Monochromatic light of wavelength 435 nm falls on a slit of width 0.095 mm. The slit is located 85 cm from a screen. How far is the center of the central bright band to the first dark band?
Why did the center of the central bright band go to therapy? Because it was feeling a bit too bright and needed some darkness in its life!
But in all seriousness, we can use the formula for the position of the central bright band to find the distance between it and the first dark band. The formula is given by:
y = (λ * L) / W
Where:
- y is the distance from the center of the central bright band to the next bright/dark band.
- λ is the wavelength of the light (435 nm = 435 x 10^-9 m).
- L is the distance between the slit and the screen (85 cm = 0.85 m).
- W is the width of the slit (0.095 mm = 0.095 x 10^-3 m).
Now let's calculate it:
y = (435 x 10^-9 m * 0.85 m) / (0.095 x 10^-3 m)
y = 3.868 x 10^-3 m
So, the distance between the center of the central bright band and the first dark band is approximately 3.868 mm. Keep it bright and funny, my friend!
To find the distance between the center of the central bright band and the first dark band, we can use the concept of diffraction.
The formula for the position of the bright bands in a single-slit diffraction pattern is given by:
y = (m * λ * L) / w
Where:
y is the distance between the central bright band and the m-th bright band,
m is the order of the bright band (0 for the central bright band, 1 for the first bright band, -1 for the first dark band, etc.),
λ is the wavelength of the light,
L is the distance between the slit and the screen, and
w is the width of the slit.
In this case, we want to find the distance between the center of the central bright band (m = 0) and the first dark band (m = -1).
Given:
λ = 435 nm (convert to meters: λ = 435 * 10^(-9) m)
w = 0.095 mm (convert to meters: w = 0.095 * 10^(-3) m)
L = 85 cm (convert to meters: L = 85 * 10^(-2) m)
Using the formula, we can calculate the distance:
y = ((-1) * (435 * 10^(-9) m) * (85 * 10^(-2) m)) / (0.095 * 10^(-3) m)
y = (-36975 * 10^(-11) m) / (9.5 * 10^(-5) m)
Simplifying the equation:
y = -3.89 * 10^(-6) m
Therefore, the distance between the center of the central bright band and the first dark band is approximately 3.89 * 10^(-6) meters.
To determine the distance between the center of the central bright band and the first dark band, we need to use the concept of diffraction.
The formula for the angular position of the bright bands in a single-slit diffraction pattern is given by:
sin(θ) = (m * λ) / w
where:
- θ is the angular position of the bright band,
- m is the order of the bright band (m = 0 represents the central bright band),
- λ is the wavelength of light, and
- w is the width of the slit.
Given:
- λ = 435 nm = 435 x 10^(-9) m (convert the wavelength from nm to meters)
- w = 0.095 mm = 0.095 x 10^(-3) m (convert the width of the slit from mm to meters)
- m = 1 (since we are calculating the distance from the central bright band to the first dark band)
Now, we can rearrange the formula to solve for θ:
θ = arcsin((m * λ) / w)
Substituting the values, we get:
θ = arcsin((1 * 435 x 10^(-9)) / (0.095 x 10^(-3)))
Using a scientific calculator, we can evaluate θ to find its value.
Next, we can calculate the distance between the center of the central bright band and the first dark band using the information given.
The distance between the screen and the slit is given as 85 cm = 0.85 m.
To find the distance between two adjacent bright bands (either bright to bright or dark to dark), we can use the formula:
y = D * tan(θ)
where:
- y is the distance between the bright bands,
- D is the distance between the slit and the screen, and
- θ is the angular position of the bright band (which we calculated earlier).
Substituting the values:
y = 0.85 * tan(θ)
Again, using a scientific calculator, we can evaluate y to get the distance between the center of the central bright band and the first dark band.
when the ray from the edge of the slit hits with the ray from the middle travels half a wavelength shorter you get a dark spot.
(width /2) sin angle = wavelength / 2
for small angles sin angle = angle = y / distance to screen
so
y = distance to screen * wavelength / width of slot
= 0.85 * 435*10^-9 / 95*10^-6