A step index fibre has a relative refractive index difference of 0.86% Estimate the critical angle at the core-cladding interface

Answer

To find the critical angle at the core-cladding interface of a step index fiber with a relative refractive index difference of 0.86%, we can follow these steps:

### Step 1: Convert the relative refractive index percentage to a decimal
The relative refractive index difference is given as 0.86%. To convert this percentage into a decimal, we divide by 100.

\[ \text{Relative refractive index difference} = \frac{0.86}{100} = 0.0086 \]

### Step 2: Use the formula for relative refractive index
The relative refractive index difference can be expressed in terms of the refractive indices of the core (\( \mu_1 \)) and cladding (\( \mu_2 \)):

\[ \frac{\mu_1 - \mu_2}{\mu_1} = 0.0086 \]

Rearranging this equation gives:

\[ \mu_1 - \mu_2 = 0.0086 \cdot \mu_1 \]

### Step 3: Express \( \mu_2 \) in terms of \( \mu_1 \)
From the rearranged equation, we can express \( \mu_2 \):

\[ \mu_2 = \mu_1 - 0.0086 \cdot \mu_1 = \mu_1 (1 - 0.0086) = \mu_1 \cdot 0.9914 \]

### Step 4: Find the ratio \( \frac{\mu_2}{\mu_1} \)
Now we can find the ratio of the refractive indices:

\[ \frac{\mu_2}{\mu_1} = 0.9914 \]

### Step 5: Calculate the critical angle using Snell's Law
The critical angle (\( \theta_C \)) can be calculated using Snell's Law, specifically:

\[ \sin(\theta_C) = \frac{\mu_2}{\mu_1} \]

Substituting the value we found:

\[ \sin(\theta_C) = 0.9914 \]

### Step 6: Find the critical angle
To find \( \theta_C \), we take the inverse sine:

\[ \theta_C = \sin^{-1}(0.9914) \]

Using a calculator, we find:

\[ \theta_C \approx 82.4^\circ \]

### Final Answer
The critical angle at the core-cladding interface is approximately \( 82.4^\circ \).

---