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Let's break this problem down step by step.

Given:
- **EMF amplitude** \( E_0 = 6.04 \, V \)
- **Phase angle** \( \phi = +30.9^\circ \)
- **Potential difference across the capacitor** \( V_C = +5.24 \, V \)

We need to find the **potential difference across the inductor** \( V_L \) (with the correct sign included).

---

### Key Concept:
For a **series RLC circuit**:
- The total voltage \( V_{\text{total}} \) across the circuit is the sum of the voltages across the resistor, inductor, and capacitor.
- The voltages across the inductor \( V_L \) and capacitor \( V_C \) are out of phase by 180°: if one voltage is at its peak, the other will be at its minimum.
- **Current** in an RLC circuit lags or leads the applied voltage by the given phase angle \( \phi \).

We will assume:
1. \( V_C \) is at its **positive peak** when the capacitor is fully charged.
2. To calculate the inductor voltage, we need to take into account the fact that \( V_L \) leads the current by \( 90^\circ \), and \( V_C \) lags it by \( 90^\circ \). 

---

### Step-by-Step Solution:

The total applied voltage at a given time can be expressed as:
\[
E(t) = E_0 \sin(\omega t + \phi)
\]

However, at the given instant, we don't need the detailed time-dependent expression—just how the voltages across the capacitor and inductor relate.

The **key phase relationship** is:  
\[
V_L - V_C = E_0 \sin(\phi)
\]
This relationship comes from how voltages across L and C sum to match the total EMF.

---

### Step 1: Calculate \( E_0 \sin(\phi) \)
First, we need to compute the component of the emf along the given phase angle:

\[
E_0 \sin(\phi) = 6.04 \, V \cdot \sin(30.9^\circ)
\]
Using:
\[
\sin(30.9^\circ) \approx 0.514
\]

\[
E_0 \sin(\phi) = 6.04 \times 0.514 = 3.10 \, V
\]

---

### Step 2: Solve for \( V_L \)
Using the equation:
\[
V_L - V_C = E_0 \sin(\phi)
\]
\[
V_L - 5.24 \, V = 3.10 \, V
\]

\[
V_L = 3.10 \, V + 5.24 \, V = 8.34 \, V
\]

---

### Final Answer:
The potential difference across the inductor is:

\[
V_L = +8.34 \, V
\]

---

Would you like further clarification or details? Here are some related questions for deeper understanding:

1. How does the phase difference between voltage and current affect RLC circuit behavior?
2. What happens when the phase angle becomes negative?
3. How do resonant circuits differ from general RLC circuits?
4. What would change if the capacitor voltage were negative at its peak?
5. How does the impedance affect the voltages across the inductor and capacitor?

**Tip:** Remember that in RLC circuits, inductors and capacitors create phase shifts of ±90°, which can be key when working with AC analysis.

An alternating source drives a series RLC circuit with an emf amplitude of 6.04 V, at a phase angle of +30.9°. When the potential difference across the capacitor reaches its maximum positive value of +5.24 V, what is the potential difference across the inductor (sign included)?

Explore how to calculate the potential difference across an inductor in a series RLC circuit with an emf amplitude of 6.04 V and phase angle of +30.9°. Learn the relationship between capacitor and inductor voltages and apply relevant formulas. This problem is ideal for college-level physics students studying AC circuits.

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