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Pre-Calc Exam Notes 85

Pre-Calc Exam Notes 85 - soidal steady state at time t is...

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Other Identities Section 3.4 85 Example3.21 Recall Snell’s law from Example 3.12 in Section 3.2: n 1 sin θ 1 = n 2 sin θ 2 . Use it to show that the p-polarization transmission Fresnel coefficient defined by t 1 2 p = 2 n 1 cos θ 1 n 2 cos θ 1 + n 1 cos θ 2 (3.45) can be written as: t 1 2 p = 2 cos θ 1 sin θ 2 sin ( θ 1 + θ 2 ) cos ( θ 1 θ 2 ) . Solution: Multiply the top and bottom of t 1 2 p by sin θ 1 sin θ 2 to get: t 1 2 p = 2 n 1 cos θ 1 n 2 cos θ 1 + n 1 cos θ 2 · sin θ 1 sin θ 2 sin θ 1 sin θ 2 = 2 ( n 1 sin θ 1 ) cos θ 1 sin θ 2 ( n 2 sin θ 2 ) sin θ 1 cos θ 1 + ( n 1 sin θ 1 ) sin θ 2 cos θ 2 = 2 cos θ 1 sin θ 2 sin θ 1 cos θ 1 + sin θ 2 cos θ 2 (by Snell’s law) = 2 cos θ 1 sin θ 2 1 2 (sin 2 θ 1 + sin 2 θ 2 ) (by the double-angle formula) = 2 cos θ 1 sin θ 2 1 2 (2 sin 1 2 (2 θ 1 + 2 θ 2 ) cos 1 2 (2 θ 1 2 θ 2 )) (by formula (3.41)) = 2 cos θ 1 sin θ 2 sin ( θ 1 + θ 2 ) cos ( θ 1 θ 2 ) Example3.22 In an AC electrical circuit, the instantaneous power p ( t ) delivered to the entire circuit in the sinu-
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Unformatted text preview: soidal steady state at time t is given by p ( t ) = v ( t ) i ( t ) , where the voltage v ( t ) and current i ( t ) are given by v ( t ) = V m cos ω t , i ( t ) = I m cos ( ω t + φ ) , for some constants V m , I m , ω , and φ . Show that the instantaneous power can be written as p ( t ) = 1 2 V m I m cos φ + 1 2 V m I m cos (2 ω t + φ ) . Solution: By deFnition of p ( t ), we have p ( t ) = V m I m cos ω t cos ( ω t + φ ) = V m I m · 1 2 (cos (2 ω t + φ ) + cos ( − φ )) (by formula (3.43)) = 1 2 V m I m cos φ + 1 2 V m I m cos (2 ω t + φ ) (since cos ( − φ ) = cos φ ) ....
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