*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 determine the phase angle between the voltage across and the current through a series RLC circuit, we need to calculate the net reactance and then use it to find the phase angle.

### 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 ($X_C$):

   $$X = X_L - X_C$$

   Given:
   - $X_C = 500 \, \text{ohms}$
   - $X_L = 250 \, \text{ohms}$

   Substitute the values:

   $$X = 250 \, \text{ohms} - 500 \, \text{ohms} = -250 \, \text{ohms}$$

   The negative sign indicates that the circuit is capacitive.

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

   The phase angle $\theta$ in a series RLC circuit is determined by the arctangent of the net reactance divided by the resistance ($R$):

   $$\theta = \arctan\left(\frac{X}{R}\right)$$

   Given:
   - $R = 1000 \, \text{ohms}$

   Substitute the values:

   $$\theta = \arctan\left(\frac{-250}{1000}\right)$$

   $$\theta = \arctan(-0.25)$$

   Using a calculator, we find:

   $$\theta \approx -14.0 \, \text{degrees}$$

   The negative angle indicates that the voltage lags the current.

### Conclusion:

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

  The phase angle is approximately $-14.0$ degrees, which means the voltage lags the current by 14.0 degrees.

### Why Other Choices Are Incorrect:

- **A: 68.2 degrees with the voltage leading the current**  
  This choice is incorrect because the calculated phase angle is not 68.2 degrees, and the voltage does not lead the current.

- **B: 14.0 degrees with the voltage leading the current**  
  This choice is incorrect because the voltage lags, not leads, the current.

- **D: 68.2 degrees with the voltage lagging the current**  
  This choice is incorrect because the phase angle is not 68.2 degrees.

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