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$\frac{\lambda_{1}}{\lambda_{2}}=\frac{n_{2}}{n_{1}}$