Physics Problem: Microwave Diffraction Through a Double Slit
Question:
Assume your microwave source produces microwaves with a frequency of 6.00 GHz (6.00 × 10⁹ Hz). The microwaves are incident on a barrier with two slits 12.0 cm apart.
- (a) At what angle is the first-order maximum?
- (b) What is the distance from the center line to the center of the first-order maximum on a screen 10.0 m beyond the slits?
- Hint: Do not use the small angle approximation.
Detailed Solution:
Step 1: Calculate Wavelength (λ)
Microwaves travel at the speed of light, so:
c = 3.00 × 10⁸ m/s, f = 6.00 × 10⁹ Hz
λ = c / f = (3.00 × 10⁸) / (6.00 × 10⁹) = 0.05 m
Step 2: Use the Double-Slit Interference Condition
For first-order maximum (n = 1), we use:
d × sin(θ) = nλ
Where:
- d = 12.0 cm = 0.12 m
- n = 1
- λ = 0.05 m
sin(θ) = λ / d = 0.05 / 0.12 ≈ 0.4167
θ ≈ sin⁻¹(0.4167) ≈ 24.6°
Step 3: Calculate Position of First-Order Maximum (y)
Use trigonometry to find y on a screen D = 10.0 m away:
tan(θ) = y / D ⇒ y = D × tan(θ)
y = 10.0 × tan(24.6°) ≈ 10.0 × 0.4571 = 4.57 m
✅ Final Answers:
(a) Angle of first-order maximum, θ = 24.6°
(b) Distance on screen from center line, y = 4.57 m
(a) Angle of first-order maximum, θ = 24.6°
(b) Distance on screen from center line, y = 4.57 m