*Question E5B08:
What is the phase angle between the voltage across and the current through a series RLC circuit if XC is 300 ohms, R is 100 ohms, and XL is 100 ohms?*  

*Correct Answer is  A: 63 degrees with the voltage lagging the current*  

### AI Response: (may contain errors)  

To determine the phase angle between the voltage across and the current through a series RLC circuit, we need to consider the reactance and resistance values given:

- $X_C = 300 \, \text{ohms}$ (capacitive reactance)
- $R = 100 \, \text{ohms}$ (resistance)
- $X_L = 100 \, \text{ohms}$ (inductive reactance)

### Step-by-step Explanation:

1. **Calculate the Net Reactance ($X$):**

   The net reactance in a series RLC circuit is given by the difference between the inductive reactance ($X_L$) and the capacitive reactance ($X_C$):

   $$X = X_L - X_C = 100 \, \text{ohms} - 300 \, \text{ohms} = -200 \, \text{ohms}$$

   A negative net reactance indicates that the circuit is capacitive.

2. **Calculate the Phase Angle ($\theta$):**

   The phase angle $\theta$ between the voltage and the current is determined by the arctangent of the ratio of the net reactance to the resistance:

   $$\theta = \arctan\left(\frac{X}{R}\right) = \arctan\left(\frac{-200}{100}\right) = \arctan(-2)$$

   Using a calculator, $\arctan(-2) \approx -63^\circ$.

3. **Interpret the Phase Angle:**

   - A negative phase angle means the voltage lags the current.
   - Therefore, the phase angle is $63^\circ$ with the voltage lagging the current.

### Why the Correct Answer is Correct:

- **Correct Answer: A: 63 degrees with the voltage lagging the current**

  This matches our calculation where the phase angle is $-63^\circ$, indicating the voltage lags the current by $63^\circ$.

### Why the Other Choices are Incorrect:

- **B: 63 degrees with the voltage leading the current**

  This is incorrect because a negative phase angle indicates lagging, not leading.

- **C: 27 degrees with the voltage leading the current**

  This is incorrect because the calculated phase angle is $63^\circ$, not $27^\circ$, and the voltage lags, not leads.

- **D: 27 degrees with the voltage lagging the current**

  This is incorrect because the phase angle is $63^\circ$, not $27^\circ$.

By understanding the relationship between reactance, resistance, and phase angle, you can determine how voltage and current interact in an RLC circuit.