Question

The 546.1 nm line in mercury is measured at an angle of 73.2° in the third-order spectrum of a diffraction grating. Calculate the number of lines per centimeter for the grating.

Answer 1

5841.42 lines per cm

Step-by-step explanation:

wavelength, λ = 546.1 nm = 546 x 10^-9 m

Angle, θ = 73.2°

order, m = 3

Let d is the separation between the slits

d x Sinθ = m x λ

d x Sin 73.2 = 3 x 546.1 x 10^-9

d x 0.957 = 1638.3 x 10^-9

d = 1.7119 x 10^-6 m

d = 1.7119 x 10^-4 cm

Let N be the number of lines per cm.

So, N = 1 /d

`N = (1)/(1.7119 * 10^(-4))`

N = 5841.42 lines per cm

Answer 2

There are 5847.95 lines per cm for the grating.

Step-by-step explanation:

Given that,

Wavelength of mercury line,
`\lambda=546.1\ nm=546.1* 10^(-9)\ m`

Angle in the third order spectrum,
`\theta=73.2^(\circ)`

Using the grating equation, we get :

`d\ \sin\theta=m\lambda`

Here, m = 3

`d=(m\lambda)/(\sin\theta)\\\\d=(3* 546.1* 10^(-9))/(\sin(73.2))\\\\d=1.71* 10^(-6)\ m`

Let there are N lines for the grating. So,

`N=(1)/(d)\\\\N=(1)/(1.71* 10^(-6)\ m)\\\\N=(1)/(1.71* 10^(-6)* 10^2\ cm)\\\\N=(1)/(1.71* 10^(-4)\ cm)\\\\N=5847.95\ \text{lines}/\text{cm}`

So, there are 5847.95 lines per cm for the grating.