Question

In Fig 35-54, two isotropic point sources ${\mathit{S}}_{\mathbf{1}}$and ${\mathit{S}}_{\mathbf{2}}$emit light at wavelength $\mathit{\lambda }\mathbf{=}\mathbf{400}\mathbf{}\mathbf{nm}$. Source ${\mathit{S}}_{\mathbf{1}}$is located at $\mathit{y}\mathbf{=}\mathbf{640}\mathbf{}\mathbf{nm}$, source ${\mathit{S}}_{\mathbf{2}}$is located at $\mathit{y}\mathbf{=}\mathbf{-}\mathbf{640}\mathbf{}\mathbf{nm}$. At point ${\mathit{P}}_{\mathbf{1}}$(at $\mathit{x}\mathbf{=}\mathbf{720}\mathbf{}\mathbf{nm}$), the wave from ${\mathit{S}}_{\mathbf{2}}$arrives ahead of the wave from ${\mathit{S}}_{\mathbf{1}}$by a phase difference of $\mathbf{0}\mathbf{.}\mathbf{600}\mathbf{\pi }$rad. (a) What multiple of $\mathit{\lambda }$gives the phase difference between the waves from the two sources as the waves from the two sources as the waves arrive at point ${\mathit{P}}_{\mathbf{2}}$, which is located at $\mathit{y}\mathbf{=}\mathbf{720}\mathbf{}\mathbf{nm}$? (The figure is not drawn to scale) (b) If the waves arrive at ${\mathit{P}}_{\mathbf{2}}$with equal amplitudes, is the interference there fully constructive, fully destructive, intermediate but closer to fully destructive, intermediate but closer to fully destructive?

Short answer

Thus,

(a) the wavelength is $2.90\lambda$.

(b) the nature is intermediate.

Step-by-step solution

Multiple of λ gives phase difference.

(a)

Since $P_1$is equidistant from $S_1$and $S_2$conclude the sources are not in phase with each other. Their phase difference is $\Delta {\varphi }_{Source}=0.60\pi \text{rad}$, which may be expressed in terms of wavelengths as

$\begin{array}{rcl}\Delta {\varphi }_{Source}& =& \frac{0.60\pi }{2\pi }\lambda \\ & =& 0.3\lambda \end{array}$

Now, $S_1$is closer to $P_2$than $S_2$is source $S_1$is 80nm from $P_2$while source$S_2$ is 1380 nm from $P_2$. Here we find a difference of $\Delta {\varphi }_{Path}=3.2\lambda$. Thus, the net difference is:

$\begin{array}{rcl}\Delta {\varphi }_{Path}& =& Delta{\varphi }_{path}-\Delta {\varphi }_{Source}\\ & =& 2.90\lambda \end{array}$

Hence, the wavelength is $2.90\lambda$.

Wave is the interference.

(b)

A whole number would mean fully constructive so the result is of the following nature: intermediate, but close to fully constructive.

Hence, the nature is intermediate.