The minimum deviations suffered by red, yellow and violet beam passing through an equilateral transparent prism are `38.4^(@), 38.7^(@)` and `39.2^(@)` respectively. Calculate the dispersive power of the medium.

Explanation

To calculate the dispersive power of the medium using the given minimum deviations for red, yellow, and violet light passing through an equilateral prism, we can follow these steps:

### Step 1: Write down the given values
- Minimum deviation for red light, \( \delta_R = 38.4^\circ \)
- Minimum deviation for yellow light, \( \delta_Y = 38.7^\circ \)
- Minimum deviation for violet light, \( \delta_V = 39.2^\circ \)
- Angle of the prism, \( A = 60^\circ \) (for an equilateral prism)

### Step 2: Use the formula for refractive index
The formula for the refractive index \( \mu \) in terms of the minimum deviation \( \delta \) is given by:
\[ \mu = \frac{\sin\left(\frac{A + \delta}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]
Since \( A = 60^\circ \), we have \( \frac{A}{2} = 30^\circ \).

### Step 3: Calculate \( \mu_R \) for red light
Substituting \( \delta_R \) into the formula:
\[ \mu_R = \frac{\sin\left(\frac{60^\circ + 38.4^\circ}{2}\right)}{\sin(30^\circ)} \]
Calculating the angle:
\[ \frac{60^\circ + 38.4^\circ}{2} = \frac{98.4^\circ}{2} = 49.2^\circ \]
Now substituting into the equation:
\[ \mu_R = \frac{\sin(49.2^\circ)}{\sin(30^\circ)} = \frac{\sin(49.2^\circ)}{0.5} \]
Calculating \( \sin(49.2^\circ) \):
\[ \mu_R \approx \frac{0.7547}{0.5} \approx 1.5094 \]

### Step 4: Calculate \( \mu_Y \) for yellow light
Using \( \delta_Y \):
\[ \mu_Y = \frac{\sin\left(\frac{60^\circ + 38.7^\circ}{2}\right)}{\sin(30^\circ)} \]
Calculating the angle:
\[ \frac{60^\circ + 38.7^\circ}{2} = \frac{98.7^\circ}{2} = 49.35^\circ \]
Now substituting into the equation:
\[ \mu_Y = \frac{\sin(49.35^\circ)}{0.5} \]
Calculating \( \sin(49.35^\circ) \):
\[ \mu_Y \approx \frac{0.7565}{0.5} \approx 1.5130 \]

### Step 5: Calculate \( \mu_V \) for violet light
Using \( \delta_V \):
\[ \mu_V = \frac{\sin\left(\frac{60^\circ + 39.2^\circ}{2}\right)}{\sin(30^\circ)} \]
Calculating the angle:
\[ \frac{60^\circ + 39.2^\circ}{2} = \frac{99.2^\circ}{2} = 49.6^\circ \]
Now substituting into the equation:
\[ \mu_V = \frac{\sin(49.6^\circ)}{0.5} \]
Calculating \( \sin(49.6^\circ) \):
\[ \mu_V \approx \frac{0.7584}{0.5} \approx 1.5168 \]

### Step 6: Calculate the dispersive power
The dispersive power \( \omega \) is given by:
\[ \omega = \frac{\mu_V - \mu_R}{\mu_Y - 1} \]
Substituting the values:
\[ \omega = \frac{1.5168 - 1.5094}{1.5130 - 1} \]
Calculating the numerator and denominator:
\[ \omega = \frac{0.0074}{0.5130} \approx 0.0144 \]

### Final Answer
The dispersive power of the medium is approximately \( 0.0144 \).

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