Problem.Set.4.soln

# Problem.Set.4.soln - Problem Set 4: (Assignment 5) For a...

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Problem Set 4: (Assignment 5) For a given system the electric field and current density are given by Assignment 5 E r , t E 0 exp z ct 2 2 L 2 cos k 0 z ct  e 1 J r, t J 0 exp z ct 2 2 L 2 cos k 0 z ct 0 e 1 (a) Calculate the magnetic flux density, B r , t . (b) Calculate the frequency dependent fields and current densities e r , , b r , ,and j r , . (c) What are the units of these three quantities in the gaussian units used by Jackson , in the SI (MKSC) units? (d) Starting with the results of part (b) evaluate e r , b r ,and j r defined in the notes. (e) Let 0 /6 and k 0 L 2 10 12 and calculate the time average of E r , t   J r , t over the period from t T /2 to t  T /2 at z  L and at z 0. (Note, a tabulated integral). (f) Plot the results of part (e) for 0 cT / L 1. Solution (a): The electric and magnetic fields are related by Faraday’s law E r , t 1 c t B r , t . The given electric field has the form E r , t f z ct e 1 therefore the curl of the electric field has the form E r , t f z ct e 2 . It follows that 1 c t B r , t f z ct e 2 or B r , t f z ct e 2 c E 0 exp z ct 2 2 L 2 cos k 0 z ct  e 2 The bracketed c should be added for SI units Solution (b) : The frequency dependent electric field amplitude is e r,  E 0 exp z ct 2 2 L 2 cos k 0 z ct  exp i t dt e 1 e 1 c E 0 exp i z c  exp u 2 2 L 2 i c u cos k 0 u du e 1 2 c E 0 exp i z c  exp u 2 2 L 2 i c k 0 u du e 1 2 c E 0 exp i z c  exp u 2 2 L 2 i c k 0 u du e 1 c L 2 E 0 exp i z c exp L 2 2 c k 0 2 exp L 2 2 c k 0 2 Similarly the frequency dependent amplitudes of the magnetic flux density are

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b r , c e 2 c L 2 E 0 exp i z c exp L 2 2 c k 0 2 exp L 2 2 c k 0 2 while the frequency dependent amplitudes of the current density are j r , e 1 c L 2 J 0 exp i z c exp L 2 2 c k 0 2 i 0 exp L 2 2 c k 0 2 i 0 Solution (c): The gaussian units for E, B, and J are statvolt/cm, gauss, and statampere/cm 2 respectively. The frequency dependent amplitudes have corresponding units statvolt s/cm,
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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.4.soln - Problem Set 4: (Assignment 5) For a...

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