Question

Following diffraction pattern was obtained using a diffraction grating using two different wavelengths ${\lambda }_{{}_1}$ and ${\lambda }_{{}_2}$. With the help of the figure identify which is the longer wavelength and their ratios.

• A. ${\lambda }_{{}_2}$ is longer than ${\lambda }_{{}_1}$ and the ratio of the longer to the shorter wavelength is 1.5
• B. ${\lambda }_{{}_1}$ is longer than ${\lambda }_{{}_2}$ and the ratio of the longer to the shorter wavelength is 1.5
• C. ${\lambda }_{{}_1}$ and ${\lambda }_{{}_2}$ are equal and their ratio is 1.0
• D. ${\lambda }_{{}_2}$ is longer than ${\lambda }_{{}_1}$ and the ratio of the longer to the shorter wavelength is 2.5

Answer 1

Answer: (C)

The equation of $n$th principal maxima for wavelength $\lambda$ is given by
$(a+b)\sin \theta =n\lambda$
where $a$ is the width of transparent portion and $b$ is that of opaque portion. The width $(a+b)$ is called the grating element. The spectral lines will overlap, i.e., they will have the same angle of diffraction if
${\lambda }_1={\lambda }_2$
When a line of wavelength ${\lambda }_1$ in order $n_1$ coincides with a line of unknown wavelength ${\lambda }_2$ in order $n_2$, then
$n_2{\lambda }_2=n_1{\lambda }_1$
or$\frac{{\lambda }_1}{{\lambda }_2}=\frac{n_2}{n_1}$