A thin beam of laser light of wavelength 514 nm passes through a diffraction grating having 3952 lines/cm.The resulting pattern is viewed on a distant curved screen that can show all bright fringes up to and including ±90.0° from the central spot.If the experiment were performed with all of the apparatus under water (which has an index of refraction of 1.33),what would be the TOTAL number of bright spots that would show up on the screen?
A) 6
B) 7
C) 12
D) 13
E) 14

Question

Quizplus · Accepted Answer

The number of bright spots in a diffraction grating can be calculated using the formula:

nλ = d sinθ

where n is the order of the bright spot (n=1,2,3...), λ is the wavelength of the light, d is the spacing between the lines in the grating, and θ is the angle between the central bright spot and the nth bright spot.

First, we need to calculate the spacing between the lines in the grating:

d = 1/3952 cm = 2.527 x 10^-4 cm

Next, we need to find the maximum angle at which we can see bright spots:

sinθ = 90°
θ = sin^-1(1) = 90°

Now we can calculate the maximum order of the bright spot that we can see:

nλ = d sinθ
n = d sinθ / λ
n = (2.527 x 10^-4 cm)(sin 90°) / (514 nm)
n = 0.000491

However, we need to take into account the refractive index of water. The wavelength of the light in water will be:

λwater = λ / nwater
λwater = 514 nm / 1.33
λwater = 386.5 nm

Now we can recalculate the maximum order of the bright spot:

nλwater = d sinθ
n = d sinθ / λwater
n = (2.527 x 10^-4 cm)(sin 90°) / (386.5 nm)
n = 0.000635

The maximum order of the bright spot that we can see is now 0.000635. We need to round up to the nearest whole number, so the maximum order is 1.

The total number of bright spots is given by:

# of bright spots = (2n + 1)^2
# of bright spots = (2(1) + 1)^2
# of bright spots = 9

However, we need to subtract the central bright spot, so the total number of bright spots that we will see on the screen is:

# of bright spots = 9 - 1
# of bright spots = 8

But we need to take into account that we can see bright fringes up to and including ±90.0° from the central spot. This means that we need to count both positive and negative orders of bright spots. So we double our result:

# of bright spots = 8 x 2
# of bright spots = 16

However, we need to subtract the central bright spot twice (once for positive orders and once for negative orders), so the final answer is:

# of bright spots = 16 - 2
# of bright spots = 14

Therefore, the answer is (D) 14.

A Thin Beam of Laser Light of Wavelength 514 Nm