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

Answer

To estimate 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-by-Step Solution:

1. **Understanding the Concept of Critical Angle**:
The critical angle (\( \theta_c \)) is defined as the angle of incidence above which total internal reflection occurs. It can be calculated using the formula:
\[ \sin(\theta_c) = \frac{\mu_r}{\mu_d} \]
where \( \mu_r \) is the refractive index of the cladding and \( \mu_d \) is the refractive index of the core.

2. **Relative Refractive Index Difference**:
The relative refractive index difference (\( \Delta \)) is given as 0.86%. This can be expressed as:
\[ \Delta = \frac{\mu_d - \mu_r}{\mu_d} \times 100 \]
Rearranging this gives:
\[ \frac{\mu_d - \mu_r}{\mu_d} = \frac{0.86}{100} = 0.0086 \]

3. **Expressing \( \mu_r \) in terms of \( \mu_d \)**:
From the rearranged equation, we can express \( \mu_r \) as:
\[ \mu_r = \mu_d \times (1 - 0.0086) = \mu_d \times 0.9914 \]

4. **Substituting \( \mu_r \) into the Critical Angle Formula**:
Now we can substitute \( \mu_r \) into the critical angle formula:
\[ \sin(\theta_c) = \frac{\mu_r}{\mu_d} = \frac{\mu_d \times 0.9914}{\mu_d} = 0.9914 \]

5. **Calculating the Critical Angle**:
Now, we can find the critical angle by taking the inverse sine:
\[ \theta_c = \sin^{-1}(0.9914) \]
Using a calculator, we find:
\[ \theta_c \approx 82.48^\circ \]

### Final Answer:
The critical angle at the core-cladding interface is approximately \( 82.48^\circ \).

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