Question

Figure 35-24a gives intensity $\mathit{l}$versus position $\mathit{x}$ on the viewing screen for the central portion of a two-slit interference pattern. The other parts of the figure give phasor diagrams for the electric field components of the waves arriving at the screen from the two slits (as in Fig. 35-13a).Which numbered points on the screen bestcorrespond to which phasor diagram?

(a) Figure 1

(b) Figure 2

(c) Figure 3

(d) Figure 4

Short answer

a) The phase difference in Figure 2 corresponds to points 3 and 5 on Figure 1

b) The phase difference in Figure 3 corresponds to points 1 and 4 on Figure 1.

c) The phase difference in Figure 4 corresponds to point 2 on Figure 1

Step-by-step solution

Given data:

Phase differences between two waves for three different cases are provided in the given phasor diagrams

Relation between phase difference and the interference pattern:

The phase difference between interfering waves for the formation of maxima

$\mathbf{∆}\mathit{\varphi }\mathbf{=}\mathbf{2}\mathit{n}\mathbf{\pi }\mathbf{}\mathbf{}\mathbf{n}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{2}\mathbf{,}\mathbf{3}\mathbf{.}\mathbf{.}\mathbf{.}$ .....(1)

Phase difference between interfering waves for the formation of minima

$\mathbf{∆}\mathit{\varphi }\mathbf{=}\left(2n+1\right)\mathbf{\pi }\mathbf{}\mathbf{}\mathbf{n}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{2}\mathbf{,}\mathbf{3}\mathbf{.}\mathbf{.}\mathbf{.}$ .....(2)

(a) Determining the interference point for the first phase difference:

The phase difference in the first figure 2 can be represented by

$∆\varphi =\left(2n+1\right)\mathrm{\pi },\mathrm{n}=0,1,2,3...$

This represents minima in the interference pattern and thus can correspond to points 3 and 5 on figure 1.

(b) Determining the interference point for the second phase difference:

The phase difference in the first figure 2 can be represented by

$∆\varphi =2n\mathrm{\pi }\mathrm{n}=0,1,2..$

This represents maxima in the interference pattern and thus can correspond to points 1 and 4 in figure 1.

(c) Determining the interference point for the second phase difference:

The phase difference in the first figure 2 can be represented by

$∆\varphi =n\frac{\mathrm{\pi }}{2},n=1,5,9...$

This represents points between maxima and minima in the interference pattern and thus can correspond to point 2 on figure 1.