A single slit is illuminated by light of wavelengths ${\mathbf{\lambda }}_{\mathbf{a}}$ and ${\mathbf{\lambda }}_{\mathbf{b}}$, chosen so that the first diffraction minimum of the ${\mathbf{\lambda }}_{\mathbf{a}}$component coincides with the second minimum of the ${\mathbf{\lambda }}_{\mathbf{b}}$component. (a) If ${\mathbf{\lambda }}_{\mathbf{b}}\mathbf{=}\mathbf{350}\mathbf{}\mathbf{nm}$ , what is ${\mathbf{\lambda }}_{\mathbf{a}}$? For what order number ${\mathit{m}}_{\mathbf{b}}$ (if any) does a minimum of the ${\mathbf{\lambda }}_{\mathbf{b}}$ component coincide with the minimum of the ${\mathbf{\lambda }}_{\mathbf{a}}$ component in the order number (b) role="math" localid="1663142824281" ${\mathit{m}}_{\mathbf{a}}\mathbf{=}\mathbf{}\mathbf{2}$ and (c) ${\mathit{m}}_{\mathbf{a}}\mathbf{=}\mathbf{}\mathbf{3}$?

The wavelength of the second light is ${\lambda }_b=350\text{nm}$

The diffraction minimum of first light for second order is $m_a=2$.

The diffraction minimum of first light for third order is$m_a=3$

The diffraction is the phenomenon in which the light bends on the edges of the slit and pattern of light is formed on a screen.

The wavelength of first light is given as:

$m_1{\lambda }_a=m_2{\lambda }_b$

Here, $m_1$ is the first order of diffraction minimum for first light and role="math" localid="1663144017967" $m_2$ is the second order of diffraction minimum for second light

Substitute all the values in equation.

$$\begin{gathered} \left(1\right){\lambda }_a=\left(2\right)\left(350\text{nm}\right) \\ {\lambda }_a=700\text{nm} \end{gathered}$$

Therefore, the wavelength of first light is $700\text{nm}$.

(b)

The order of the diffraction minima of second light is given as:

$m_a{\lambda }_a=m_b{\lambda }_b$

Substitute all the values in equation.

$$\begin{gathered} \left(2\right)\left(700\text{nm}\right)=m_b\left(350\text{nm}\right) \\ {m}_b=4 \end{gathered}$$

Therefore, the order of diffraction minima for second light is width="8" height="15" role="math">4.

(c)

The order of the diffraction minima of second light is given as:

$m_a{\lambda }_a=m_b{\lambda }_b$

Substitute all the values in equation.

$$\begin{gathered} \left(3\right)\left(700\text{nm}\right)=m_b\left(350\text{nm}\right) \\ {m}_b=6 \end{gathered}$$

Therefore, the order of diffraction minima for second light islocalid="1663144938596" $6$.