A laser emitting at 632.8nm illuminates a double slit. A screen is positioned 2.00 m from the slits. Interference fringes are observed with a separation of 1.0 cm.
(a) Determine the separation of the slits.
(b) Determine the angle of the first-order dark fringes.
hughj
To determine the separation of the slits, we can use the equation for the interference pattern:
d * sin(θ) = m * λ
where:
d is the separation of the slits
θ is the angle between the central maximum and the fringe (in radians)
m is the order of the fringe (in this case, m = 1)
λ is the wavelength
Let's calculate the separation of the slits:
(a) Separation of the slits (d):
Given: λ = 632.8 nm (wavelength), θ = not given, m = 1
Since the angles of the dark fringes are not given, we need to consider the small-angle approximation:
sin(θ) ≈ θ (for small angles)
Using the equation for the interference pattern, we can rearrange it to solve for d:
d ≈ (m * λ) / θ
Assuming that the first-order dark fringe is at a small angle, let's consider θ to be very small. Thus, we can approximate sin(θ) as θ.
Therefore, d ≈ (m * λ) / θ
Let's plug in the values:
d ≈ (1 * 632.8 nm) / θ
Now, since the separation of the slits should give a result in meters, and θ is in radians, let's convert the wavelength and then solve for d:
λ = 632.8 nm = 632.8 × 10^(-9) m
d ≈ (1 * 632.8 × 10^(-9) m) / θ
We are missing the value of θ, so we cannot calculate the exact value for the slit separation without it.
(b) Angle of the first-order dark fringes (θ):
To determine the angle of the first-order dark fringes, we can use the equation:
λ = d * sin(θ)
Rearranging the equation, we can solve for θ:
θ = arcsin(λ / d)
Plugging in the given values:
λ = 632.8 nm = 632.8 × 10^(-9) m
d = not given
Since we don't have the value for the slit separation, we cannot calculate the exact angle of the first-order dark fringes without it.
To find the solutions to these questions, we can use the principles of interference and diffraction.
(a) To determine the separation of the slits, we can use the formula for the fringe separation in the double-slit interference pattern:
Δy = λL / d
where Δy is the fringe separation, λ is the wavelength of light, L is the distance from the slits to the screen, and d is the slit separation.
In this case, we know that the fringe separation is 1.0 cm (or 0.01 m), the wavelength of light (λ) is 632.8 nm (or 6.328 x 10^-7 m), and the distance from the slits to the screen (L) is 2.00 m. Plugging these values into the formula, we can solve for the slit separation (d):
d = λL / Δy
= (6.328 x 10^-7 m) * (2.00 m) / (0.01 m)
≈ 1.27 x 10^-4 m
Therefore, the separation of the slits is approximately 1.27 x 10^-4 m.
(b) To determine the angle of the first-order dark fringes, we can use the formula for the angular position of fringes in a double-slit interference pattern:
tanθ = (mλ) / d
where θ is the angle of the fringe, m is the order of the fringe (in this case, the first-order), λ is the wavelength of light, and d is the slit separation.
In this case, we know that the wavelength of light (λ) is 632.8 nm (or 6.328 x 10^-7 m), the slit separation (d) is approximately 1.27 x 10^-4 m, and we are looking for the angle of the first-order dark fringes (m = 1). Plugging these values into the formula, we can solve for the angle (θ):
θ = arctan((mλ) / d)
= arctan((1) * (6.328 x 10^-7 m) / (1.27 x 10^-4 m))
≈ 0.296°
Therefore, the angle of the first-order dark fringes is approximately 0.296°.