1) A diffraction grating with 2.500 x 10^3 lines/cm is used to examine the sodium spectrum. Calculate the angular separation of the two closely spaced yellow lines of sodium (588.995 and 589.592) in each of the first three orders.
2) A diffraction grating has 1950 lines/cm. The angle between the first order maximum and the zeroth-order maximum for monochromatic light is 27 degrees. What is the wavelength of the light?
Thanks!
n*lambda = d sin theta should get both problems.
1) To calculate the angular separation of the two closely spaced yellow lines of sodium, we need to use the formula for angular separation in a diffraction grating:
θ = λ / d
Where:
- θ is the angular separation
- λ is the wavelength of light
- d is the spacing between the lines on the grating
First, convert the line spacing from lines per centimeter to lines per millimeter by dividing by 10:
d = 2,500 lines/cm / 10 = 250 lines/mm
Now, we can calculate the angular separation for each order using the given wavelengths of the yellow lines of sodium:
For the first order:
θ1 = λ1 / d = 588.995 nm / 250 lines/mm
For the second order:
θ2 = λ2 / d = 589.592 nm / 250 lines/mm
For the third order:
θ3 = λ3 / d = λ2 / d = 589.592 nm / 250 lines/mm
Finally, calculate the angular separations for each order.
2) To find the wavelength of light given the angle between the first-order maximum and the zeroth-order maximum, we can use the formula for the relationship between wavelength, line spacing, and angle in a diffraction grating:
sin(θ) = m * λ / d
Where:
- θ is the given angle (in radians)
- m is the order of the maximum (in this case, m = 1 for the first-order maximum)
- λ is the wavelength of light
- d is the spacing between the lines on the grating
First, convert the given angle to radians by multiplying by π/180:
θ = 27° * π/180 = 0.471 radians
Substituting the given values into the equation, we have:
sin(0.471 radians) = 1 * λ / 1950 lines/cm
Solving for λ, the wavelength of light:
λ = sin(0.471 radians) * 1950 lines/cm
Finally, we can calculate the wavelength of the light.