Question

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

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

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).

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

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}

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

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.