In the circuit shown below, the amplitudes of the voltage across the resistor and the capacitor are equal. What is the value of the angular frequency \(\omega_0\) (in rad/s)? (Round off the answer to one decimal place.)
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In a series \(RC\) circuit, the resistor and capacitor have equal voltage magnitudes when the reactance equals the resistance: \(|X_C|=R\Rightarrow \omega=1/(RC)\).
Step 1: For a series \(RC\) circuit, the magnitudes of voltages are
\[
|V_R|=I R,\qquad |V_C|=I\left|\frac{1}{j\omega C}\right|=\frac{I}{\omega C}.
\]
Given \(|V_R|=|V_C|\Rightarrow R=\dfrac{1}{\omega_0 C}\), hence
\[
\omega_0=\frac{1}{RC}.
\]
Step 2: Substitute \(R=1\,\text{k}\Omega=1000\,\Omega\) and \(C=100\,\mu\text{F}=1\times10^{-4}\,\text{F}\):
\[
\omega_0=\frac{1}{(1000)(1\times10^{-4})}=\frac{1}{0.1}=10\ \text{rad/s}.
\]
Rounded to one decimal place: \(\boxed{10.0}\).
