A beam of microwaves with = 0.7 mm is incident upon a 12 cm slit. At a distance of 2 m from the slit, what is the approximate width of the slit's image?
To calculate the approximate width of the slit's image, we can use the concept of diffraction. The diffracted pattern occurs when a beam of coherent light passes through a narrow slit and spreads out. The angular spread of the diffracted pattern depends on the wavelength and the size of the slit.
In this case, we are dealing with a beam of microwaves with a wavelength (λ) of 0.7 mm and a slit size of 12 cm. The distance from the slit to the image plane is 2 m.
First, convert the slit size from centimeters to millimeters:
12 cm = 120 mm
Next, we can use the formula for the angular spread of the diffraction pattern:
sin(θ) = λ / a
Where:
- θ is the angular spread of the diffraction pattern
- λ is the wavelength of the microwaves
- a is the slit size
Rearranging the formula, we get:
a = λ / sin(θ)
We can rearrange the formula to solve for θ:
θ = arcsin(λ / a)
Using the given values, we can calculate the angular spread of the diffraction pattern (θ) using the sine inverse (arcsin) function.
θ = arcsin(0.7 mm / 120 mm)
Now, we need to find the approximate width of the slit's image. The width of the image is given by:
w = 2 * D * tan(θ)
Where:
- w is the approximate width of the slit's image
- D is the distance from the slit to the image plane
- θ is the angular spread of the diffraction pattern
Plugging in the values:
w = 2 * 2000 mm * tan(arcsin(0.7 mm / 120 mm))
Finally, calculate the value of w using your preferred method, such as a scientific calculator or computer software capable of trigonometric functions. The resulting value will be the approximate width of the slit's image at a distance of 2 m from the slit.