Light is incident normally on a diffraction grating having 6000 lines/cm. Third order line is observed at an angle of 60°. What is the wavelength of light?

Correct option-1

Concept:

Diffraction Grating:

• A diffraction grating is an optical instrument that consists of a large number of evenly spaced parallel slits that produce an interference pattern similar to but sharper than that of a double slit.

• A diffraction grating is used to disperse a beam of various wavelengths into a spectrum of associated lines as it works on the principle of diffraction of light.
• Diffraction of light is an optical phenomenon in which the light gets bends around corners or edges of an object such that it spreads out and illuminates areas where a shadow is expected.
• This effect is only observable when the size of the opening or edge is comparable with the wavelength of the incident light.
• So, the amount of bending depends on the relative size of the wavelength of light to the size of the opening or corner. 
• The schematic diagram of the diffraction grating is shown below-

The condition for principal maxima is given by-

\(\left( a+b \right)\sin {{\theta }_{m}}=m\lambda \)..(i)

Where,

m = order in diffraction spectrum

\(\lambda\) = wavelength of the incident beam

a = width of the slits and 

b = distance between slits

• The quantity (a+b) represents the grating constant or the reciprocal of (N) number of lines ruled per centimeter of the grating.
• i.e. \(\left( a+b \right)=\frac{1}{N(\text{lines/cm)}}\)

Calculation:

Given-

m = 3 (second-order)

\(N=6000\text{ lines/cm}\)

\(\Rightarrow (a+b)=\left( \frac{1}{6000} \right)cm=\frac{{{10}^{5}}}{6}\)Å

From the condition for principal maxima, we have

\(\left( a+b \right)\sin {{\theta }_{m}}=m\lambda \)..(i)

On substituting the given values in equation(i), we get

\(\frac{{{10}^{5}}}{6}\times \sin 60^0 =3\times\lambda\)

\(\Rightarrow \lambda =\left( \frac{{{10}^{5}}}{6}\times \frac{\sqrt3}{2}\times \frac{1}{3} \right)\)Å

\(\Rightarrow \lambda =\left( \frac{1.732\times {{10}^{5}}}{36} \right)\)Å

\(\therefore \lambda =4811\) Å

Hence, Option-1 is the correct answer.