A thin glass prism of `mu = 1.5` is immersed in water of `mu =1.33` . The ratio of deviation of the ray in water to that in air for the same prism is

Options

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

Answer

To solve the problem of finding the ratio of deviation of a ray in water to that in air for a thin glass prism, we can follow these steps:

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

### Step 2: Calculate the deviation in air
For the prism in air, the refractive index of air is approximately 1. Therefore, the relative refractive index \( n \) of the glass prism with respect to air is:
\[ n_{\text{air}} = \frac{\mu_{\text{glass}}}{\mu_{\text{air}}} = \frac{1.5}{1} = 1.5 \]
Now, substituting this into the deviation formula gives:
\[ D_{\text{air}} = (1.5 - 1) \cdot A = 0.5 \cdot A \]

### Step 3: Calculate the deviation in water
For the prism immersed in water, the refractive index of water is given as 1.33. Thus, the relative refractive index \( n \) of the glass prism with respect to water is:
\[ n_{\text{water}} = \frac{\mu_{\text{glass}}}{\mu_{\text{water}}} = \frac{1.5}{1.33} \]
Now substituting this into the deviation formula gives:
\[ D_{\text{water}} = \left(\frac{1.5}{1.33} - 1\right) \cdot A \]
Calculating \( \frac{1.5}{1.33} - 1 \):
\[ D_{\text{water}} = \left(1.5 - 1.33\right) \cdot \frac{A}{1.33} = \frac{0.17}{1.33} \cdot A \]

### Step 4: Find the ratio of deviations
Now, we can find the ratio of the deviation in water to the deviation in air:
\[ \text{Ratio} = \frac{D_{\text{water}}}{D_{\text{air}}} = \frac{\left(\frac{0.17}{1.33} \cdot A\right)}{(0.5 \cdot A)} \]
The \( A \) cancels out:
\[ \text{Ratio} = \frac{0.17}{1.33 \cdot 0.5} = \frac{0.17}{0.665} \]
Calculating this gives:
\[ \text{Ratio} \approx 0.255 \]

### Step 5: Approximate the ratio
To express this as a fraction:
\[ \text{Ratio} \approx \frac{1}{3} \]

### Final Answer
Thus, the ratio of the deviation of the ray in water to that in air for the same prism is approximately:
\[ \frac{1}{3} \]