Final answer:
The angular positions of the third and fourth principal maxima are approximately 0.4427° and 0.5903°, respectively. The separation of these maxima on a screen 2.0 m from the slits is approximately 4.64 cm.
Step-by-step explanation:
To find the angular positions of the third and fourth principal maxima, we can use the formula for the angle of the principal maxima in a multiple-slit interference pattern:
sin(θn) = nλ / d
where θn is the angle of the nth principal maximum, n is the order of the maximum, λ is the wavelength of the light, and d is the separation between the slits. For the third principal maximum, n = 3, so:
sin(θ3) = (3 * 580 nm) / 0.25 mm
Solving for θ3, we find that θ3 ≈ 0.4427°. Similarly, for the fourth principal maximum, n = 4:
sin(θ4) = (4 * 580 nm) / 0.25 mm
Which yields θ4 ≈ 0.5903°.
To find the separation of these maxima on a screen 2.0 m from the slits, we can use the formula for the separation of the principal maxima:
y = (λL) / d
where y is the separation of the maxima on the screen, L is the distance between the slits and the screen, and d is the separation between the slits. Plugging in the values, we get:
y = (580 nm * 2.0 m) / 0.25 mm
Which gives us y ≈ 4.64 cm.