A thin prism of glass is placed in air and water respectively. If `n_(0) = (3)/(2)` and `n_(w) = (4)/(3)`, then the ratio of deviation produced by the prism for a small angle of incidence when placed in air and water separately is:

Options

A. `9 : 8`
B. `4 :3`
C. `3 :4`
D. `4 :1`

Explanation

To solve the problem, we need to calculate the angle of deviation produced by a thin prism of glass when placed in air and water, and then find the ratio of these deviations.

### Step 1: Understand the formula for angle of deviation
For a thin prism, the angle of deviation (Δ) can be expressed as:
\[ \Delta = (\mu - 1) \cdot A \]
where \( \mu \) is the refractive index of the prism material with respect to the medium, and \( A \) is the angle of the prism.

### Step 2: Calculate the deviation in air
Given that the refractive index of the prism in air (\( n_0 \)) is \( \frac{3}{2} \):
\[ \Delta_{air} = (n_0 - 1) \cdot A = \left(\frac{3}{2} - 1\right) \cdot A = \left(\frac{1}{2}\right) \cdot A = \frac{A}{2} \]

### Step 3: Calculate the deviation in water
Now, we need to find the deviation when the prism is placed in water. The refractive index of water (\( n_w \)) is \( \frac{4}{3} \). The effective refractive index of the prism with respect to water is given by:
\[ \mu_{g/w} = \frac{n_0}{n_w} = \frac{\frac{3}{2}}{\frac{4}{3}} = \frac{3}{2} \cdot \frac{3}{4} = \frac{9}{8} \]
Now, we can calculate the deviation in water:
\[ \Delta_{water} = \left(\mu_{g/w} - 1\right) \cdot A = \left(\frac{9}{8} - 1\right) \cdot A = \left(\frac{1}{8}\right) \cdot A = \frac{A}{8} \]

### Step 4: Find the ratio of deviations
Now, we can find the ratio of the deviation in air to the deviation in water:
\[ \text{Ratio} = \frac{\Delta_{air}}{\Delta_{water}} = \frac{\frac{A}{2}}{\frac{A}{8}} = \frac{A}{2} \cdot \frac{8}{A} = \frac{8}{2} = 4 \]

Thus, the ratio of deviation produced by the prism for a small angle of incidence when placed in air and water is:
\[ \text{Ratio} = 4:1 \]

### Final Answer
The ratio of deviation produced by the prism for a small angle of incidence when placed in air and water is \( 4:1 \).

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