b.In an optical ﬁbre cable, the power drops by a factor of 10 approximately every 27 km. If the transmit power is 0.325 W (3.25 × 10–1 W), how far will a signal be transmitted before the power is attenuated to 3.25 μW?

Include a table showing the signal power (in W) versus distance in 27 km intervals (similar to Table 1.2 in Section 1.4.4) as part of your answer.

If a high-quality coaxial cable (LDF5-50A) operating at 200 Mbps is used instead of the optical fibre, how would you expect the calculated distance value to change? Briefly explain your answer (see Section 1.4.4). For this last part, you are neither required to include another table nor a new calculation.

### Part 1: Optical Fiber Signal Attenuation

The problem states that the power drops by a factor of 10 approximately every 27 km, and we are tasked with finding how far a signal will travel before the transmit power reduces from 0.325 W (3.25 × 10⁻¹ W) to 3.25 μW (3.25 × 10⁻⁶ W).

#### Step-by-step Solution

The power attenuation follows a logarithmic pattern because it drops by a factor of 10 over regular intervals (27 km). The general formula for attenuation is:

\[
P(d) = P_0 \times 10^{-\frac{d}{D}}
\]

Where:
- \(P(d)\) is the power at distance \(d\).
- \(P_0\) is the initial transmit power (0.325 W).
- \(D\) is the distance interval over which the power drops by a factor of 10 (27 km).
- \(d\) is the distance traveled by the signal.

We are tasked with finding the distance \(d\) when the power attenuates to 3.25 μW (3.25 × 10⁻⁶ W).

\[
3.25 \times 10^{-6} = 0.325 \times 10^{-\frac{d}{27}}
\]

To solve for \(d\), we follow these steps:

1. Divide both sides by 0.325:
   \[
   \frac{3.25 \times 10^{-6}}{0.325} = 10^{-\frac{d}{27}}
   \]
   Simplifying:
   \[
   10^{-5} = 10^{-\frac{d}{27}}
   \]

2. Since the bases are the same (both are powers of 10), we can set the exponents equal:
   \[
   -5 = -\frac{d}{27}
   \]

3. Solving for \(d\):
   \[
   d = 5 \times 27 = 135 \text{ km}
   \]

Thus, the signal will be attenuated to 3.25 μW after traveling **135 km**.

#### Table: Signal Power vs. Distance in 27 km Intervals

| Distance (km) | Power (W)         |
|---------------|-------------------|
| 0             | 0.325             |
| 27            | 0.0325            |
| 54            | 0.00325           |
| 81            | 0.000325          |
| 108           | 0.0000325         |
| 135           | 0.00000325 (3.25 μW) |

### Part 2: Using a High-Quality Coaxial Cable

If a high-quality coaxial cable (such as LDF5-50A) operating at 200 Mbps is used instead of optical fiber, the calculated distance would likely **decrease**. This is because coaxial cables generally experience higher attenuation compared to optical fibers. While optical fibers can transmit signals over long distances with minimal loss, coaxial cables suffer from more significant attenuation over the same distance, particularly at high data rates like 200 Mbps.

Thus, with the coaxial cable, the signal would need amplification or regeneration at shorter intervals compared to the optical fiber, leading to a shorter maximum transmission distance before reaching the same power level.

---

Would you like further details or have any questions?

Here are five related questions:
1. How does signal attenuation affect communication quality over long distances?
2. What are the typical attenuation factors for different types of cables?
3. How does increasing the data rate impact signal attenuation in both optical fibers and coaxial cables?
4. How do repeaters or amplifiers help maintain signal strength over long distances?
5. What are some common applications where optical fibers outperform coaxial cables?

**Tip:** Always consider the type of medium used for signal transmission, as different cables (optical fiber vs coaxial) exhibit distinct attenuation and bandwidth capabilities.

In an optical fibre cable, the power drops by a factor of 10 approximately every 27 km. If the transmit power is 0.325 W (3.25 × 10–1 W), how far will a signal be transmitted before the power is attenuated to 3.25 μW? Include a table showing the signal power (in W) versus distance in 27 km intervals. If a high-quality coaxial cable (LDF5-50A) operating at 200 Mbps is used instead of the optical fibre, how would you expect the calculated distance value to change? Briefly explain your answer.

Explore how signal power in an optical fiber drops by a factor of 10 every 27 km. Solve for how far a signal will travel before the power attenuates from 0.325 W to 3.25 μW. Includes a table showing power at 27 km intervals and compares optical fibers to coaxial cables like LDF5-50A.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation