1) Yellow light is incident on a single-slit 0.0315 mm wide. On a screen 70.0 cm away, a dark band appears 13.0 mm from the centre of the bright central band. Calculate the wavelength of the light.
tanφ = x/L =13•10^-3/0.7 =1.86•10^-2 ,
φ = 1.06º => sin φ =1.86•10^-2.
b•sin φ = k•λ,
λ = b•sin φ/k =0.0315•10^-3•1.86•10^-2/1 =5.85•10^-7 m
To calculate the wavelength of the light, we can use the formula for single-slit diffraction:
w * sin(θ) = m * λ
where:
w is the width of the slit (0.0315 mm or 0.0315 x 10^-3 m)
θ is the angle of the dark band from the central bright band
m is the order number of the dark band (in this case, m = 1)
λ is the wavelength of the light (what we want to find)
First, let's convert the width of the slit to meters:
w = 0.0315 mm = 0.0315 x 10^-3 m
Next, let's convert the distance to the screen to meters:
d = 70.0 cm = 70.0 x 10^-2 m
Now, we can use the formula to find the wavelength of the light:
w * sin(θ) = m * λ
Rearranging the formula, we have:
λ = w * sin(θ) / m
Substituting the given values:
λ = (0.0315 x 10^-3) * sin(θ) / 1
To find the angle of the dark band, we can use trigonometry. We know that the distance from the center of the bright central band to the dark band is 13.0 mm or 13.0 x 10^-3 m.
sin(θ) = opposite / hypotenuse
sin(θ) = (13.0 x 10^-3 m) / (70.0 x 10^-2 m)
Now we can substitute the value of sin(θ) into the formula for λ:
λ = (0.0315 x 10^-3 m) * [(13.0 x 10^-3 m) / (70.0 x 10^-2 m)]
Simplifying the expression:
λ = (0.0315 x 13.0) / 70.0
Now, we can calculate the value of λ:
λ = 0.00583 m
Finally, we can express the answer in scientific notation:
λ = 5.83 x 10^-3 m
Therefore, the wavelength of the light is 5.83 x 10^-3 meters.