Two identical charged spheres are suspended by strings of equal lengths. The strings make an angle `theta` with each other. When suspended in water the angle remains the same. If density of the material of the sphere is 1.5 g/cc, the dielectric constant of water will be ________ (Take density of water = 1 g/cc)

Answer

To solve the problem, we need to find the dielectric constant of water based on the given conditions. Here's a step-by-step solution:

### Step 1: Understand the Forces Acting on the Spheres
When the charged spheres are suspended, they experience two main forces:
1. The gravitational force acting downwards (weight).
2. The electric force acting horizontally due to the repulsion between the identical charges on the spheres.

### Step 2: Set Up the Force Balance
Let:
- \( \rho \) = density of the material of the sphere = 1.5 g/cc
- \( \rho_w \) = density of water = 1 g/cc
- \( V_g \) = volume of the sphere
- \( F_E \) = electric force between the spheres
- \( F_g \) = weight of the sphere = \( \rho \cdot V_g \cdot g \)

The angle \( \theta \) between the strings can be expressed in terms of the forces:
\[ \tan(\theta) = \frac{F_E}{F_g} \]

### Step 3: Analyze the Situation in Air and Water
When the spheres are suspended in air, the force balance can be written as:
\[ \tan(\theta) = \frac{F_E}{\rho \cdot V_g \cdot g} \]

When the spheres are submerged in water, the effective weight of the spheres changes due to the buoyant force:
\[ F_g' = \rho \cdot V_g \cdot g - \rho_w \cdot V_g \cdot g = (\rho - \rho_w) \cdot V_g \cdot g \]

The angle remains the same, so we have:
\[ \tan(\theta) = \frac{F_E}{(\rho - \rho_w) \cdot V_g \cdot g} \]

### Step 4: Set the Two Equations Equal
Since \( \tan(\theta) \) is the same in both cases, we can set the two equations equal to each other:
\[ \frac{F_E}{\rho \cdot V_g \cdot g} = \frac{F_E}{(\rho - \rho_w) \cdot V_g \cdot g} \]

### Step 5: Simplify the Equation
Cancelling \( F_E \) and \( V_g \cdot g \) from both sides gives:
\[ \frac{1}{\rho} = \frac{1}{\rho - \rho_w} \]

### Step 6: Substitute Known Values
Substituting \( \rho = 1.5 \) g/cc and \( \rho_w = 1 \) g/cc:
\[ \frac{1}{1.5} = \frac{1}{1.5 - 1} \]

### Step 7: Solve for Dielectric Constant \( K \)
From the above equation, we can express the electric force in terms of the dielectric constant \( K \):
\[ K = \frac{\rho}{\rho - \rho_w} \]
Substituting the values:
\[ K = \frac{1.5}{1.5 - 1} = \frac{1.5}{0.5} = 3 \]

### Conclusion
Thus, the dielectric constant of water is:
\[ \boxed{3} \]