Question

An interference is observed due to two coherent sources $\text{A}\&\text{B}$ having zero phase difference separated by a distance $4\lambda$ along the $y-$axis where $\lambda$ is the wavelength of the source. A detector $\text{D}$ is moving on the positive $x-$axis. The number of points on the $x-$axis including the points $x=0$ & $x=\infty$ at which maxima will be deserved is-

• A. $3$
• B. $4$
• C. $5$
• D. $2$

Answer 1

The correct option is A $3$


For maxima: The path difference between the light rays of two coherent sources $S_1$ and $S_2$ must be

$\Delta x=n\lambda$

Where $n=$ Positive integer

From the figure, from $△S_1{S}_{2}D$ we get,

$(S_{1}D{)}^2=(S_1{S}_2{)}^2+(S_{2}D{)}^2$

$(S_{1}M+MD{)}^2=(S_1{S}_2{)}^2+\left(S_{2}D{)}^2\right[\because S_{1}M=\Delta x]$

$(\Delta x+x_n{)}^2=(4\lambda {)}^2+(x_n{)}^2$

$(n\lambda +x_n{)}^2=(4\lambda {)}^2+(x_n{)}^2$

$\Rightarrow x_n=\frac{16{\lambda }^2-n^2{\lambda }^2}{2n\lambda }$

Where $x_n=$ Order of the maxima, and $n=1,2,\mathrm{3...}$

$\Rightarrow x_1=\frac{16{\lambda }^2-{\lambda }^2}{2\lambda }=\frac{15}{2}\lambda$

$\Rightarrow x_2=\frac{16{\lambda }^2-4{\lambda }^2}{4\lambda }=3\lambda$

$\Rightarrow x_3=\frac{16{\lambda }^2-9{\lambda }^2}{6\lambda }=\frac{7}{6}\lambda$

$\Rightarrow x_4=\frac{16{\lambda }^2-16{\lambda }^2}{2\lambda }=0$

$\therefore$ Number of points having maxima will be, $n=3$

<!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> Hence, $\left(A\right)$ is the correct answer.

Why this question? To understand the concept of maxima and minima having path difference $\Delta x$