Q17Engineering Physics

Question

Answer 1

A plane transmission diffraction grating operates entirely on the optical principle of multiple-slit Fraunhofer diffraction. The exact condition for the formation of principal maxima (bright lines) is rigidly governed by the grating equation.

Given Optical Parameters

Number of grating lines per centimeter ( $N$ ): $4250 lines/cm$
Order of the observed spectrum ( $n$ ): $2$ (Second order)
Angle of diffraction ( $θ$ ): $3 0^{\circ}$

Step 1: Calculating the Grating Element

The grating element, mathematically denoted as $(a + b)$ , is the absolute physical distance between two consecutive transparent slits on the grating. It is simply the reciprocal of the exact number of lines per unit length.

(a + b) = \frac{1}{N} = \frac{1}{4250} cm

Step 2: Applying the Grating Equation

The fundamental grating equation for principal maxima is:

(a + b) sin θ = nλ

We rigorously rearrange the equation to strictly solve for the unknown wavelength, $λ$ :

λ = \frac{( a + b ) sin θ}{n} = \frac{sin θ}{n N}

Substituting the exact given values directly:

λ = \frac{sin ( 3 0 ^{\circ} )}{2 \times 4250} cm

Since $sin (3 0^{\circ}) = 0.5$ :

λ = \frac{0.5}{8500} cm = 5.88235 \times 1 0^{- 5} cm

Step 3: Unit Conversion

To systematically express this highly precise result in standard Angstroms ( $\overset{˚}{A}$ ), we utilize the absolute conversion factor $1 \overset{˚}{A} = 1 0^{- 8} cm$ :

λ = 5.882 \times 1 0^{- 5} \times 1 0^{8} \overset{˚}{A} = 5882 \overset{˚}{A}

Q17Engineering Physics

Question

Answer

Given Optical Parameters

Step 1: Calculating the Grating Element

Step 2: Applying the Grating Equation

Step 3: Unit Conversion