Above figure, shows an $$ac$$ generator connected to a “black box” through a pair of terminals. The box contains an $$RLC$$ circuit, possibly even a multiloop circuit, whose elements and connections we do not know. Measurements outside the box reveal that, $$\xi \left ( t \right )=\left ( 75.00 \right )\sin \omega _{d}t$$ and $$i\left ( t \right )=\left ( 1.20A \right )\sin\left ( \omega _{d}t+42.0^{0} \right )$$. (a) What is the power factor? (b) Does the current lead or lag the $$emf$$? (c) Is the circuit in the box largely inductive or largely capacitive? (d) Is the circuit in the box in resonance? (e) Must there be a capacitor in the box? (f) An inductor? (g) A resistor? (h) At what average rate is energy delivered to the box by the generator? (i) Why don’t you need to know $$\omega _{d}$$ to answer all these questions?
(a) The power factor is $$\cos \phi$$, where, $$\phi$$ is the phase constant defined by the expression $$i=I\sin \left ( \omega t-\phi \right )$$. Thus, $$\phi=-42.0^{0}$$ and $$\cos \phi =\cos \left ( -42.0^{0} \right )=0.743$$.
(b) Since $$\phi <0$$, $$\omega t-\phi >\omega t$$. The current leads the emf.
(c) The phase constant is related to the reactance difference by $$\tan \phi=\left ( X_{L}-X_{C} \right )/R$$. We have
$$\tan \phi =\tan \left ( -42.0^{0} \right )=-0.900$$,
a negative number. Therefore, $$X_{L}-X_{C}$$ is negative, which leads to $$X_{C}>X_{L}$$. The circuit in the box is predominantly capacitive.
(d) If the circuit were in resonance $$X_{L}$$ would be the same as $$X_{C}$$, $$\tan \phi$$ would be zero, and $$\phi$$ would be zero. Since $$\phi$$ is not zero, we conclude the circuit is not in resonance.
(e) Since $$\tan \phi$$ is negative and finite, neither the capacitive reactance nor the resistance are zero. This means the box must contain a capacitor and a resistor.
(f) The inductive reactance may be zero, so there need not be an inductor.
(g) Yes, there is a resistor.
(h) The average power is,
$$P_{avg}=\frac{1}{2}\varepsilon _{m}I\cos \phi =\frac{1}{2}\left ( 75.0V \right )\left ( 1.20A \right )\left ( 0.743 \right )=33.4W$$.
(i) The answers above depend on the frequency only through the phase constant $$\phi$$, 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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