An isolated metallic object is charged in vacuum to a potential `V_(0)` using a suitable source, its electrostatic energy being `W_(0)`. It is then disconnected from the source and immersed in a large volume of dielectric with dielectric constant K. The electrostatic energy of the sphere in the dielectric is :

Options

A. `K^(2)W_(0)`
B. `K W_(0)`
C. `(W_(0))/(K^(2))`
D. `(W_(0))/(K)`

Answer

To solve the problem step by step, we will analyze the situation before and after the metallic object is immersed in the dielectric medium.

### Step 1: Understand the Initial Conditions
Initially, the isolated metallic object is charged in vacuum to a potential \( V_0 \). The electrostatic energy associated with this charge is given as \( W_0 \).

### Step 2: Relate Charge, Capacitance, and Energy
The charge \( q \) on the object can be expressed in terms of its capacitance \( C_0 \) and potential \( V_0 \):
\[ q = C_0 \cdot V_0 \]
The electrostatic energy \( W_0 \) can be expressed as:
\[ W_0 = \frac{1}{2} C_0 V_0^2 \]

### Step 3: Insert the Object into Dielectric
When the metallic object is immersed in a dielectric medium with dielectric constant \( K \), the capacitance of the object changes. The new capacitance \( C \) becomes:
\[ C = K \cdot C_0 \]
Since the charge \( q \) remains constant (as the object is isolated), we can express the new potential \( V \) in terms of the new capacitance:
\[ V = \frac{q}{C} = \frac{C_0 V_0}{K \cdot C_0} = \frac{V_0}{K} \]

### Step 4: Calculate the New Electrostatic Energy
The new electrostatic energy \( W \) in the dielectric can be calculated using the new capacitance and potential:
\[ W = \frac{1}{2} C V^2 \]
Substituting the expressions for \( C \) and \( V \):
\[ W = \frac{1}{2} (K \cdot C_0) \left(\frac{V_0}{K}\right)^2 \]
Simplifying this expression:
\[ W = \frac{1}{2} K C_0 \cdot \frac{V_0^2}{K^2} = \frac{1}{2} \frac{C_0 V_0^2}{K} \]
Since \( \frac{1}{2} C_0 V_0^2 = W_0 \), we can write:
\[ W = \frac{W_0}{K} \]

### Final Answer
Thus, the electrostatic energy of the sphere in the dielectric is:
\[ W = \frac{W_0}{K} \]