# Robert_W_Brown;_Yu-Chung_N_Cheng;_E_Mark_Haacke;(z-lib.org).pdf - Adobe ReaderFile Edit View Window Help(77 of 978)Fill & SignToolsCommentОpen43100%Sign InExport PDF3.2. The Rotating Frame for an RF Field43Create PDFEquation (3.21) displays a constant term plus two oscillating terms whose effects averageto zero over times that are half multiples of the rf period or, more relevantly, for all timeslarge compared to the rf period. The definition for an average over a time interval T is givenbyEdit PDFAdobe Acrobat ProEasily edit text and images in PDFdocuments(3.22)< f(t) >dt f (t)Start NowWith the details left as aproblem for the reader, the primed-frame time averaged value of(3.21) isSend Files1primedStore Files(3.23)for constant binProblem 3.2f (t)Show that the average (3.22) over time interval T for f(t) = sin (nwt)cos (nwt) for any positive integer n is zero whenora) T 2mT/w for any positive integeror T =mT/w for even n)or when the following limit is takenb) Т> 2п/wThe reduced amplitude bn in (3.23) is noteworthy. It implies that only half of theoriginal linearly polarized field (3.15) amplitude is available in the rotating reference framefor tipping the spin. Also, (3.23) is still valid for a time dependent bn(t), provided it changesonly over time scales much larger than the rf period 2T/w. It has been noted earlier thatLlin(.

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Problem 3.2
Show that the average (3.22) over time interval T for f(t) = sin (nωt) or f(t) =
cos (nωt) for any positive integer n is zero when
a) T = 2mπ/ω for any positive integer m (or T = mπ/ω for even n)
or when the following limit is taken
b) T ≫ 2π/ω

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Step 1

Formula for average of any function f(t) is shown below

Step 2

For part (a) of the given problem,  function and time period is shown below

Step 3

plugging sinusoidal function(sine term) and time period in...

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