Monochromatic light of wavelength 500\text{ nm} is incident normally on a plane diffractio… — JAMB UTME

Monochromatic light of wavelength $500\text{ nm}$ is incident normally on a plane diffraction grating. If the second-order principal maximum is observed at an angle of $30^\circ$ to the normal, calculate the number of lines per millimeter ruled on the grating.

Answer: 500 lines/mm

Answer

500 lines/mm

Using the diffraction grating equation

d \sin\theta = n\lambda

with

n = 2

\lambda = 5.0 \times 10^{-7}\text{ m}

, and

\sin(30^\circ) = 0.5

gives a slit separation of

d = 2.0 \times 10^{-6}\text{ m}

. Converting to grating ruling density per millimeter yields

\frac{10^{-3}\text{ m}}{2.0 \times 10^{-6}\text{ m}} = 500\text{ lines/mm}

Step-by-Step Solution

Identify the given physical parameters and convert wavelength to standard SI meters.

Order

n = 2

, diffraction angle

\theta = 30^\circ

, wavelength

\lambda = 500 \times 10^{-9}\text{ m} = 5.0 \times 10^{-7}\text{ m}

Standard SI unit conversion is required to perform wave equation calculations accurately.

Apply the diffraction grating equation

d \sin\theta = n\lambda

to compute the grating spacing

d

d = \frac{2 \times 5.0 \times 10^{-7}\text{ m}}{\sin 30^\circ} = \frac{1.0 \times 10^{-6}}{0.5} = 2.0 \times 10^{-6}\text{ m}

The condition for principal constructive interference maxima is

d \sin\theta = n\lambda

Calculate the number of lines per millimeter by dividing

1\text{ mm}

(

10^{-3}\text{ m}

) by the grating spacing

d

N_{\text{mm}} = \frac{10^{-3}\text{ m}}{2.0 \times 10^{-6}\text{ m}} = 500\text{ lines/mm}

Grating density (lines per unit length) is the reciprocal of the slit separation

d

Key Concept

Diffraction Grating Principal Maxima Condition

Estimated Time:2m 0s

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