To find the least distance from the central bright maximum where the bright fringes of the two wavelengths (800 nm and 600 nm) coincide, we can use the formula for the position of bright fringes in a double-slit experiment:
Formula for Bright Fringe Position
The position of the bright fringes is given by:
y = (m * λ * L) / d
- y = position of the fringe on the screen
- m = order of the fringe (0, 1, 2, ...)
- λ = wavelength of light
- L = distance from the slits to the screen
- d = distance between the slits
Given Values
- Wavelength 1 (λ₁) = 800 nm = 800 x 10-9 m
- Wavelength 2 (λ₂) = 600 nm = 600 x 10-9 m
- Distance to screen (L) = 1.4 m
- Slit separation (d) = 0.28 mm = 0.28 x 10-3 m
Finding Coinciding Fringes
To find the least distance where the bright fringes coincide, we need to find the least common multiple (LCM) of the wavelengths in terms of their fringe orders:
For the first wavelength (800 nm):
y₁ = (m₁ * 800 x 10-9 * 1.4) / (0.28 x 10-3)
For the second wavelength (600 nm):
y₂ = (m₂ * 600 x 10-9 * 1.4) / (0.28 x 10-3)
To find the least distance where both y₁ and y₂ are equal, we set:
m₁ * 800 = m₂ * 600
Finding Values of m₁ and m₂
Rearranging gives:
m₁ / m₂ = 600 / 800 = 3 / 4
Let m₂ = 4k and m₁ = 3k for some integer k. The smallest integer k is 1:
Calculating the Position
Now, substituting m₁ and m₂ back into the fringe position formulas:
For m₁ = 3:
y₁ = (3 * 800 x 10-9 * 1.4) / (0.28 x 10-3)
y₁ = 0.012 m = 12 mm
For m₂ = 4:
y₂ = (4 * 600 x 10-9 * 1.4) / (0.28 x 10-3)
y₂ = 0.012 m = 12 mm
Final Result
The least distance from the central bright maximum where the bright fringes of the two wavelengths coincide is 12 mm.
