Question

For \( V = 120 \, \text{V} \), find the resonant current \( I_0 \).

Hint:

At resonance in an AC circuit, the resonant current \( I_0 \) is given by \( I_0 = \frac{V}{R} \), where \( V \) is the voltage and \( R \) is the resistance.

Answer 1

Step 1: Understanding the formula for resonant current.
At resonance, the impedance of the circuit is purely resistive, and the current is determined by Ohm’s law: \[ I = \frac{V}{R} \] where \( V \) is the voltage and \( R \) is the resistance. At resonance, this equation gives the resonant current \( I_0 \).
Step 2: Formula.
Substituting \( V = 120 \, \text{V} \) into the formula: \[ I_0 = \frac{120}{R} \] Thus, the resonant current is \( \frac{120}{R} \), where \( R \) is the resistance of the circuit.
Step 3: Conclusion.
The resonant current is \( \frac{V}{R} \), which is determined by the applied voltage and the resistance.

Answer 2

Step 1: At resonance, an AC circuit behaves purely like a resistor, so Ohm's law applies directly: I = V ÷ R.

Step 2: Here the voltage is given as V = 120 V.

Step 3: So the resonant current is I₀ = 120 ÷ R, where R is the circuit's resistance.

Answer 3

Starting from general impedance.
In an AC circuit, the overall impedance is \( Z = \sqrt{R^2 + (X_L - X_C)^2} \), where \( X_L \) and \( X_C \) are the inductive and capacitive reactances.
At resonance, \( X_L = X_C \) by definition, so the reactance term cancels and \( Z \) collapses to just \( R \).
With impedance equal to resistance, Ohm's law directly gives the resonant current: \[ I_0 = \frac{V}{Z} = \frac{V}{R} = \frac{120}{R} \]