Light falls on a double slit with slit separation
of 2.19 × 10
−6 m, and the first bright fringe is
seen at an angle of 12.7
◦
relative to the central
maximum.
What is the wavelength of the light?
Answer in units of nm
λ = d• tanα = 2.19•10^-6 • 0.225 =4.94 •10^-7 m = 494 nm
It says incorrect. Can you explain the process?
The coordinate of the k-maximum (separation between the central and the first maxima) is
x(max) = k•λ•L/d,
where L is the separation between the slits and the screen, d is the separation between the slits.
Now, imagine the triangle : the midpoint between slits (taking into account that the slits are very close to each other, this point is the slit), the central maximum on the screen, and the first bright fringe. In this triangle
tan α = x(1max)/L
x(1max) = L •tan α
Then for k=1
x(1max) = λ•L/d,
L •tan α =λ•L/d,
tan α = λ/d,
To find the wavelength of the light, we can use the equation for the fringe separation in a double-slit interference pattern:
λ = (d * sin(θ)) / m
Where:
λ = wavelength of light
d = slit separation
θ = angle of the bright fringe relative to the central maximum
m = order of the fringe (in this case, m = 1 for the first bright fringe)
Given values:
d = 2.19 × 10^(-6) m
θ = 12.7 degrees = 12.7 * (π/180) radians (converting to radians)
m = 1
Substituting the values into the equation:
λ = (2.19 × 10^(-6) * sin(12.7 * (π/180))) / 1
Now we can solve for λ by calculating the right side of the equation.