Material `A` has critical angle `i_(A)`, and material `B` has critical angle `i_(B)(i_(B)gti_(A))`. Then which of the following is true
(i) Light can be totally internally reflected when it passes from `B` to `A`
(ii) Light can be totally internally relected when it passes from `A` to `B`
(iii) Critical angle for total internal reflection is `i_(B)-i_(A)`
(iv) Critical angle between `A` and `B` is `sin^(-1)((sini_(A))/(sin i_(B)))`

Options

A. (i) and (iii)
B. (i) and (iv)
C. (ii) and (iii)
D. (ii) and (iv)

Answer

To solve the problem, we need to analyze the given statements based on the critical angles of materials A and B.

### Step 1: Understand the Critical Angle
The critical angle \( i \) for a material is defined as the angle of incidence above which total internal reflection occurs. It is related to the refractive index \( \mu \) of the material by the formula:
\[ \sin(i) = \frac{1}{\mu} \]
For material A, we have:
\[ \sin(i_A) = \frac{1}{\mu_A} \]
For material B, we have:
\[ \sin(i_B) = \frac{1}{\mu_B} \]

### Step 2: Compare the Critical Angles
Given that \( i_B > i_A \), we can deduce:
\[ \sin(i_B) > \sin(i_A) \]
This implies:
\[ \frac{1}{\mu_B} > \frac{1}{\mu_A} \quad \Rightarrow \quad \mu_B < \mu_A \]
Thus, material A is denser than material B.

### Step 3: Analyze the Statements
1. **Statement (i)**: Light can be totally internally reflected when it passes from B to A.
- This is **false** because total internal reflection occurs when light travels from a denser medium (A) to a rarer medium (B).

2. **Statement (ii)**: Light can be totally internally reflected when it passes from A to B.
- This is **true** because light is moving from a denser medium (A) to a rarer medium (B).

3. **Statement (iii)**: Critical angle for total internal reflection is \( i_B - i_A \).
- This is **false**. The critical angle is not defined as the difference between the two critical angles.

4. **Statement (iv)**: Critical angle between A and B is \( \sin^{-1}\left(\frac{\sin(i_A)}{\sin(i_B)}\right) \).
- This is **true**. From Snell's law at the boundary between two media, we can derive this relationship.

### Conclusion
From the analysis, the true statements are:
- Statement (ii): Light can be totally internally reflected when it passes from A to B.
- Statement (iv): Critical angle between A and B is \( \sin^{-1}\left(\frac{\sin(i_A)}{\sin(i_B)}\right) \).

Thus, the correct options are (ii) and (iv).