Diffraction Grating Formula

A diffraction grating is constructed by scratching a flat piece of transparent material with multiple parallel lines. The material can be scratched with a great number of scratches per cm. The grating to be utilized, for example, contains 6,000 lines per cm. The scratches are opaque, but the spaces between them allow light to pass through. When light falls on a diffraction grating, it forms a multiplicity for the source with a parallel slit.

What is Diffraction Grating?

A diffraction grating is a periodic optical component that separates light into many beams that go in different directions. It’s an alternative to using a prism to study spectra. When light strikes the grating, the split light will often have maxima at an angle θ.

The rays will fall in a parallel bundle on the grating. The wavefront will be perpendicular to the rays and parallel to the grating since the rays and wavefront constitute an orthogonal set. Huygens’ Principle is relevant in this case.

According to it, each transparent slit acts as a new source, and each point on a wavefront acts as a new source, resulting in cylindrical wavefronts spreading out from each.

If a peak continually falls on a valley, the waves cancel and there is no light at that location. Also, if peaks constantly fall on peaks and valleys regularly fall on valleys, the light is brighter at that place. Diffraction is an alternative to using a prism to detect spectra.

Formula for Diffraction Grating

Consider two rays that originate from the line at an angle θ to the straight line. If the difference in their two path lengths is an integral multiple of their wavelength λ, constructive interference will occur, which is given as:

nλ = d sin θ

where,

n = 1, 2, 3, …, is an integer known as order of the grating, 

λ is the wavelength, 

d is the distance between the two spectra and 

θ is the angle.

Also, the distance between two consecutive slits (lines) of the grating is called a grating element. Grating element ‘d’ is calculated as:

Grating element, d = Length of grating/Number of lines

Sample Problems

Problem 1: Determine the slit spacing of a diffraction grating of width 2 cm and produces a deviation of 30° in the second-order with the light of wavelength 500 nm. 

Solution:

Given that,

The order, n = 2,

The angle of deviation, θ = 30° and

The wavelength, λ = 500 nm = 500 × 10^-9 m.

Then, by the diffraction grating formula:

nλ = d sin θ

2 × 500 × 10^-9 m = d × sin 30°

d = 2 × 10^-6 m

Problem 2: Find the number of slits per centimetre for monochromatic light of the wavelength of 600 nm strikes a grating and produces the fourth-order bright line at a 30° angle. 

Solution:

Given that,

The order, n = 4,

The angle of deviation, θ = 30° and

The wavelength, λ = 600 nm = 600 × 10^-9 m.

Then, by the diffraction grating formula:

nλ = d sin θ

4 × 600 × 10^-9 m = d × sin 30°

or

d = 4.8 × 10^-6 m

Now, the number of slits per centimeter is given as:

x = 1 / 4.8 × 10^-6 m

  = 2.08 × 10⁵/ m

  = 2.08 × 10⁵ / 10² cm

  = 2.08 × 10³ / cm

  = 2080 / cm

Problem 3: A grating containing 5000 slits per centimetre is illuminated with monochromatic light and produces the second-order bright line at a 30° angle. Determine the wavelength of the light used? (1 Å = 10-10 m)

Solution:

Given that,

The order, n = 2,

The angle of deviation, θ = 30° and

Number of slits per cm, N = 5000

This implies, the distance between slits, d = 1/N = 1/5000 cm = 5 × 10^-4 cm = 5 × 10^-6 m

Then, by the diffraction grating formula:

nλ = d sin θ

2 × λ = 5 × 10^-6 m × sin 30°

λ = 1.25 × 10^-6 m 

  = 1250 Å 

Problem 4: What is the slit spacing of a diffraction grating of width 1 cm and produces a deviation of 30° in the fourth-order with the light of wavelength 1000 nm. 

Solution:

Given that,

The order, n = 4,

The angle of deviation, θ = 30° and

The wavelength, λ = 1000 nm = 1000 × 10^-9 m.

Then, by the diffraction grating formula:

nλ = d sin θ

4 × 1000 × 10^-9 m = d × sin 30°

d = 8 × 10^-6 m

Problem 5: Find the distance between the slits in a diffraction grating of width 1 cm and produces a deviation of 30° in the Second-order with the light of wavelength 300 nm. 

Solution:

Given that,

The order, n = 2,

The angle of deviation, θ = 30° and

The wavelength, λ = 300 nm = 300 × 10^-9 m.

Then, by the diffraction grating formula:

nλ = d sin θ

2 × 300 × 10^-9 m = d × sin 30°

d = 1.2 × 10^-5 m