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Chapter 30: Problem 9

What is the impedance of a series RLC circuit when the frequency of time- varying emf is set to the resonant frequency of the circuit?

Short Answer

Expert verified
Answer: At the resonant frequency of an RLC circuit, the impedance is equivalent to the resistance (R) of the resistor in the circuit.

Step by step solution

01

Calculate the resonant frequency

In a series RLC circuit, the resonant frequency (f_r) is given by the formula: f_r = \frac{1}{2\pi\sqrt{LC}} Where L is the inductance (measured in Henries) and C is the capacitance (measured in Farads).
02

Calculate the inductive reactance at resonant frequency

Inductive reactance (X_L) is the opposition offered by the inductor coil at a specific frequency (f) and is given by the formula: X_L = 2\pi f L At the resonant frequency, replace f with f_r: X_L = 2\pi f_r L
03

Calculate the capacitive reactance at resonant frequency

Capacitive reactance (X_C) is the opposition offered by the capacitor at a specific frequency (f) and is given by the formula: X_C = \frac{1}{2\pi f C} At the resonant frequency, replace f with f_r: X_C = \frac{1}{2\pi f_r C}
04

Find the net reactance at resonant frequency

The net reactance (X_net) of the RLC circuit is the difference between inductive reactance and capacitive reactance: X_net = X_L - X_C At the resonant frequency, we know that the inductive and capacitive reactance are equal: X_L = X_C Therefore, X_net = 0
05

Calculate the impedance at resonant frequency

In a series RLC circuit, the impedance (Z) is given by the formula: Z = \sqrt{R^2 + X_{net}^2} At the resonant frequency, the net reactance is zero (X_net = 0), so the impedance is solely determined by the resistor: Z = R In conclusion, at the resonant frequency of an RLC circuit, the impedance is equivalent to the resistance (R) of the resistor in the circuit.

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