Problem.Set.7.soln.2009

Problem.Set.7.soln.2009 - Problem Set 7. ( Adapted from...

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Unformatted text preview: Problem Set 7. ( Adapted from Jackson 9 . 10 ) The transitional charge and current densities for the radiative transition from the m = 0,2 p state in hydrogen to the 1 s ground state are (with the neglect of spin),  ( r ,  ,  , t ) = 2 e 6 a o 4 r exp ( − 3 r 2 a o ) Y 00 Y 10 exp (− i  o t ) =  o ( r ) exp (− i  o t ) ; J ( r ,  ,  , t ) = − iv o 2 [ r ̂ 2 + a o z z ̂ ]  ( r ,  ,  , t ) . where a o = 4  o  2 / me 2 = 0.529 × 10 − 10 m is the Bohr radius,  o = 3 e 2 / 32  o  a o is the frequency difference of the levels, and v o = e 2 / 4  o  =  c ≈ c / 137 is the Bohr orbit speed. a) Find the general expression for the vector potential, A ( r , t ) , and the  ( r , t ) for this localized radiation source. b) Evaluate all the nonvanishing radiation multipole contributions in the long wavelength limit. c) Find the magnetic induction, B ( r , t ) , and the electric field, E ( r , t ) , in the radiation limit. Solution: a ) Given that J ( r ,  ,  , t ) = J o ( r ) exp (− i  o t ) A ( r , t ) = 1 c ∫∫∫ ∫ J ( r ′ , t ′ ) G ( r − r ′ , t − t ′ ) d 3 r ′ dt ′ = 1 c ∫∫∫ ∫ J ( r ′ , t ′ ) Θ ( t − t ′ ) c | r − r ′ |  (| r − r ′ | − c ( t − t ′ )) d 3 r ′ dt ′ = exp (− i  o t ) c ∫∫∫ J ( r ′ ) exp ( i  o | r − r ′ |/ c ) | r − r ′ | d 3 r ′ = A o ( r ) exp (− i  o t ) with G ( r − r ′ ,  o ) = exp ( i  o | r − r ′ |/ c ) | r − r ′ | = 4  k ∑ l = ∞ ∑ m =− l l j l ( kr < )[ ij l ( kr > ) − n l ( kr > )] Y lm (  ,  ) Y lm (  ′ ,  ′ ) ∗ ; k =  o / c Since the source is localized near r ′ << c /  =  / 2  (the long wavelength approximation where k =  / c = 2  /  ) and r ′ < r (the observation point): A o ( r ) = 4  k c ∑ l = ∞ ∑ m =− l l [ ij l ( kr ) − n l ( kr )] Y lm (  ,  ) ∫∫∫ J ( r ′ ) j l ( kr ′ ) Y lm (  ′ ,  ′ ) ∗ r ′ 2 d  ′ dr ′ ....
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This note was uploaded on 12/19/2010 for the course PHYS 423 taught by Professor G. during the Spring '10 term at Missouri S&T.

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Problem.Set.7.soln.2009 - Problem Set 7. ( Adapted from...

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