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

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

where:
- $\phi$ = phase angle
- $X_L$ = inductive reactance
- $X_C$ = capacitive reactance
- $R$ = resistance

Given:
- $X_C = 300 \, \Omega$
- $R = 100 \, \Omega$
- $X_L = 100 \, \Omega$

First, we calculate the net reactance ($X$):

$$
X = X_L - X_C = 100 \, \Omega - 300 \, \Omega = -200 \, \Omega
$$

Now, we can substitute this into the tangent formula:

$$
\tan(\phi) = \frac{-200 \, \Omega}{100 \, \Omega} = -2
$$

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

$$
\phi = \tan^{-1}(-2)
$$

Calculating this gives:

$$
\phi \approx -63^\circ
$$

This indicates that the voltage lags the current by 63 degrees, as a negative angle in this context means the voltage is behind the current.

### Explanation of the Correct Answer:
- **Correct Answer: A (63 degrees with the voltage lagging the current)**  
  This is correct because the negative reactance indicates that the circuit is capacitive, causing the voltage to lag behind the current.

### Explanation of Incorrect Choices:
- **B (63 degrees with the voltage leading the current)**  
  This is incorrect because a positive phase angle would indicate that the voltage leads the current, which is not the case here.

- **C (27 degrees with the voltage leading the current)**  
  This is incorrect as it misrepresents both the magnitude of the phase angle and the relationship between voltage and current.

- **D (27 degrees with the voltage lagging the current)**  
  This is incorrect because the calculated phase angle is 63 degrees, not 27 degrees.

### Summary of Useful Formulas:
- To find the phase angle in an RLC circuit:
  $$\tan(\phi) = \frac{X_L - X_C}{R}$$
- Remember that:
  - If $X_L > X_C$, the circuit is inductive (voltage leads current).
  - If $X_C > X_L$, the circuit is capacitive (voltage lags current). 

This understanding will help you tackle similar questions on the exam effectively.