Question

Is there an interference maximum, a minimum, an intermediate state closer to a maximum, or an intermediate state closer to a minimum at point P in Fig. 35-10 if the path length difference of the two rays is

(a)$\mathbf{2}\mathbf{.}\mathbf{2}\mathit{\lambda }$, (b)$\mathbf{3}\mathbf{.}\mathbf{5}\mathit{\lambda }$, (c) $\mathbf{1}\mathbf{.}\mathbf{8}\mathit{\lambda }$, and (d) $\mathbf{1}\mathbf{.}\mathbf{0}\mathit{\lambda }$?

For each situation, give the value of associated with the maximum orminimum involved.

Short answer

(a) There is an intermediate state at point $P$ close to the maxima for $m=2$ when the path difference is$2.2\lambda$.

(b) There is a minimum at point P for$m=3$when the path difference is$3.5\lambda$.

(c) There is an intermediate state at point $P$ close to the maxima for$m=2$when the path difference is$1.8\lambda$.

(d) There is a maxima at point P for $m=1$ when the path difference is $1.0\lambda$.

Step-by-step solution

Given data:

Interference from a pair of slits.

Interference fringe path difference:

The path difference of two rays creating abright fringe of order$\mathit{m}$for slit separationlocalid="1663156893374" $\mathit{d}$ ,screen distance$\mathit{D}$ and wavelength localid="1663156010374" $\mathit{\lambda }$is

localid="1663156168036" $\mathbf{∆}\mathit{L}\mathbf{=}\mathit{m}\mathit{\lambda }$

path difference of two rays creating a dark fringe of order $\mathit{m}$ for slit separationlocalid="1663156062331" $\mathit{D}$ ,screen distance and wavelength $\mathit{\lambda }$is

$\mathbf{∆}\mathit{L}\mathbf{=}\left(m+\frac{1}{2}\right)\mathit{\lambda }$ .....(2)

(a) Determining fringe order for path difference 2.2λ 

From equation (1), path difference for the second order bright fringe is $2\lambda$and from equation (2) the path difference for the second order dark fringe is role="math" localid="1663156500745" $\left(2+\frac{1}{2}\right)\lambda =2.5\lambda$.

Thus, the point for which the path difference is $2.2\lambda$ is an intermediate state closer to the second order maxima.

(b) Determining fringe order for path difference 3.5λ  :

From equation (2) the path difference for the third order dark fringe is,

$\left(3+\frac{1}{2}\right)\lambda =3.5\lambda$

Thus, the point for which the path difference is $3.5\lambda$ is the third order minima.

(c) Determining fringe order for path difference 1.8λ :

From equation (1), path difference for the second order bright fringe$2\lambda$ is and from equation (2) the path difference for the first order dark fringe is,

$\left(1+\frac{1}{2}\right)\lambda =1.5\lambda$

Thus, the point for which the path difference is $1.8\lambda$ is an intermediate state closer to the second order maxima.

(d) Determining fringe order for path difference  1.0λ:

From equation (1) the path difference for the first order bright fringe is$1\lambda$ .

Thus, the point for which the path difference is $1.0\lambda$ is the first order maxima.