Photons are incident from vacuum on a transparent material with a refractive index n for a given wavelength. Determine the momentum of the incident photon, if its wavelength in the material is equal to `lambda`.

Options

A. `nh//lambda`
B. `h//lambda`
C. `h/(nlambda)`
D. `h//lambda(n+1)`

Answer

To determine the momentum of the incident photon when it transitions from vacuum to a transparent material with a refractive index \( n \) and a wavelength \( \lambda \) in the material, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between wavelength and momentum:**
The momentum \( p \) of a photon can be expressed in terms of its wavelength \( \lambda' \) in vacuum using the formula:
\[ p = \frac{h}{\lambda'} \]
where \( h \) is Planck's constant.

2. **Identify the wavelength in the material:**
The wavelength of the photon in the material is given as \( \lambda \). The relationship between the wavelength in vacuum \( \lambda' \) and the wavelength in the material \( \lambda \) is given by:
\[ \lambda = \frac{\lambda'}{n} \]
where \( n \) is the refractive index of the material.

3. **Express \( \lambda' \) in terms of \( \lambda \):**
Rearranging the equation gives:
\[ \lambda' = n \lambda \]

4. **Substitute \( \lambda' \) into the momentum formula:**
Now, substituting \( \lambda' \) into the momentum formula:
\[ p = \frac{h}{\lambda'} = \frac{h}{n \lambda} \]

5. **Final expression for the momentum of the incident photon:**
Therefore, the momentum of the incident photon is:
\[ p = \frac{h}{n \lambda} \]

### Final Answer:
The momentum of the incident photon is given by:
\[ p = \frac{h}{n \lambda} \]