Marks: 5
In an experiment conducted using the Young's double-slit apparatus, the separation between the slits is 20 µm. A first-order constructive interference fringe appears at an angle of 2.5o from the zeroth order (central) fringe.
A. What wavelength of light is used in the experiment?
B. At what angle would the fourth-order (m = 4) bright fringe appear?
C. At what angle would the fourth-order (m=4) dark fringe appear?
To answer these questions, we can use the formula for the angle of the fringe in the Young's double-slit experiment:
θ = m * λ / d
where:
θ is the angle of the fringe,
m is the order of the fringe,
λ is the wavelength of light used, and
d is the separation between the slits.
A. To find the wavelength of light used in the experiment, we can rearrange the formula:
λ = θ * d / m
Given:
θ = 2.5 degrees = 0.0436 radians,
d = 20 µm = 20 × 10^(-6) m, and
m = 1 (first-order fringe).
Substituting these values into the formula, we get:
λ = (0.0436 * 20 × 10^(-6)) / 1
Calculating this, we find that the wavelength of light used in the experiment is approximately 8.72 × 10^(-7) m or 872 nm (nanometers).
B. Now let's find the angle at which the fourth-order (m = 4) bright fringe appears. We can use the same formula:
θ = m * λ / d
Given:
m = 4,
λ = 872 nm, and
d = 20 µm = 20 × 10^(-6) m.
Substituting these values, we have:
θ = (4 * 872 × 10^(-9)) / 20 × 10^(-6)
Evaluating this expression, we find that the angle of the fourth-order (m = 4) bright fringe is approximately 0.174 degrees.
C. Finally, let's find the angle at which the fourth-order (m = 4) dark fringe appears. In a Young's double-slit experiment, the dark fringes occur at the same angles as the bright fringes but shifted by half a wavelength. Therefore, we can use the same formula, but with an adjusted value of m:
θ = m * λ / d
Given:
m = 4,
λ = 872 nm, and
d = 20 µm = 20 × 10^(-6) m.
Substituting these values, we have:
θ = (4 * 872 × 10^(-9)) / 20 × 10^(-6)
Calculating this expression, we find that the angle of the fourth-order (m = 4) dark fringe is approximately 0.174 degrees.