https://r2.erweima.ai/mathbot/image/1719540712876vsx68g2i.png

Let's analyze and solve the problem in the image. The text is in Vietnamese and discusses the energy of a photon in different media. Here is the translation and step-by-step solution.

**Translation:**
> A monochromatic light beam in vacuum has a wavelength \( \lambda \) and the energy of one photon in the beam is \( \epsilon \). When traveling in another transparent medium, the wavelength of the monochromatic light beam becomes \( \frac{\lambda}{\sqrt{2}} \). What is the energy of the photon then?
>
> A. \( 0.5\epsilon \)
> B. \( \epsilon \)
> C. \( \epsilon\sqrt{2} \)
> D. \( \frac{\epsilon}{\sqrt{2}} \)

**Solution:**

1. **Initial Energy in Vacuum:**
   - The energy of a photon in a vacuum is given by \( E = h \nu \), where \( h \) is Planck's constant and \( \nu \) is the frequency.
   - Since \( \nu = \frac{c}{\lambda} \) (where \( c \) is the speed of light in vacuum and \( \lambda \) is the wavelength), the energy can also be expressed as \( E = \frac{hc}{\lambda} \).
   - Given that the energy of one photon in the beam is \( \epsilon \), we have \( \epsilon = \frac{hc}{\lambda} \).

2. **Energy in Another Medium:**
   - When the light travels in another medium with a different wavelength \( \lambda' \), the relationship between energy and wavelength still holds: \( E' = \frac{hc}{\lambda'} \).
   - Here, \( \lambda' = \frac{\lambda}{\sqrt{2}} \).
   - Substituting \( \lambda' \) into the energy equation: \( E' = \frac{hc}{\frac{\lambda}{\sqrt{2}}} = \frac{hc \sqrt{2}}{\lambda} \).

3. **Relating to Initial Energy:**
   - Recall that \( \epsilon = \frac{hc}{\lambda} \).
   - Thus, \( E' = \sqrt{2} \cdot \frac{hc}{\lambda} = \epsilon \sqrt{2} \).

4. **Conclusion:**
   - The energy of the photon in the new medium is \( \epsilon \sqrt{2} \).

The correct answer is:

C. \( \epsilon \sqrt{2} \)

Would you like further details or have any additional questions?

Here are 5 related questions you might find interesting:
1. How does the refractive index of a medium affect the speed of light within it?
2. What is the relationship between photon energy and frequency?
3. How does wavelength change when light transitions between different media?
4. Can the energy of a photon change when it moves from one medium to another?
5. How is Planck's constant used in determining photon energy?

**Tip:** Remember that the energy of a photon is directly proportional to its frequency and inversely proportional to its wavelength.

Understand how photon energy changes when light travels from vacuum to another medium with different wavelength. Learn the step-by-step solution and apply the concept of Planck's constant and wavelength relationships.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation