Heated sodium is often used in lamps because it has a strong emission line at 58

Question

Heated sodium is often used in lamps because it has a strong emission line at 589.3 nm. A diffraction grating with an unknown line spacing is placed 50.00 cm in front of a screen. The grating is then illuminated with sodium light creating an interference pattern of bright yellow fringes on the screen. The first two bright fringes are measured to be 5.93 cm and 12.12 cm away from the central maximum. Determine the distance between consecutive slits of the diffraction grating. To state this information in a more familiar way, also indicate how many lines per millimeter the diffraction grating contains. How far away from the central maximum is the next bright fringe (after the one at 12.12 cm) on the screen? How many total bright yellow fringes will be seen on the screen? (Be sure to include the central maximum and to account for the fringes both to the right and left of the central axis.)

Explanation / Answer

wave length, lamda=589.3 nm

distance between slit and screen, D=50cm

position of bright fringes,y1=5.93cm and y2=12.12cm

a)

use,

d*sin(theta)=n*lambda

d*(y1/D)=1*lambda

d*(5.93*10^-2)/(50*10^-2) = 1*589.3*10^-9

===> d=4.97*10^-6 m

slit width, d=4.97*10^-6 m

and

lines per millimeter, n=1/(4.97*10^-6)

n=201207 lines/m

n=201.21 lines/mm

b)

d*sin(theta)=n*lambda

4.97*10^-6*(y3/(50*10^-2))=3*589.3*10^-9

===> y3=17.78 cm

position of the third fringe is, y3=17.78 cm

c)

again,

4.97*10^-6*sin(90))=n*589.3*10^-9

====> n=8.4

therefore, no of bright fringe, n=8

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