https://r2.erweima.ai/mathbot/image/1726256400028tg3m3bm6.jpg

The exercise discusses spectrophotometry and light dispersion. You are asked to calculate the refractive indices \(n_1\) and \(n_2\) for two wavelengths and deduce the values of constants \(A\) and \(B\) from the equation:

\[
n(\lambda) = A + \frac{B}{\lambda^2}
\]

### Given Information:
- Wavelengths:
  - \( \lambda_1 = 578 \, \text{nm} \)
  - \( \lambda_2 = 436 \, \text{nm} \)
  
- Propagation speeds in the medium:
  - \( c_1 = 221190 \, \text{km/s} \)
  - \( c_2 = 219970 \, \text{km/s} \)

- Speed of light in vacuum:
  \[
  c_0 = 299792 \, \text{km/s}
  \]

### Steps to Solve:

1. **Calculate the refractive indices \(n_1\) and \(n_2\)** for each wavelength using the formula:
   \[
   n = \frac{c_0}{c}
   \]
   where \(c_0\) is the speed of light in vacuum and \(c\) is the speed in the material for the given wavelength.

2. **Set up equations for \(n(\lambda_1)\) and \(n(\lambda_2)\)** from the formula:
   \[
   n(\lambda) = A + \frac{B}{\lambda^2}
   \]
   Solve for the constants \(A\) and \(B\).

Let me calculate the refractive indices first, then solve for \(A\) and \(B\).The refractive indices are calculated as follows:

- \( n_1 = 1.355 \) for \( \lambda_1 = 578 \, \text{nm} \)
- \( n_2 = 1.363 \) for \( \lambda_2 = 436 \, \text{nm} \)

### Next steps:
Set up the two equations based on the formula \( n(\lambda) = A + \frac{B}{\lambda^2} \) for the two wavelengths:

\[
n_1 = A + \frac{B}{\lambda_1^2}
\]
\[
n_2 = A + \frac{B}{\lambda_2^2}
\]

Solve these equations to find the constants \( A \) and \( B \). I will calculate this next.It seems like I can’t do more advanced data analysis right now. Please try again later.

In the meantime, here are the next steps you can follow manually to solve for \( A \) and \( B \):

1. **Set up the equations**:
   - \( 1.355 = A + \frac{B}{(578)^2} \)
   - \( 1.363 = A + \frac{B}{(436)^2} \)

2. **Solve these two simultaneous equations** to find the constants \( A \) and \( B \).

Let me know if you have any questions or if you'd like more assistance!

---

Here are 5 related questions for deeper understanding:
1. What is the physical significance of the refractive index in a medium?
2. How does light dispersion occur in different materials?
3. Why is the refractive index wavelength-dependent?
4. How can spectrophotometry be used to determine molecular concentration?
5. What are some real-world applications of refractive index measurements?

**Tip**: Solving simultaneous equations like this can often be simplified using substitution or matrix methods.

Calculate the refractive indices n1 and n2 for the given wavelengths λ1 = 578 nm and λ2 = 436 nm, and deduce the constants A and B from the equation n(λ) = A + B/λ². The given speeds of light in the medium are c1 = 221190 km/s and c2 = 219970 km/s. Use the speed of light in a vacuum c0 = 299792 km/s.

Calculate the refractive indices n1 and n2 for wavelengths λ1 = 578 nm and λ2 = 436 nm. Deduce constants A and B from the equation n(λ) = A + B/λ², using the given speeds of light c1 and c2 in the medium. Detailed step-by-step guide to solving this advanced physics problem.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation