The ratio of angle of minimum deviation of a prism in air and when dipped in water will be `(._(a)mu_(g)=3//2 and ._(a)mu_(w)=4//3)`

Options

A. `1//8`
B. `1//2`
C. `3//4`
D. `1//4`

Answer

To solve the problem of finding the ratio of the angle of minimum deviation of a prism in air and when dipped in water, we can follow these steps:

### Step 1: Understand the formula for angle of minimum deviation
The angle of minimum deviation (D) for a prism can be expressed as:
\[ D = (n - 1)A \]
where \( n \) is the refractive index of the prism with respect to the medium and \( A \) is the angle of the prism.

### Step 2: Calculate the angle of minimum deviation in air
Given the refractive index of glass with respect to air (\( \mu_g \)) is \( \frac{3}{2} \):
\[ D_{air} = \left(\frac{3}{2} - 1\right)A = \left(\frac{1}{2}\right)A = \frac{A}{2} \]

### Step 3: Calculate the refractive index of glass with respect to water
To find the refractive index of glass with respect to water, we use the formula:
\[ \mu_{g/w} = \frac{\mu_g}{\mu_w} \]
where \( \mu_g = \frac{3}{2} \) (refractive index of glass with respect to air) and \( \mu_w = \frac{4}{3} \) (refractive index of water with respect to air).

Calculating this:
\[ \mu_{g/w} = \frac{\frac{3}{2}}{\frac{4}{3}} = \frac{3}{2} \times \frac{3}{4} = \frac{9}{8} \]

### Step 4: Calculate the angle of minimum deviation in water
Using the refractive index of glass with respect to water:
\[ D_{water} = \left(\frac{9}{8} - 1\right)A = \left(\frac{1}{8}\right)A = \frac{A}{8} \]

### Step 5: Find the ratio of angles of minimum deviation
Now we can find the ratio of the angle of minimum deviation in air to that in water:
\[ \frac{D_{air}}{D_{water}} = \frac{\frac{A}{2}}{\frac{A}{8}} = \frac{A/2}{A/8} = \frac{1/2}{1/8} = \frac{8}{2} = 4 \]

### Conclusion
Thus, the ratio of the angle of minimum deviation of the prism in air to that when dipped in water is:
\[ \frac{D_{air}}{D_{water}} = 4 \]