Light waves with two different wavelengths, 632 nm and 474 nm, pass simultaneously through a single slit whose width is 7.63 x 10-5 m and strike a screen 1.30 m from the slit. Two diffraction patterns are formed on the screen. What is the distance (in cm) between the common center of the diffraction patterns and the first occurrence of the spot where a dark fringe from one pattern falls on top of a dark fringe from the other pattern?
Let’s use the condition of diffraction minimum for one split of the width b
b•sinα =k1•λ1
b•sinα =k2•λ2,
Since we have superposition of two maxima
b•sinα is the same for two wavelengths,
k1•λ1= k2•λ2,
λ1/λ2 = k1/k2, 632/474 = 4/3.
Therefore k1=4, k2=3.
Now
sinα = k1•λ1/b =4•632•10^-9/ 7.63•10^-5 =0.033.
As the angle is very small tanα = sinα = 0.033.
tan α= x/L,
x =L• tanα =1.3•0.033 = 0.043 m = 4/3 cm
My mistake.
It should be
k1•λ1= k2•λ2,
λ1/λ2 = k2/k1, 632/474 = 4/3.
k1 =3 k2 = 4
sinα = k1•λ1/b =3•632•10^-9/ 7.63•10^-5 =0.0248
x=L• tanα =L• sinα =1.3•0.0248 =0.0323 m = 3.23 cm
To find the distance between the center of the diffraction patterns and the first occurrence of the spot where a dark fringe from one pattern falls on top of a dark fringe from the other pattern, we need to use the formula for the position of dark fringes in a single slit diffraction pattern.
The formula for the position of dark fringes in a single slit diffraction pattern is given by:
y = (m * λ * L) / d
Where:
- y is the distance from the center of the pattern to the m-th dark fringe
- m is the order of the dark fringe (m = 1, 2, 3, ...)
- λ is the wavelength of the light wave
- L is the distance from the slit to the screen
- d is the width of the slit
Let's calculate the position of the first dark fringe for each wavelength and then find the difference between them.
For the first wavelength, λ1 = 632 nm = 632 x 10^-9 m:
y1 = (1 * 632 x 10^-9 m * 1.30 m) / (7.63 x 10^-5 m)
y1 ≈ 8.320 cm
For the second wavelength, λ2 = 474 nm = 474 x 10^-9 m:
y2 = (1 * 474 x 10^-9 m * 1.30 m) / (7.63 x 10^-5 m)
y2 ≈ 5.624 cm
The difference between the positions is:
Δy = y1 - y2
Δy ≈ 8.320 cm - 5.624 cm
Δy ≈ 2.696 cm
Therefore, the distance between the common center of the diffraction patterns and the first occurrence of the spot where a dark fringe from one pattern falls on top of a dark fringe from the other pattern is approximately 2.696 cm.
To find the distance between the center of the diffraction patterns and the first occurrence of overlapping dark fringes, we need to determine the angular positions of the dark fringes for each wavelength and then calculate the distance using trigonometry.
First, let's determine the angular position of the dark fringes for each wavelength using the single slit diffraction formula:
θ = (λ / d) * sin(φ)
where:
θ is the angular position,
λ is the wavelength of light,
d is the width of the slit, and
φ is the order of the dark fringe.
For the first wavelength (632 nm):
θ1 = (632 x 10^-9 m / (7.63 x 10^-5 m)) * sin(φ)
For the second wavelength (474 nm):
θ2 = (474 x 10^-9 m / (7.63 x 10^-5 m)) * sin(φ)
The distance between the center of the diffraction patterns and the first occurrence of overlapping dark fringes (d) can be calculated using the tangent function:
d = tan(θ1) - tan(θ2)
Now, let's substitute the values into the equations:
θ1 = (632 x 10^-9 m / (7.63 x 10^-5 m)) * sin(1)
θ2 = (474 x 10^-9 m / (7.63 x 10^-5 m)) * sin(1)
d = tan(θ1) - tan(θ2)
Calculating the values:
θ1 = 0.08244 radians
θ2 = 0.06183 radians
d = tan(0.08244) - tan(0.06183)
d ≈ 0.0157 radians
Finally, we can convert the angle to centimeters by multiplying it by the distance to the screen (1.30 m):
distance (in cm) = d * 1.30 m * 100 cm/m
distance ≈ 2.04 cm
Therefore, the distance between the center of the diffraction patterns and the first occurrence of the spot where a dark fringe from one pattern falls on top of a dark fringe from the other pattern is approximately 2.04 cm.