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Unformatted text preview: 57. (a) The power factor is cos φ, where φ is the phase constant deﬁned by the expression i = I sin(ωt − φ).
Thus, φ = −42.0◦ and cos φ = cos(−42.0◦ ) = 0.743.
(b) Since φ < 0, ωt − φ > ωt. The current leads the emf.
(c) The phase constant is related to the reactance diﬀerence by tan φ = (XL − XC )/R. We have
tan φ = tan(−42.0◦ ) = −0.900, a negative number. Therefore, XL − XC is negative, which leads to
XC > XL . The circuit in the box is predominantly capacitive.
(d) If the circuit were in resonance XL would be the same as XC , tan φ would be zero, and φ would be
zero. Since φ is not zero, we conclude the circuit is not in resonance.
(e) Since tan φ is negative and ﬁnite, neither the capacitive reactance nor the resistance are zero. This
means the box must contain a capacitor and a resistor. The inductive reactance may be zero, so
there need not be an inductor. If there is an inductor its reactance must be less than that of the
capacitor at the operating frequency.
(f) The average power is
Pavg = 1
Em I cos φ = (75.0 V)(1.20 A)(0.743) = 33.4 W .
2 (g) The answers above depend on the frequency only through the phase constant φ, which is given. If
values were given for R, L and C then the value of the frequency would also be needed to compute
the power factor. ...
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