The permittivity `in_(0)` of vacuum is `8.86 xx 10^(-12) C^(2)//N-m^(2)` and the dielectric constant of water is 81. The permittivity of water in `C^(2)//N-m^(2)` is -

Options

A. `81 xx 8.86 xx 10^(-12)`
B. `8.86 xx 10^(-12)`
C. `(8.86 xx 10^(-12))//81`
D. `81//(8.86 xx 10^(-12))`

Answer

To find the permittivity of water, we can use the relationship between the permittivity of a medium and the permittivity of free space (vacuum). The relationship is given by:

\[ \epsilon = \epsilon_r \cdot \epsilon_0 \]

Where:
- \(\epsilon\) is the permittivity of the medium (water in this case),
- \(\epsilon_r\) is the relative permittivity (dielectric constant) of the medium,
- \(\epsilon_0\) is the permittivity of free space (vacuum).

### Step 1: Identify the given values
- The permittivity of free space, \(\epsilon_0 = 8.86 \times 10^{-12} \, \text{C}^2/\text{N m}^2\)
- The dielectric constant of water, \(\epsilon_r = 81\)

### Step 2: Substitute the values into the formula
Now, we can substitute the values into the equation:

\[ \epsilon = \epsilon_r \cdot \epsilon_0 = 81 \cdot (8.86 \times 10^{-12} \, \text{C}^2/\text{N m}^2) \]

### Step 3: Calculate the permittivity of water
Now we perform the multiplication:

\[ \epsilon = 81 \cdot 8.86 \times 10^{-12} \]

Calculating this gives:

\[ \epsilon = 718.26 \times 10^{-12} \, \text{C}^2/\text{N m}^2 \]

### Step 4: Express in standard form
We can express this in standard form:

\[ \epsilon = 7.1826 \times 10^{-10} \, \text{C}^2/\text{N m}^2 \]

Thus, the permittivity of water is approximately:

\[ \epsilon \approx 7.18 \times 10^{-10} \, \text{C}^2/\text{N m}^2 \]

### Final Answer:
The permittivity of water is \( \approx 7.18 \times 10^{-10} \, \text{C}^2/\text{N m}^2 \).

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