Microwaves (520cm wavelength) enter a long narrow window of a building (which is opaque to microwaves except or the window). If the window is 36 cm wide, what is the distance from the central maximum to the first order minimum of a diffraction patter produced by the window on a wall 6.50 m away?
This question can be solved by the diffraction equation. sinTheta= wavelength/slitwidth. Then use the Theta to calcualte the distance from the geometry.
An interesting variation of this occurred in the 1980's, spys being spys, microwaves were used to listen through windows and sometimes walls to humans speaking. Thus, building vulnerable to sensitive to snooping had microwave shields (screens) installed in the windows and walls.
http://www.newscientist.com.nyud.net:8090/mobile/article/dn8208
To calculate the distance from the central maximum to the first-order minimum of the diffraction pattern produced by the window, we can use the formula:
d * sin(θ) = m * λ
where:
- d is the width of the window
- θ is the angle between the central maximum and the first-order minimum
- m is the order number of the minimum (in this case, m=1)
- λ is the wavelength of the microwaves
First, let's convert the width of the window from centimeters to meters:
d = 36 cm = 0.36 meters
Now we can rearrange the formula to solve for θ:
θ = sin^(-1)((m * λ) / d)
Plugging in the values:
- d = 0.36 meters
- m = 1
- λ = 520 cm (convert to meters by dividing by 100: 520/100 = 5.2 meters)
θ = sin^(-1)((1 * 5.2) / 0.36)
Now, we can calculate the value of θ:
θ = sin^(-1)(14.44)
Using a calculator, we find that θ ≈ 0.259 radians.
Finally, to find the distance from the central maximum to the first-order minimum on the wall, we can use trigonometry:
distance = opposite side / tan(θ)
The opposite side is the distance from the window to the wall, given as 6.50 meters:
distance = 6.50 m / tan(0.259)
Using a calculator, we find that the distance is approximately 16.48 meters.
Therefore, the distance from the central maximum to the first-order minimum of the diffraction pattern is approximately 16.48 meters.