A beam o monochromatic light of wavelength `lambda` is reflected from air into water to refractive index 4/3. The wavelength of light beam inside water will be

Options

A. `9/16xxlambda`
B. `3lambda//4`
C. `lambdaxx4/3`
D. `lambda`

Answer

To solve the problem of finding the wavelength of a monochromatic light beam inside water, we can follow these steps:

### Step 1: Understand the relationship between wavelength, frequency, and velocity
The speed of light in a medium is given by the equation:
\[ v = f \cdot \lambda \]
where:
- \( v \) is the speed of light in the medium,
- \( f \) is the frequency of the light,
- \( \lambda \) is the wavelength of the light.

### Step 2: Recognize that frequency remains constant
When light passes from one medium to another (in this case, from air to water), the frequency of the light does not change. Therefore, we can say:
\[ f_{\text{air}} = f_{\text{water}} \]

### Step 3: Relate the speed of light in different media to the refractive index
The speed of light in a medium is related to the speed of light in a vacuum and the refractive index \( n \) of the medium:
\[ v = \frac{c}{n} \]
where:
- \( c \) is the speed of light in vacuum,
- \( n \) is the refractive index of the medium.

### Step 4: Write the relationship for wavelength in different media
Since the frequency remains constant, we can write:
\[ f = \frac{v_{\text{air}}}{\lambda_{\text{air}}} = \frac{v_{\text{water}}}{\lambda_{\text{water}}} \]
This leads to the equation:
\[ \lambda_{\text{water}} = \frac{v_{\text{water}}}{f} \]

### Step 5: Substitute the speed of light in water
Using the relationship for speed in terms of refractive index:
\[ \lambda_{\text{water}} = \frac{c/n}{f} \]

### Step 6: Relate the wavelength in water to the wavelength in air
From the relationship of speed and wavelength:
\[ \lambda_{\text{water}} = \frac{\lambda_{\text{air}}}{n} \]
Given that \( n = \frac{4}{3} \), we can substitute this value:
\[ \lambda_{\text{water}} = \frac{\lambda}{\frac{4}{3}} = \frac{3\lambda}{4} \]

### Final Answer
Thus, the wavelength of the light beam inside water will be:
\[ \lambda_{\text{water}} = \frac{3\lambda}{4} \]