A single slit fraunhoffer diffraction pattern is formed with white light. For what wavelength of light the third secondary maximum in diffraction pattern coincides with the second secondary maximum in the pattern for red light of wavelength 6500A?

To solve the problem of determining the wavelength of light for which the third secondary maximum in a single slit Fraunhofer diffraction pattern coincides with the second secondary maximum for red light (6500 Å), we need to delve into the principles of diffraction and the mathematical relationships governing it.

Understanding Single Slit Diffraction

In a single slit diffraction experiment, light passing through a narrow slit spreads out and creates a pattern of bright and dark fringes on a screen. The positions of these maxima and minima are determined by the slit width and the wavelength of the light used.

Key Formula for Maxima and Minima

The condition for the minima in a single slit diffraction pattern is given by:

• a sin(θ) = mλ

Where:

• a = width of the slit
• θ = angle of diffraction
• m = order of the minimum (m = 1, 2, 3,...)
• λ = wavelength of the light

The secondary maxima occur between these minima, and their positions can be approximated, but they are not as straightforward to calculate as the minima. However, the positions of the secondary maxima can be derived from the minima positions.

Finding the Coinciding Maxima

In this scenario, we want the third secondary maximum of the unknown wavelength light to coincide with the second secondary maximum of red light (6500 Å). The secondary maxima occur approximately at:

• y ≈ (m + 0.5)λ

Where m is the order of the secondary maximum (0, 1, 2,...). For the second secondary maximum, m = 1, and for the third secondary maximum, m = 2.

Setting Up the Equation

For red light (λ_red = 6500 Å), the position of the second secondary maximum can be expressed as:

• y_red = (1 + 0.5) * 6500 Å = 9750 Å

For the unknown wavelength (let's denote it as λ_x), the position of the third secondary maximum is:

• y_x = (2 + 0.5) * λ_x = 2.5 * λ_x

Equating the Two Positions

To find the wavelength λ_x, we set the two positions equal to each other:

• 2.5 * λ_x = 9750 Å

Now, solving for λ_x gives:

• λ_x = 9750 Å / 2.5 = 3900 Å

Conclusion

The wavelength of light for which the third secondary maximum in the diffraction pattern coincides with the second secondary maximum for red light (6500 Å) is 3900 Å. This wavelength falls within the ultraviolet range of the electromagnetic spectrum, which is shorter than visible light.