Question:
For the circuit given below, choose the angular frequency \(\omega_0\) (in rad/s) at which the voltage across the capacitor has maximum amplitude?
For the circuit given below, choose the angular frequency \(\omega_0\) (in rad/s) at which the voltage across the capacitor has maximum amplitude?
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A series \(RC\) acts as a low-pass voltage divider for the capacitor: the capacitor voltage is largest at DC and decreases monotonically with frequency.
Updated On: Aug 26, 2025
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The Correct Option is D
Solution and Explanation
Step 1: For a series \(RC\) circuit with source amplitude \(V_s\), the capacitor voltage amplitude is \[ |V_C| = V_s \frac{|Z_C|}{|Z_R+Z_C|} = V_s \frac{\tfrac{1}{\omega C}}{\sqrt{R^2+\left(\tfrac{1}{\omega C}\right)^2}}. \]
Step 2: Let \[ x = \frac{1}{\omega C}. \] Then \[ |V_C| = V_s \frac{x}{\sqrt{R^2+x^2}}. \] As \(\omega \to 0\) (i.e., \(x \to \infty\)): \[ |V_C| \to V_s \frac{x}{x\sqrt{1+\tfrac{R^2}{x^2}}} \to V_s. \] For any finite \(\omega > 0\), \(x\) is finite and \(|V_C| < V_s\).
Hence, the maximum amplitude occurs at \[ \omega_0 = 0 \ \text{rad/s (DC)}. \] Final Answer: \(\boxed{0}\).
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