Microwaves of wavelength 4.50 cm enter a long, narrow window in a building that is otherwise essentially opaque to the incoming waves. If the window is 36.5 cm wide, what is the distance from the central maximum to the first-order minimum along a wall 6.70 m from the window?
a sintheta=1*lambda
but sinTheta=tantheta=distance/6.70
.0365*distance/6.7=lambda=.0450
solve for distance in meters.
To calculate the distance from the central maximum to the first-order minimum, we can use the formula for the position of the minimum in a single-slit diffraction:
sin(θ) = m * λ / w
Where:
θ is the angle between the incident wave and the direction of the minimum,
m is the order of the minimum (1 for the first-order),
λ is the wavelength of the wave, and
w is the width of the window.
We can rearrange the formula to solve for the angle θ:
θ = sin^(-1)(m * λ / w)
In our case, we have a second-order minimum (m = 1), a wavelength of 4.50 cm (0.045 m), and a window width of 36.5 cm (0.365 m).
θ = sin^(-1)(1 * 0.045 / 0.365)
θ ≈ sin^(-1)(0.1233)
θ ≈ 7.09 degrees
Now, we can calculate the distance from the window to the first-order minimum:
d = x * tan(θ)
Where:
d is the distance from the window to the first-order minimum,
x is the distance from the window to the wall,
and θ is the angle obtained above.
In our case, x is given as 6.70 m.
d = 6.70 * tan(7.09)
d ≈ 0.828 m
Therefore, the distance from the central maximum to the first-order minimum along the wall 6.70 m from the window is approximately 0.828 meters.
To find the distance from the central maximum to the first-order minimum, we can use the concept of diffraction. Diffraction is the bending or spreading of waves as they encounter an obstacle or pass through an opening.
In this case, the microwave waves are passing through a narrow window. The distance from the central maximum (brightest point) to the first-order minimum (first dark fringe) can be determined using the formula:
d*sin(θ) = m * λ
Where:
- d is the width of the window (36.5 cm = 0.365 m),
- θ is the angle between the central maximum and the first-order minimum,
- m is the order of the minimum (in this case, m = 1), and
- λ is the wavelength of the microwave (4.50 cm = 0.0450 m).
We need to find θ to determine the distance from the central maximum to the first-order minimum. We can use trigonometry to solve for θ using the following formula:
tan(θ) = (distance from the window)/(distance from the wall)
Given:
- Distance from the window = 6.70 m
Rearranging the formula, we get:
θ = arctan((distance from the window)/(distance from the wall))
θ = arctan((6.70 m)/(6.70 m))
Since the distance from the window is equal to the distance from the wall, θ will be small and can be considered as the approximation of θ ≈ D / L.
θ ≈ (36.5 cm) / (6.70 m)
Now, we have the value of θ. We can substitute it into the diffraction formula to find the distance from the central maximum to the first-order minimum.
d * sin(θ) = m * λ
(0.365 m) * sin(θ) = (1) * (0.0450 m)
Now, solving for sin(θ):
sin(θ) = (1 * 0.0450 m) / (0.365 m)
sin(θ) ≈ 0.1233
Finally, we can calculate the distance from the central maximum to the first-order minimum:
distance = d * sin(θ)
distance = (0.365 m) * (0.1233)
distance ≈ 0.0450 m
Therefore, the distance from the central maximum to the first-order minimum is approximately 0.0450 meters.