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To solve this problem, we need to analyze the forces acting on the charged sphere that’s suspended in a homogeneous electric field.

### Given Data:
- Mass of the sphere, \( m = 2 \, \text{g} = 0.002 \, \text{kg} \)
- Angle with the vertical, \( \theta = 37^\circ \)
- Electric field strength, \( E = 10^3 \, \text{N/C} \)
- Gravitational acceleration, \( g = 10 \, \text{m/s}^2 \)

The sphere is in equilibrium, meaning the forces are balanced. There are two forces acting on it:
1. **Gravitational Force (Weight),** \( F_g = mg \).
2. **Electric Force,** \( F_e = qE \), where \( q \) is the charge we need to find.

There’s also the **tension** in the string, \( T \), which has components balancing these two forces.

### Step-by-Step Solution

1. **Calculate the Gravitational Force:**
   \[
   F_g = mg = 0.002 \, \text{kg} \times 10 \, \text{m/s}^2 = 0.02 \, \text{N}
   \]

2. **Resolve the Tension Force:**
   Since the sphere is in equilibrium, the components of the tension \( T \) balance out the electric force and gravitational force:
   - **Vertical Component of Tension (balances weight):**
     \[
     T \cos \theta = F_g
     \]
   - **Horizontal Component of Tension (balances electric force):**
     \[
     T \sin \theta = F_e = qE
     \]

3. **Calculate \( T \) Using the Vertical Component:**
   \[
   T = \frac{F_g}{\cos \theta} = \frac{0.02}{\cos 37^\circ}
   \]
   Using \( \cos 37^\circ \approx 0.8 \):
   \[
   T = \frac{0.02}{0.8} = 0.025 \, \text{N}
   \]

4. **Find the Electric Force \( F_e \):**
   Now use the horizontal component:
   \[
   F_e = T \sin \theta = 0.025 \times \sin 37^\circ
   \]
   Using \( \sin 37^\circ \approx 0.6 \):
   \[
   F_e = 0.025 \times 0.6 = 0.015 \, \text{N}
   \]

5. **Calculate the Charge \( q \):**
   From \( F_e = qE \):
   \[
   q = \frac{F_e}{E} = \frac{0.015}{10^3} = 1.5 \times 10^{-5} \, \text{C} = 15 \, \mu\text{C}
   \]

### Answer
The charge \( q \) on the sphere is:
\[
\boxed{15 \, \mu\text{C}}
\]

So, the correct option is **(B) 15 µC**.

---

Would you like a more detailed explanation on any part, or do you have any questions?

Here are five related questions to consider:
1. How would the answer change if the angle \( \theta \) were different?
2. What happens to the equilibrium if the electric field strength increases?
3. How would the solution change if the mass of the sphere were doubled?
4. Can this method be used if the electric field is non-uniform?
5. What if there was an additional force acting at an angle; how would you resolve it?

**Tip:** When dealing with forces in equilibrium, always start by resolving components along the axes where forces are acting directly.

Una esfera electrizada se encuentra en reposo dentro de un campo eléctrico homogéneo, según indica el gráfico. Si la esfera de 2 g se encuentra en equilibrio mecánico, determine la cantidad de carga eléctrica que presenta. (g = 10 m/s²)

Determine the electric charge on a 2 g sphere in mechanical equilibrium within a homogeneous electric field of 10³ N/C, inclined at 37°. This solution involves equilibrium forces, trigonometry, and electrostatics principles, suitable for high school physics students.

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