The speed of light in media `M_1` and `M_2` are `1.5 xx 10^8 m//s and 2.0 xx 10^8 m//s` respectively. A ray of light enters from medium `M_1` to `M_2` at an incidence angle `i`. If the ray suffers total internal reflection, the value of `i` is.

Options

A. Equal to or less than `sin^(-1)((3)/(5))`
B. Equal to or greater than `sin^(-1)((3)/(4))`
C. less than `sin^(-1)((2)/(3))`
D. Equal to `sin^(-1)((2)/(3))`

Answer

To solve the problem, we need to find the angle of incidence \( i \) at which total internal reflection occurs when light travels from medium \( M_1 \) to medium \( M_2 \).

### Step-by-Step Solution:

1. **Identify the speeds of light in the media**:
- Speed of light in medium \( M_1 \) (\( v_1 \)) = \( 1.5 \times 10^8 \, \text{m/s} \)
- Speed of light in medium \( M_2 \) (\( v_2 \)) = \( 2.0 \times 10^8 \, \text{m/s} \)

2. **Calculate the refractive indices of the media**:
- The refractive index \( \mu_1 \) of medium \( M_1 \) can be calculated using the formula:
\[ \mu_1 = \frac{c}{v_1} \]
where \( c = 3.0 \times 10^8 \, \text{m/s} \).
\[ \mu_1 = \frac{3.0 \times 10^8}{1.5 \times 10^8} = 2 \]

- The refractive index \( \mu_2 \) of medium \( M_2 \) is calculated similarly:
\[ \mu_2 = \frac{c}{v_2} \]
\[ \mu_2 = \frac{3.0 \times 10^8}{2.0 \times 10^8} = 1.5 \]

3. **Determine the critical angle for total internal reflection**:
- The critical angle \( \theta_c \) can be found using Snell's law:
\[ \sin \theta_c = \frac{\mu_2}{\mu_1} \]
\[ \sin \theta_c = \frac{1.5}{2} = 0.75 \]

4. **Calculate the critical angle**:
- To find the critical angle \( \theta_c \):
\[ \theta_c = \sin^{-1}(0.75) \]
- Using a calculator, we find:
\[ \theta_c \approx 48.6^\circ \]

5. **Determine the angle of incidence for total internal reflection**:
- For total internal reflection to occur, the angle of incidence \( i \) must be greater than the critical angle:
\[ i \geq \theta_c \]
- Therefore, the angle of incidence \( i \) must satisfy:
\[ i \geq 48.6^\circ \]

### Final Answer:
The value of the angle of incidence \( i \) must be greater than or equal to approximately \( 48.6^\circ \).

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