Light of wavelength 649nm illuminates two slits, which are 1.6mm apart. A fringe pattern appears on a screen 2.7m from the slits.
What is the distance between dark fringe number 4 and dark fringe number 5?
To find the distance between two dark fringes, we first need to determine the distance between adjacent dark fringes. This can be done using the equation for the location of the mth dark fringe in a double-slit interference pattern:
y = (m * λ * L) / d
where:
y is the distance between the central maximum and the mth dark fringe,
m is the order of the dark fringe,
λ is the wavelength of light,
L is the distance between the slits and the screen,
and d is the distance between the slits.
Given:
λ = 649 nm (or 649 x 10^-9 m),
d = 1.6 mm (or 1.6 x 10^-3 m),
L = 2.7 m,
and we want to find the distance between the 4th and 5th dark fringes.
Calculating for the distance between adjacent dark fringes:
y(5) - y(4) = [(5 * λ * L) / d] - [(4 * λ * L) / d]
= [(5 * 649 x 10^-9 m * 2.7 m) / (1.6 x 10^-3 m)] - [(4 * 649 x 10^-9 m * 2.7 m) / (1.6 x 10^-3 m)]
Simplifying the calculation:
y(5) - y(4) = (4.325 * 10^-2 m) - (3.45625 * 10^-2 m)
= 0.86875 * 10^-2 m
= 8.6875 * 10^-3 m
Therefore, the distance between the 4th and 5th dark fringes is approximately 8.6875 * 10^-3 meters.
To find the distance between dark fringe number 4 and dark fringe number 5, we can use the formula for the fringe separation:
D = λL / d
where:
D is the fringe separation,
λ is the wavelength of light,
L is the distance between the slits and the screen, and
d is the distance between the two slits.
Given:
wavelength (λ) = 649 nm = 649 × 10^(-9) m
distance between slits (d) = 1.6 mm = 1.6 × 10^(-3) m
distance between slits and screen (L) = 2.7 m
Now, substitute the given values into the formula:
D = (649 × 10^(-9) m) × (2.7 m) / (1.6 × 10^(-3) m)
D ≈ 1.1 × 10^(-3) m
Thus, the distance between dark fringe number 4 and dark fringe number 5 is approximately 1.1 × 10^(-3) meters.