*Question E5B07:
What is the phase angle between the voltage across and the current through a series RLC circuit if XC is 500 ohms, R is 1 kilohm, and XL is 250 ohms?*  

*Correct Answer is  C: 14.0 degrees with the voltage lagging the current*  

### AI Response: (may contain errors)  

To find the phase angle between the voltage across and the current through a series RLC circuit, we can use the following relationships:

1. **Impedance Calculation**: The total impedance $Z$ in a series RLC circuit is given by:

   $$Z = R + j(X_L - X_C)$$

   where:
   - $R$ = resistance (in ohms)
   - $X_L$ = inductive reactance (in ohms)
   - $X_C$ = capacitive reactance (in ohms)
   - $j$ = imaginary unit

2. **Phase Angle Calculation**: The phase angle $\phi$ can be calculated using:

   $$\tan(\phi) = \frac{X_L - X_C}{R}$$

### Given Values:
- $X_C = 500 \, \Omega$
- $R = 1000 \, \Omega$ (1 kilohm)
- $X_L = 250 \, \Omega$

### Step-by-Step Calculation:

1. **Calculate the net reactance**:

   $$X_L - X_C = 250 - 500 = -250 \, \Omega$$

2. **Calculate the tangent of the phase angle**:

   $$\tan(\phi) = \frac{-250}{1000} = -0.25$$

3. **Find the phase angle**:

   To find $\phi$, we take the arctangent:

   $$\phi = \tan^{-1}(-0.25)$$

   This gives us approximately:

   $$\phi \approx -14.0^\circ$$

### Interpretation:
- A negative phase angle indicates that the voltage lags the current, which is typical in a circuit where the capacitive reactance is greater than the inductive reactance.

### Conclusion:
- The correct answer is **C: 14.0 degrees with the voltage lagging the current**.

### Why Other Choices Are Incorrect:
- **A: 68.2 degrees with the voltage leading the current**: This would imply a highly inductive circuit, which is not the case here since $X_C$ is greater than $X_L$.
- **B: 14.0 degrees with the voltage leading the current**: This is incorrect because a negative phase angle indicates lagging, not leading.
- **D: 68.2 degrees with the voltage lagging the current**: While it correctly states that the voltage lags, the angle is incorrect based on our calculations.

### Summary:
- The phase angle is determined by the relationship between resistance and the net reactance. In this case, the capacitive reactance dominates, leading to a lagging voltage.