GIVEN: Light of wavelength 585 nm falls on a slit 6.66×10−2 mm wide.
Part A: On a very large distant screen, how many totally dark fringes (indicating complete cancellation) will there be, including both sides of the central bright spot?
*Answer: ?????
Part B: At what angle will the dark fringe that is most distant from the central bright fringe occur?
*Answer: I'm asuming that this will be the equation to use: d*sinA=(N+.5)*L , but I need help finding N in part A to complete this.
A) How many integer values of N satisfy the "dark line" equation
d*sinA = N*L
with |sin A| < 1 ?
N = (d/L) sin A = 113.8 sin A
N can range from -113 to 113 while |sin A| remains below 1. That allows for 226 lines. That includes negative integer values of N.
A central bright fringe occurs at N = 0.
For N=113 (the farthest possible dark fringe from center),
sin A = (113) (L/d) = 0.99257
A = 83 degrees
Part A: To determine the number of totally dark fringes on a distant screen, we need to use the formula for the location of dark fringes in a single-slit diffraction pattern:
y = (m * λ * L) / w
Where:
- y is the distance from the central bright spot to the mth dark fringe
- λ is the wavelength of light
- L is the distance from the slit to the screen
- w is the width of the slit
To find the number of dark fringes on both sides of the central bright spot, we can plug in m = 1 and solve for y. We'll assume L is very large so that we can ignore the contribution of the distance:
y = (1 * 585 nm * L) / (6.66×10^(-2) mm)
Now, let's convert the units to be consistent:
y = (1 * 585 × 10^(-9) m * L) / (6.66×10^(-5) m)
Simplifying:
y = 0.087793 m * L
Since we have ignored the distance to the screen (L), it cancels out and doesn't affect the number of dark fringes. Therefore, the number of totally dark fringes, including both sides of the central bright spot, is infinite.
So, the answer for Part A is: ∞ (infinity)
Part B: To find the angle for the dark fringe most distant from the central bright fringe, we can use the equation you mentioned:
d * sin(θ) = (N + 0.5) * λ
Where:
- d is the width of the slit
- θ is the angle of incidence for the dark fringe
- N is the order of the dark fringe
- λ is the wavelength of light
To find N, let's substitute d = 6.66×10^(-2) mm, θ = 90 degrees (since it's the most distant dark fringe), and λ = 585 nm:
(6.66×10^(-2) mm) * sin(90 degrees) = (N + 0.5) * 585 nm
Simplifying:
(6.66×10^(-2)) * 1 = (N + 0.5) * 585 × 10^(-9)
Dividing both sides by (585 × 10^(-9)):
0.113846 = (N + 0.5)
Subtracting 0.5 from both sides:
0.113346 = N
So, the dark fringe most distant from the central bright fringe occurs at approximately the Nth order (N = 0.113346).
Please note that since the value of N is not a whole number, it may not correspond to an actual dark fringe.
To find the answers to Part A and Part B, we need to use the principles of interference and diffraction of light through a single slit.
Part A: The number of totally dark fringes, including both sides of the central bright spot, can be determined using the formula:
N = (2 * w * L) / λ
where N is the number of dark fringes, w is the width of the slit, L is the distance from the slit to the screen, and λ is the wavelength of light.
Given:
- w = 6.66×10^(-2) mm = 6.66×10^(-5) cm
- λ = 585 nm = 585×10^(-7) cm
To calculate the distance L, we need more information. If L is not given, we cannot determine the exact number of dark fringes.
Part B: To find the angle at which the dark fringe that is most distant from the central bright fringe occurs, we can use the formula:
d * sin(A) = (N + 0.5) * λ
where d is the distance between the slit and the screen, A is the angle at which the dark fringe occurs, N is the order number of the dark fringe (0 for the central bright fringe), and λ is the wavelength of light.
The value of N can be determined once we have the answer to Part A.
So, to summarize:
- We cannot determine the exact number of dark fringes without knowing the distance L.
- We cannot determine the angle of the most distant dark fringe without knowing the number of dark fringes.