qualifying Examination Problems...

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Physics Qualifying Examination Problems 1–6 Thursday, January 10, 2008 1–5 pm Problems 7-12 Friday, January 11, 2008 1-5 pm 1. Solve each problem. 2. Start each problem solution on a fresh page. You may use multiple pages per problem. 3. Turn in at least one page per problem, even if you give no solution. At the top of each solution page put the problem number (1–12) and your Social Security number, but not your name or any other information.
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1. A parallel-plate capacitor has circular disk electrodes of radius a that are separated by a distance d ( ). A resistor of resistance R connects the plates along their d << a common axis. At t = 0, the capacitor is suddenly given charge Q 0 . Neglect all edge effects. (a) What is the capacitance of the capacitor? (b) Obtain the time-dependent charge on the capacitor plates. (c) Find the electric field between the plates. (d) Obtain the magnetic field between the plates as a function of position and time. (e) Derive an expression for the Poynting vector S G as a function of position between the plates.
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2. A hydrogen atom is placed in a uniform electric field ( ) E t G in the z -direction. ( ) E t G is 0 for t < 0 and / 0 ( ) t E t E e τ = G G for t > 0. The atom is initially in the ground state. (a) Find the probability for the atom to have made a transition to the 2 s state as t → ∞ . (b) Find the probability for the atom to have made a transition to one 2 p state as t → ∞ . (c) Assuming that 0 E G is small, how small should it be for the calculation to be valid? The following radial wave functions of hydrogen may be useful to solve the problem: 10 3/2 20 3/2 21 3/2 1 2exp( / ) 1 (2 / )exp( / 2 ) (2 ) 1 ( / 3 )exp( / 2 ) (2 ) R r a a R r a r a a R r a r a a = = = where a is Bohr’s radius. The relevant spherical harmonics are
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