Answer :
Fringe spacing (n=1) for 20GHz microwaves passing through 10cm apart slits onto a screen 2m away: 0.3m.
Given:
Frequency [tex](\( f \)) = 20 GHz = \( 20 \times 10^9 \) Hz[/tex]
Distance between the slits [tex](\( d \))[/tex] = 0.10 m
Distance from the slits to the screen [tex](\( L \)) = 2.00 m[/tex]
We can calculate the wavelength [tex](\( \lambda \))[/tex] of the microwaves using the formula:
[tex]\[ \lambda = \frac{c}{f} \][/tex]
Where [tex]\( c \) is the speed of light (\( 3 \times 10^8 \) m/s).[/tex]
[tex]\[ \lambda = \frac{3 \times 10^8 \, \text{m/s}}{20 \times 10^9 \, \text{Hz}} \][/tex]
[tex]\[ \lambda = 0.015 \, \text{m} \][/tex]
Now, we can use the formula for fringe spacing [tex](\( y \)):[/tex]
[tex]\[ y = \frac{\lambda L}{d} \][/tex]
Substituting the known values:
[tex]\[ y = \frac{0.015 \, \text{m} \times 2.00 \, \text{m}}{0.10 \, \text{m}} \][/tex]
[tex]\[ y = \frac{0.03 \, \text{m}^2}{0.10 \, \text{m}} \][/tex]
[tex]\[ y = 0.3 \, \text{m} \][/tex]
So, the fringe spacing [tex](when \( n=1 \)) on the screen 2.00 m behind the slits is \( 0.3 \, \text{m} \).[/tex]
The question involves calculating the fringe spacing for 20GHz microwaves passing through a double-slit experiment. The wavelength is first calculated using the speed of light and frequency, and then the fringe spacing is found using the double-slit interference formula, resulting in a fringe spacing of 0.3 m or 30 cm for the first-order fringe.
The student is asking about calculating the fringe spacing on a projection screen in a double-slit experiment involving microwaves of a 20 GHz frequency. To find the fringe spacing, we can use the formula for double-slit interference:
y = (λD) / d
First, we need to calculate the wavelength (λ) of the 20GHz microwaves. The speed of light (c) is 3 x 10^8 m/s, and the microwave frequency (f) is 20GHz, which is 20 x 10^9 Hz. Using the formula λ = c/f, we get the wavelength.
λ [tex]= (3 \times 10^8 m/s) / (20 \times 10^9 Hz) = 1.5 \times 10^-2\ m[/tex]
Now, using the above wavelength [tex](1.5 \times 10^-2 m)[/tex], the distance to the screen (D) as 2.00 m, and the distance between the slits (d) as 10 cm (which is 0.10 m), we can calculate the fringe spacing for n = 1.
[tex]y = (1.5 \times 10^-2\ m \times 2.00 m) / 0.10 m = 0.3 m[/tex]
Thus, the fringe spacing (y when n = 1) on the screen is 0.3 m or 30 cm.
