If `f(V)` and `f_(R)` are the focal lengths of a convex lens for violet and red light respectively and `F_(V)`and `F_(R)` are the focal lengths of concave lens for violet and red light respectively, then we have

Options

A. `f_(V)gtf_(R)`and `F_(V)gtF_(R)`
B. `f_(V)ltf_(R)`and `F_(V)ltF_(R)`
C. `f_(V)gtf_(R)`and `F_(V)gtF_(R)`
D. `f_(V)gtf_(R)`and `F_(V)ltF_(R)`

Answer

To solve the problem, we need to analyze the relationship between the focal lengths of lenses (both convex and concave) for different colors of light (violet and red).

1. **Understanding Focal Lengths**:
- The focal length of a lens is influenced by the refractive index (μ) of the lens material, which varies with the wavelength of light.
- For a convex lens, the focal length (f) is given by the formula:
\[ \frac{1}{f} = (μ - 1) \left( \frac{1}{R_1} + \frac{1}{R_2} \right) \]
- For a concave lens, the focal length (F) is similarly defined but results in a negative value for focal length since it diverges light.

2. **Refractive Index and Wavelength**:
- The refractive index (μ) of a material decreases with increasing wavelength (λ). This means that as the wavelength increases, the refractive index becomes lower.
- Therefore, for red light (which has a longer wavelength than violet light), the refractive index (μ_R) will be less than that for violet light (μ_V).

3. **Comparing Focal Lengths**:
- Since the refractive index for red light is lower, the focal length for red light will be greater than that for violet light in a convex lens:
\[ f_R > f_V \]
- For concave lenses, the same principle applies:
\[ F_R > F_V \]
- This indicates that the focal length of the concave lens for red light is also greater than that for violet light.

4. **Conclusion**:
- Combining the observations for both types of lenses, we can summarize:
- For convex lenses: \( f_R > f_V \)
- For concave lenses: \( F_R > F_V \)

Thus, we conclude that the focal lengths of both convex and concave lenses for red light are greater than those for violet light.

### Summary of Relationships:
- \( f_R > f_V \) (Convex lens)
- \( F_R > F_V \) (Concave lens)