a narrow slit is placed in front of a convex lens of focal length 25cm and is illuminated normally with a parallel beam of light of wavelength 600nm. the first diffraction minima on either side of the central maximum are separated by 6*10^-3 m. Find the width of slit in nm.

To solve the problem of finding the width of the slit when a narrow slit is illuminated by a parallel beam of light, we can use the principles of diffraction. The situation described involves a single slit and a convex lens, with the light wavelength and the distance between the first diffraction minima provided. Let's break this down step by step.

Understanding the Diffraction Pattern

When light passes through a narrow slit, it spreads out and creates a diffraction pattern on a screen. The central maximum is the brightest part, and the minima are the points where the light intensity drops to zero. The positions of these minima can be calculated using the formula:

• y = (m * λ * L) / a

Where:

• y = distance from the central maximum to the m-th minimum
• m = order of the minimum (1 for the first minimum)
• λ = wavelength of light (600 nm or 600 x 10^-9 m)
• L = distance from the slit to the screen
• a = width of the slit

Given Data

From the problem, we know:

• Wavelength, λ = 600 nm = 600 x 10^-9 m
• Distance between the first minima on either side = 6 x 10^-3 m

Since the distance between the first minima on either side of the central maximum is 6 x 10^-3 m, the distance from the central maximum to the first minimum (y) is half of this value:

• y = 3 x 10^-3 m

Finding the Slit Width

Now, we can rearrange the formula to solve for the slit width (a). For the first minimum (m = 1), the equation becomes:

• y = (1 * λ * L) / a

Rearranging gives us:

• a = (λ * L) / y

However, we need to find the distance L from the lens to the screen. The lens focuses the light, and we can use the lens formula to relate the focal length (f) and the distance L. Since the focal length of the lens is given as 25 cm (0.25 m), we can assume that L is approximately equal to f for small angles, so:

• L ≈ 0.25 m

Calculating the Slit Width

Now we can substitute the values into the rearranged equation:

• a = (600 x 10^-9 m * 0.25 m) / (3 x 10^-3 m)

Calculating this gives:

• a = (150 x 10^-9 m) / (3 x 10^-3 m)
• a = 50 x 10^-6 m

Finally, converting this to nanometers:

• a = 50 x 10^-6 m = 50,000 nm

Final Result

The width of the slit is therefore 50,000 nm. This result illustrates how diffraction patterns can provide insights into the physical dimensions of slits used in optical experiments.