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Unformatted text preview: 182 Chapt. 23 The Electric Field (with θ in radians of course) which is exact to 2nd order in small θ since the Taylor’s expansion for sin θ is sin θ = ∞ summationdisplay ℓ =0 ( − 1) ℓ θ 2 ℓ +1 (2 ℓ + 1)! (Ar-264). The angle θ in radians is larger than sin( θ ) at 10 ◦ by 0 . 51 %, 20 ◦ by 2 . 1 %, 30 ◦ by 4 . 7 %, 45 ◦ by 11 %, and 60 ◦ by 21 %. So the small angle approximation is pretty good even for not-so-small angles. With the small angle approximation, the DE becomes I d 2 θ dt 2 = − pEθ which is the simple harmonic oscillator DE. The general solution of this linear 2nd order DE (linear because the solution function and its derivatives appear linearly and 2nd order because the highest derivative is 2n order) is θ = A cos( ωt ) + B sin( ωt ) , where ω here is not the angular velocity (not dθ/dt ), but the angular frequency and is given by ω = radicalbigg pE I ....
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This note was uploaded on 11/16/2011 for the course PHY 2053 taught by Professor Buchler during the Fall '06 term at University of Florida.
- Fall '06