If the input and output power of an optical fibre of length 150 m are `10 mu W` and `9 mu W` respectively then loss in dB/km is approximately

Options

A. `-1`
B. `-2`
C. `-3`
D. `-4`

Answer

To find the loss in dB/km for the given optical fiber, we can follow these steps:

### Step 1: Understand the formula for power loss in dB
The power loss (attenuation) in decibels (dB) can be calculated using the formula:
\[ \text{Loss (dB)} = 10 \log_{10} \left( \frac{P_{\text{out}}}{P_{\text{in}}} \right) \]
where \( P_{\text{out}} \) is the output power and \( P_{\text{in}} \) is the input power.

### Step 2: Substitute the given values
From the question, we have:
- \( P_{\text{in}} = 10 \, \mu W \)
- \( P_{\text{out}} = 9 \, \mu W \)

Substituting these values into the formula:
\[ \text{Loss (dB)} = 10 \log_{10} \left( \frac{9 \, \mu W}{10 \, \mu W} \right) \]

### Step 3: Simplify the fraction
The fraction simplifies to:
\[ \frac{9 \, \mu W}{10 \, \mu W} = 0.9 \]

### Step 4: Calculate the logarithm
Now we calculate the logarithm:
\[ \text{Loss (dB)} = 10 \log_{10} (0.9) \]

Using a calculator or logarithm table, we find:
\[ \log_{10} (0.9) \approx -0.045757 \]

### Step 5: Multiply by 10
Now, we multiply by 10:
\[ \text{Loss (dB)} = 10 \times (-0.045757) \approx -0.45757 \, \text{dB} \]

### Step 6: Convert to dB/km
Since the length of the optical fiber is given as 150 m, we need to convert this to kilometers:
\[ 150 \, \text{m} = 0.15 \, \text{km} \]

To find the loss per kilometer, we divide the total loss by the length in kilometers:
\[ \text{Loss (dB/km)} = \frac{-0.45757 \, \text{dB}}{0.15 \, \text{km}} \approx -3.0505 \, \text{dB/km} \]

### Final Answer
The loss in dB/km is approximately:
\[ \text{Loss} \approx -3 \, \text{dB/km} \]