A beam of microwaves with wavelength= 1 mm is incident upon a 9 cm slit. At a distance of 2 m from the slit, what is the approximate width of the slit's image?
The approximate width of the slit's image is 0.2 mm.
To determine the approximate width of the slit's image, we can use the concept of diffraction. The formula to calculate the angular width of the central maximum is given by:
θ = 1.22 * λ / a
Where:
- θ is the angular width of the central maximum,
- λ is the wavelength of the microwaves, and
- a is the width of the slit.
Given:
- λ = 1 mm = 0.001 m
- a = 9 cm = 0.09 m
Let's calculate θ using the formula first:
θ = 1.22 * 0.001 / 0.09
θ ≈ 0.0136 radians
To find the approximate width of the slit's image at a distance of 2 m, we can use the small-angle approximation, which states that for small angles:
tan(θ) ≈ θ
Now, we can calculate the approximate width of the slit's image:
width_of_image = 2 * tan(θ) * 2
width_of_image ≈ 2 * 0.0136 * 2
width_of_image ≈ 0.0544 meters
Therefore, the approximate width of the slit's image at a distance of 2 m is approximately 0.0544 meters.
To find the approximate width of the slit's image, we can use the concept of diffraction. Diffraction occurs when a wave encounters an obstacle or aperture and bends around it, resulting in a spreading out of the wavefront.
The width of the slit's image can be determined using the formula for the angular width of the central maximum in a diffraction pattern:
θ ≈ λ / d
where:
θ is the angular width of the central maximum,
λ is the wavelength of the microwaves, and
d is the width of the slit.
Let's calculate it step by step:
1. Convert the wavelength of 1 mm to meters:
λ = 1 mm = 1 × 10^(-3) m
2. Convert the width of the slit from centimeters to meters:
d = 9 cm = 9 × 10^(-2) m
3. Substitute the values into the formula:
θ ≈ (1 × 10^(-3) m) / (9 × 10^(-2) m)
4. Simplify the expression:
θ ≈ 1.11 × 10^(-2) radians
Now, to find the approximate width of the slit's image at a distance of 2 m from the slit, we can use the small angle approximation, which states that for small angles:
θ ≈ x / D
where:
θ is the angular width of the central maximum,
x is the width of the slit's image, and
D is the distance from the slit to the screen (2 m in this case).
Rearranging the equation to solve for x:
x ≈ θ * D
x ≈ (1.11 × 10^(-2) radians) * 2 m
x ≈ 2.22 × 10^(-2) m
Therefore, the approximate width of the slit's image at a distance of 2 m from the slit is 2.22 × 10^(-2) meters.