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: Identify the Input and Output Power
We have:
- Input Power (P_in) = 10 µW
- Output Power (P_out) = 9 µW

### Step 2: Calculate the Power Loss in Decibels
The formula for power loss in decibels (dB) is given by:

\[ \text{Loss (dB)} = 10 \log_{10} \left(\frac{P_{\text{out}}}{P_{\text{in}}}\right) \]

Substituting the values:

\[ \text{Loss (dB)} = 10 \log_{10} \left(\frac{9 \, \mu W}{10 \, \mu W}\right) \]

### Step 3: Simplify the Ratio
The ratio of output power to input power is:

\[ \frac{9 \, \mu W}{10 \, \mu W} = 0.9 \]

### 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: Calculate the Loss in dB
Now, substituting back into the equation:

\[ \text{Loss (dB)} = 10 \times (-0.045757) \approx -0.45757 \, \text{dB} \]

### Step 6: Convert Length from Meters to Kilometers
The length of the optical fiber is given as 150 meters. We convert this to kilometers:

\[ 150 \, \text{m} = 150 \times 10^{-3} \, \text{km} = 0.15 \, \text{km} \]

### Step 7: Calculate the Loss in dB/km
Now we can find the loss in dB/km:

\[ \text{Loss (dB/km)} = \frac{\text{Loss (dB)}}{\text{Length (km)}} = \frac{-0.45757 \, \text{dB}}{0.15 \, \text{km}} \]

Calculating this gives:

\[ \text{Loss (dB/km)} \approx -3.0505 \, \text{dB/km} \]

### Step 8: Round the Result
Rounding this to a suitable number of significant figures, we get:

\[ \text{Loss (dB/km)} \approx -3 \, \text{dB/km} \]

### Final Answer
Thus, the loss in dB/km is approximately **-3 dB/km**.

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