The velocity of light in the core of a step index fibre `2xx10^(8) m//s` and the critical angle at the core-cladding interface is `80^(@)`.Find the numerical aperture and the acceptance angle for the fibre in air.The velocity of light in vacuum is `3xx10^(8)m//s`

Answer

To solve the problem, we will follow these steps:

### Step 1: Calculate the Refractive Index of the Core
The refractive index (μ) of the core can be calculated using the formula:

\[ \mu_d = \frac{c}{v} \]

where:
- \(c = 3 \times 10^8 \, \text{m/s}\) (velocity of light in vacuum)
- \(v = 2 \times 10^8 \, \text{m/s}\) (velocity of light in the core)

Substituting the values:

\[ \mu_d = \frac{3 \times 10^8}{2 \times 10^8} = 1.5 \]

### Step 2: Use the Critical Angle to Find the Refractive Index of the Cladding
The critical angle (θ_c) is related to the refractive indices of the core and cladding by Snell's law:

\[ \sin(\theta_c) = \frac{\mu_r}{\mu_d} \]

where:
- \(\theta_c = 80^\circ\)
- \(\mu_d = 1.5\) (from Step 1)

Rearranging the formula to find \(\mu_r\):

\[ \mu_r = \mu_d \cdot \sin(\theta_c) \]

Calculating \(\sin(80^\circ)\):

\[ \sin(80^\circ) \approx 0.9848 \]

Now substituting the values:

\[ \mu_r = 1.5 \cdot 0.9848 \approx 1.4772 \]

### Step 3: Calculate the Numerical Aperture (NA)
The numerical aperture (NA) is given by the formula:

\[ \text{NA} = \sqrt{\mu_d^2 - \mu_r^2} \]

Substituting the values:

\[ \text{NA} = \sqrt{(1.5)^2 - (1.4772)^2} \]

Calculating:

\[ \text{NA} = \sqrt{2.25 - 2.1855} = \sqrt{0.0645} \approx 0.254 \]

### Step 4: Calculate the Acceptance Angle (α)
The acceptance angle (α) can be calculated using:

\[ \alpha = \sin^{-1}(\text{NA}) \]

Substituting the value of NA:

\[ \alpha = \sin^{-1}(0.254) \]

Calculating:

\[ \alpha \approx 14.74^\circ \]

### Final Answers
- Numerical Aperture (NA) ≈ 0.254
- Acceptance Angle (α) ≈ 14.74°

---