Four monochromatic and coherent sources of light,emitting waves in phase of wavelength $\lambda$,are placed at the points $x = 0, d, 2d$,and $3d$ on the $x$-axis. Then:

$A$ light source,which emits two wavelengths $\lambda_1=400 \ nm$ and $\lambda_2=600 \ nm$,is used in a Young's double slit experiment. If recorded fringe widths for $\lambda_1$ and $\lambda_2$ are $\beta_1$ and $\beta_2$ and the number of fringes for them within a distance $y$ on one side of the central maximum are $m_1$ and $m_2$,respectively,then
$(A)$ $\beta_2 > \beta_1$
$(B)$ $m_1 > m_2$
$(C)$ From the central maximum,$3^{\text{rd}}$ maximum of $\lambda_2$ overlaps with $5^{\text{th}}$ minimum of $\lambda_1$
$(D)$ The angular separation of fringes for $\lambda_1$ is greater than $\lambda_2$

In Young's double-slit experiment, light of wavelength $4000 \, Å$ is used to produce bright fringes of width $0.6 \, mm$ at a distance of $2 \, m$ from the slits. If the whole apparatus is immersed in a liquid of refractive index $1.5$, what will be the new fringe width in $mm$?

In a $YDSE$,if the slits are of unequal width:

State two conditions for obtaining sustained interference of light. In Young's double-slit experiment,using light of wavelength $400 \, nm$,interference fringes of width $'X'$ are obtained. If the wavelength of light is increased to $600 \, nm$ and the separation between the slits is halved,find the ratio of the distances between the screen and the slits in the two arrangements if the fringe width remains the same.

In a Young's double-slit experiment,the slits are separated by $0.12 \, mm$ and the screen is at a distance of $1 \, m$. Find the distance of the $3^{rd}$ dark fringe from the center of the screen in $cm$. Given $\lambda = 6000 \, \mathring{A}$.