5.6 ELECTRIC CIRCUIT PROBLEMS
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8. A circuit has in series an electromotive force given by \( E(t)=E_{0} \sin \omega t \mathrm{~V}, \mathrm{a} \) resistor of \( R \Omega \), an inductor of \( L \mathrm{H} \), and a capacitor of \( C \) farads.
(a) Show that the steady-state current is
\[
i=\frac{E_{0}}{Z}\left(\frac{R}{Z} \sin \omega t-\frac{X}{Z} \cos \omega t\right),
\]
where \( X=L \omega-1 / C \omega \) and \( Z=\sqrt{X^{2}+R^{2}} \). The quantity \( X \) is called the reactance of the circuit and \( Z \) is called the impedance.
(b) Using the result of part (a) show that the steady-state current may be written
\[
i=\frac{E_{0}}{Z} \sin (\omega t-\phi),
\]
where \( \phi \) is determined by the equations
\[
\cos \phi=\frac{R}{Z}, \quad \sin \phi=\frac{X}{Z} .
\]
Thus show that the steady-state current attains its maximum absolute value \( E_{0} / Z \) at times \( t_{n}+\phi / \omega \), where
\[
t_{n}=\frac{1}{\omega}\left[\frac{(2 n-1) \pi}{2}\right] \quad(n=1,2,3, \ldots),
\]
are the times at which the electromotive force attains its maximum absolute value \( E_{0} \).
(c) Show that the amplitude of the steady-state current is a maximum when
\[
\omega=\frac{1}{\sqrt{L C}}
\]
For this value of \( \omega \) electrical resonance is said to occur.
(d) If \( R=20, L=\frac{1}{4}, C=10^{-4} \), and \( E_{0}=100 \), find the value of \( \omega \) that gives rise to electrical resonance and determine the amplitude of the steady-state current in this case.
