Monochromatic light of wavelength `lambda_1` travelling in medium of refractive index `n_1` enters a denser medium of refractive index `n_2`. The wavelength in the second medium is

Options

A. (a) `lambda_1(n_1/n_2)`
B. (b) `lambda(n_2)/(n_1)`
C. (c) `lambda_1`
D. (d) `lambda((n_2-n_1)/(n_1))`

Explanation

To find the wavelength of monochromatic light in a denser medium, we can use the relationship between the wavelength, refractive index, and the speed of light. Here’s a step-by-step solution:

### Step 1: Understand the relationship between wavelength, speed of light, and refractive index
The speed of light in a medium is given by:
\[ v = \frac{c}{n} \]
where:
- \( v \) is the speed of light in the medium,
- \( c \) is the speed of light in vacuum (approximately \( 3 \times 10^8 \) m/s),
- \( n \) is the refractive index of the medium.

### Step 2: Determine the wavelength in the first medium
The wavelength of light in the first medium (\( \lambda_1 \)) is related to the speed of light and the frequency (\( f \)) of the light:
\[ \lambda_1 = \frac{v_1}{f} \]
where \( v_1 \) is the speed of light in the first medium.

### Step 3: Calculate the speed of light in the first medium
Using the refractive index \( n_1 \):
\[ v_1 = \frac{c}{n_1} \]

### Step 4: Relate the frequency in both media
The frequency of light remains constant when it enters a different medium:
\[ f = \frac{v_1}{\lambda_1} = \frac{v_2}{\lambda_2} \]
where \( v_2 \) is the speed of light in the second medium and \( \lambda_2 \) is the wavelength in the second medium.

### Step 5: Calculate the speed of light in the second medium
Using the refractive index \( n_2 \):
\[ v_2 = \frac{c}{n_2} \]

### Step 6: Substitute to find the wavelength in the second medium
From the frequency relationship:
\[ \frac{v_1}{\lambda_1} = \frac{v_2}{\lambda_2} \]
Substituting the expressions for \( v_1 \) and \( v_2 \):
\[ \frac{\frac{c}{n_1}}{\lambda_1} = \frac{\frac{c}{n_2}}{\lambda_2} \]

### Step 7: Rearranging the equation
Cancelling \( c \) from both sides gives:
\[ \frac{1}{n_1 \lambda_1} = \frac{1}{n_2 \lambda_2} \]
Rearranging this gives:
\[ \lambda_2 = \lambda_1 \cdot \frac{n_1}{n_2} \]

### Final Answer
The wavelength in the second medium is:
\[ \lambda_2 = \lambda_1 \cdot \frac{n_1}{n_2} \]

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