Question

Figure 35-25 shows two sources s₁ and s₂ that emit radio waves of wavelength$\mathit{\lambda }$in all directions. The sources are exactly in phase and are separated by a distance equal to 1.5$\mathit{\lambda }$ . The vertical broken line is the perpendicular bisector of the distance between the sources.

(a) If we start at the indicated start point and travel along path 1, does the interference produce a maximum all along the path, a minimum all along the path, or alternating maxima and minima? Repeat for

(b) path 2 (along an axis through the sources) and

(c) path 3 (along a perpendicular to that axis).

Short answer

(a) The interference produces maxima all along path 1.

(b) The interference produces minima all along path 2.

(c) The interference produces alternating maxima and minima all along path 3.

Step-by-step solution

Given data

Distance between the two sources = 1.5$\lambda$

Interference fringe path difference

The path difference of two rays creating a bright fringe of order m for slit separation d, screen distance D, and wavelength $\lambda$is

$\mathit{\Delta }\mathit{L}\mathbf{=}\mathit{m}\mathit{\lambda }$ …(i)

The path difference of two rays creating a dark fringe of order m for slit separation d, screen distance D and wavelength $\lambda$is

$\mathit{\Delta }\mathit{L}\mathbf{=}\left(m+\frac{1}{2}\right)\mathit{\lambda }$…(ii)

(a) Determining the path difference between light from the two sources in path 1

For any point y path 1, the path difference between light rays from the two sources is

$\begin{array}{rcl}\Delta L& =& \sqrt{{\left(0.75\lambda \right)}^2+y^2}-\sqrt{{\left(0.75\lambda \right)}^2+y^2}\\ & =& 0\\ & & \end{array}$

This is equal to equation (i) with m=0. Thus path 1 has bright fringes all along.

(b) Determining the path difference between light from the two sources in path 2

For any point on path 2 at a distance a from source 2, the path difference between light rays from the two sources is

$$\begin{gathered} \Delta L=\left(a+1.5\lambda \right)-a \\ =1.5\lambda \end{gathered}$$

This is equal to equation (ii) with m=1. Thus path 1 has dark fringes all along.

(c) Determining the path difference between light from the two sources in path 3 

For any point on path 3 at a horizontal distance a from source 2 and vertical distance y, the path difference between light rays from the two sources is

$\Delta L=\sqrt{{\left(a+1.5\lambda \right)}^2+y^2}-\sqrt{a^2+y^2}$

This function has a maxima$\mathrm{\Delta L}=1.5\mathrm{\lambda }$at y=0 and tends to 0 as$y\to \infty$. Thus there is a dark fringe at y=0, a bright fringe when $\Delta L$reduces to $1\lambda$, another dark fringe when $\Delta L$reduces to 0.5$\lambda$and finally a bright fringe as $y\to \infty$. Thus there are alternating bright and dark fringes.