(a) In a series LCR circuit connected across an AC source of variable frequency, obtain the expression for its impedance and draw a plot showing its variation with frequency of the AC source.
(b) What is the phase difference between the voltages across inductor and the capacitor at resonance in the LCR circuit ?
(c ) When and inductor is connected to a 200 V DC voltage, a current of 1Aflows trough it. when the same inductor is connected to a 200 V, 50 Hz AC source, only 0.5 A current flows. Explain, why ? Also, calculate the self inductance of the inductor.

Explanation

### Solution:

**(a)** To derive the expression for the impedance \( Z \) in a series LCR circuit connected to an AC source, we start with the definitions of the voltages across the resistor, inductor, and capacitor.

1. **Voltage across the Resistor \( V_R \)**:
\[ V_R = I_m R \]
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