Relative permittivity of water is 81.If `epsilon_w` and `epsilon_0` are permittivity 's of water and vacuum respectively, then:

Options

A. `epsilon_0=9epsilon_w`
B. `epsilon_0=81epsilon_w`
C. `epsilon_w=9epsilon_0`
D. `epsilon_w=81epsilon_0`

Answer

To solve the problem, we need to establish the relationship between the permittivity of water (\( \epsilon_w \)) and the permittivity of vacuum (\( \epsilon_0 \)), given that the relative permittivity of water (\( \epsilon_r \)) is 81.

### Step-by-Step Solution:

1. **Understand the Definition of Relative Permittivity**:
The relative permittivity (\( \epsilon_r \)) of a material is defined as the ratio of the permittivity of that material (\( \epsilon_w \)) to the permittivity of vacuum (\( \epsilon_0 \)).
\[ \epsilon_r = \frac{\epsilon_w}{\epsilon_0} \]

2. **Substitute the Given Value**:
We know from the problem that the relative permittivity of water is 81. Therefore, we can write:
\[ 81 = \frac{\epsilon_w}{\epsilon_0} \]

3. **Rearranging the Equation**:
To find the relationship between \( \epsilon_w \) and \( \epsilon_0 \), we can rearrange the equation:
\[ \epsilon_w = 81 \cdot \epsilon_0 \]

4. **Conclude the Result**:
From the rearranged equation, we find that the permittivity of water is:
\[ \epsilon_w = 81 \epsilon_0 \]

### Final Answer:
Thus, the correct relationship is:
\[ \epsilon_w = 81 \epsilon_0 \]
This corresponds to option 4.

---