24P3Q09

9ai)

When two (or more) waves overlap, the resultant displacement (at any point and instant) is the (vector) sum of the displacements due to each individual wave (at that point and instant).

9aii)

Light waves with constant phase difference(s).

9bi)

1.5λ

COMMENT: The first minima occurs where path difference is 0.5λ. The second minima occurs where path difference is 1.5λ.

9bii)

The path difference at R is 1.5λ, which is an odd number of half wavelengths.

The light waves from A and B therefore superpose in anti-phase at R, resulting in a destructive interference.

9biii)

\displaystyle \begin{aligned}\Delta y&=\frac{L\lambda }{d} 2.9\times {{10}^{-3}}&=\frac{(2.70)\lambda }{0.60\times {{10}^{-3}}} \lambda &=6.44\times {{10}^{-7}}\text{ m}\end{aligned}

9biv)

Since the double slits are also two single slits, the resulting interference pattern is a combination of both double-slit and a single-slit interference pattern.

As a single slit, a central maximum would have been formed, with intensity peaking at the centre, and decreases along either sides.

As a result, the intensity of the bright fringes formed by double-slit interference will also peak at the centre, and decreases along either sides.

COMMENT: The intensity of the double-slit interference fringes is modulated by the single-slit diffraction envelope.

c)

\displaystyle \mathrm{I}\propto {{A}^{2}}

Light from slit A has intensity I and amplitude A.

Light from slit B has intensity \displaystyle \frac{\mathrm{I}}{2} and amplitude \displaystyle \frac{A}{\sqrt{2}}.

Amplitude at P has amplitude \displaystyle A+\frac{A}{\sqrt{2}}=1.7071A.

Intensity at P is thus \displaystyle {{1.7071}^{2}}\mathrm{I}=2.91\mathrm{I}

di)

\displaystyle \tan \theta =\frac{1.12}{2.70}\text{ }\Rightarrow \text{ }\theta =22.53{}^\circ

\displaystyle \begin{aligned}d\sin \theta &=n\lambda  (1.67\times {{10}^{-6}})\sin 22.53{}^\circ &=(1)\lambda  \lambda &=6.40\times {{10}^{-7}}\text{ m}\end{aligned}

dii)

Assuming 2.9 mm and 1.12 m are both measured using the same instrument with the same resolution, the percentage uncertainty of 2.9 mm is much higher than that of 1.12m.

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