Let's analyze the circuit and calculate the total impedance ($Z_T$) and total current ($I_T$).

Given Data:

• $E_T = 120$ volts (Total voltage)
• $I_R = 10$ A (Current through the resistor)
• $R = 12$ ohms (Resistance)
• $I_L = 5$ A (Current through the inductor)
• $X_L = 24$ ohms (Inductive reactance)
• $I_C = 5$ A (Current through the capacitor)
• $X_C = 24$ ohms (Capacitive reactance)

Steps to Solve:

1. Calculate the Impedance of each component:

   • For the resistor $R$, the impedance is simply $Z_R = R = 12$ ohms.
   • For the inductor, the impedance is $Z_L = jX_L = j24$ ohms.
   • For the capacitor, the impedance is $Z_C = -jX_C = -j24$ ohms.

2. Determine the impedance of the inductive and capacitive branches:

   • The impedance of the inductive branch $Z_L$ is $j24$ ohms.
   • The impedance of the capacitive branch $Z_C$ is $-j24$ ohms.

   Since these are in parallel, the combined impedance $Z_{L \parallel C}$ can be calculated as: $\frac{1}{Z_{L \parallel C}} = \frac{1}{Z_L} + \frac{1}{Z_C}$ $\frac{1}{Z_{L \parallel C}} = \frac{1}{j24} + \frac{1}{-j24}$ $\frac{1}{Z_{L \parallel C}} = \frac{-j + j}{576} = 0$ This indicates that $Z_{L \parallel C}$ is theoretically infinite, but practically, the impedance cancels out as they are in resonance. However, considering the question is simplified, $Z_{L \parallel C} = \infty$, so it does not affect the overall impedance directly.

3. Total impedance $Z_T$: Since the inductive and capacitive impedances cancel each other out due to resonance, the total impedance $Z_T$ of the circuit is simply the impedance of the resistor $Z_R$: $Z_T = Z_R = 12 \, \text{ohms}$

4. Calculate the Total Current $I_T$: $I_T = \frac{E_T}{Z_T} = \frac{120 \, \text{V}}{12 \, \Omega} = 10 \, \text{A}$

Final Answers:

• Total Impedance $Z_T$: $12$ ohms
• Total Current $I_T$: $10$ A

Would you like further details or have any questions?

Related Questions:

1. How would the total impedance change if the resistance value increased?
2. What would be the impact on total current if the inductive reactance doubled?
3. How do you calculate the power dissipated in the resistor?
4. What is the phase angle between the current and voltage in this circuit?
5. How does resonance affect the overall impedance in an RLC circuit?
6. Can you determine the reactive power in the inductive and capacitive branches?
7. How would the circuit behavior change if the voltage source frequency was altered?
8. What is the significance of resonance in AC circuits?

Tip:

When dealing with RLC circuits, always check for resonance conditions, as they can significantly impact the total impedance and current.