*Question E5A08:
What is the phase relationship between the current through and the voltage across a series resonant circuit at resonance?*  

*Correct Answer is  C: The voltage and current are in phase*  

### AI Response: (may contain errors)  

In a series resonant circuit, the phase relationship between the current and voltage is crucial for understanding how the circuit operates at resonance.

### Correct Answer: C - The voltage and current are in phase

1. **Why the correct answer is correct:**
   - At resonance, the inductive reactance ($X_L$) and capacitive reactance ($X_C$) are equal, which means they cancel each other out. This results in the circuit behaving purely resistively. 
   - When the circuit is purely resistive, the voltage across the circuit and the current through the circuit are in phase, meaning they reach their maximum and minimum values simultaneously.

### Why the other choices are incorrect:

2. **Choice A: The voltage leads the current by 90 degrees**
   - This is true for a purely inductive circuit, where the current lags the voltage by 90 degrees. However, at resonance, the circuit is not purely inductive.

3. **Choice B: The current leads the voltage by 90 degrees**
   - This is true for a purely capacitive circuit, where the current leads the voltage by 90 degrees. Again, at resonance, the circuit is not purely capacitive.

4. **Choice D: The voltage and current are 180 degrees out of phase**
   - This would imply that when the voltage is at a maximum, the current is at a minimum, which is characteristic of a circuit with a purely resistive load but not at resonance. At resonance, they are in phase, not out of phase.

### Useful Concepts:
- **Resonance Condition:** At resonance in a series RLC circuit, the condition is given by:
  $$X_L = X_C$$
  where:
  - $X_L = 2\pi f L$ (inductive reactance)
  - $X_C = \frac{1}{2\pi f C}$ (capacitive reactance)
  
- **Impedance at Resonance:** The total impedance ($Z$) at resonance is minimized and is equal to the resistance ($R$) of the circuit:
  $$Z = R$$

Understanding these concepts will help you grasp the behavior of series resonant circuits and prepare effectively for the Extra Class exam.