In a certain spectrum produced by a glass prism of dispersive power `0.0305,` it is found that the refractive index for the red ray is `1.645.`and that for the violet ray is `1.665`. What is the refractive index for the yellow ray?

Explanation

To find the refractive index for the yellow ray, we can use the formula related to the dispersive power of the prism. The dispersive power (ω) is defined as:

\[ \omega = \frac{\mu_v - \mu_r}{\mu_y - 1} \]

Where:
- \(\mu_v\) = refractive index of violet light
- \(\mu_r\) = refractive index of red light
- \(\mu_y\) = refractive index of yellow light

Given:
- \(\omega = 0.0305\)
- \(\mu_r = 1.645\)
- \(\mu_v = 1.665\)

We need to find \(\mu_y\).

### Step 1: Substitute the known values into the dispersive power formula.

\[ 0.0305 = \frac{1.665 - 1.645}{\mu_y - 1} \]

### Step 2: Calculate the difference in refractive indices for violet and red light.

\[ 1.665 - 1.645 = 0.020 \]

### Step 3: Substitute this difference back into the equation.

\[ 0.0305 = \frac{0.020}{\mu_y - 1} \]

### Step 4: Cross-multiply to solve for \(\mu_y - 1\).

\[ 0.0305(\mu_y - 1) = 0.020 \]

### Step 5: Divide both sides by 0.0305 to isolate \(\mu_y - 1\).

\[ \mu_y - 1 = \frac{0.020}{0.0305} \]

### Step 6: Calculate the right side.

\[ \mu_y - 1 \approx 0.65574 \]

### Step 7: Add 1 to both sides to find \(\mu_y\).

\[ \mu_y \approx 0.65574 + 1 = 1.65574 \]

### Step 8: Round the answer to an appropriate number of significant figures.

\[ \mu_y \approx 1.656 \]

Thus, the refractive index for the yellow ray is approximately **1.656**.

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