Chapter 35 — Interference

14 points · 3 extra credit

Chapter 35 — Interference

Core: 14 points · Extra credit: 3 points. Show your reasoning; partial credit is given.

1. 1. (1 pt •) Two loudspeakers driven by the same amplifier emit identical waves of wavelength $\lambda=480\,\text{nm}$ (treat as coherent light sources for this problem). At a certain point the waves from the two sources have traveled path lengths that differ by $\Delta r = 1.44\,\mu\text{m}$.

   Is the interference at that point constructive or destructive? Justify with the ratio $\Delta r/\lambda$.

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2. 2. (2 pts ••) In a Young's double-slit experiment the slits are separated by $d=0.20\,\text{mm}$ and the screen is $L=1.8\,\text{m}$ away. Monochromatic light of wavelength $\lambda=550\,\text{nm}$ is used.

   (a) Find the spacing between adjacent bright fringes on the screen.

   (b) How far from the central maximum is the tenth-order ($m=10$) bright fringe?

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3. 3. (2 pts ••) Coherent light of wavelength $\lambda=633\,\text{nm}$ passes through two slits separated by $d=0.12\,\text{mm}$.

   (a) At what angle $\theta$ (measured from the central axis) does the second-order bright fringe appear?

   (b) If the slit separation were instead $d=2.5\,\mu\text{m}$, how many bright fringes in total would appear on a distant screen? (Count all orders that satisfy $|\sin\theta|\le 1$.)

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4. 4. (2 pts ••) Two identical coherent sources illuminate a screen; at the center of the pattern (zero path difference) the combined intensity is $I_0$. At a point $P$ their waves arrive with a path difference $\Delta r$, giving a combined intensity $I=I_0\cos^2(\phi/2)$, where $\phi$ is the phase difference.

   (a) Find the phase difference $\phi$ (in radians and degrees) when $\Delta r = 0.10\,\lambda$, and evaluate $I/I_0$ there.

   (b) For what smallest nonzero path difference (as a fraction of $\lambda$) does the intensity fall to one-quarter of its central-maximum value?

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5. 5. (2 pts ••) A soap film ($n=1.33$) of uniform thickness $t=320\,\text{nm}$ is surrounded by air and illuminated at normal incidence with white light.

   (a) Explain why light reflected from the front surface undergoes a half-cycle ($\pi$) phase shift while light reflected from the back surface does not.

   (b) Which visible wavelength ($400$–$700\,\text{nm}$) is most strongly reflected? Identify its color.

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6. 6. (2 pts ••) A lens is to be coated with a single layer of magnesium fluoride ($n=1.38$) to minimize reflection of $\lambda=550\,\text{nm}$ light at normal incidence. The lens glass has $n=1.52$.

   (a) Both reflections (air→coating and coating→glass) occur at a surface of higher index, so each reflected ray gets a $\pi$ phase shift. Explain whether these two shifts add or cancel, and use that to write the correct condition on $2nt$ for destructive interference (a dark, non-reflecting result) of the reflected light.

   (b) Find the minimum (nonzero) coating thickness.

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7. 7. (3 pts •••) In a Newton's-rings apparatus a plano-convex lens of radius of curvature $R=10.0\,\text{m}$ rests on a flat glass plate, and the gap is illuminated from above with sodium light, $\lambda=589\,\text{nm}$. Viewed in reflection, a pattern of concentric dark rings appears.

   (a) By considering the two reflected rays from the top and bottom of the air gap, show that the dark rings occur where the gap thickness satisfies $2t=m\lambda$ ($m=0,1,2,\dots$), and hence that the radius of the $m$-th dark ring is $r_m=\sqrt{m\lambda R}$. (Use $t \approx r^2/2R$ for the thin gap.)

   (b) Compute the radius of the fifth dark ring ($m=5$).

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8. Extra Credit (3 pts •••)

   Two slits act as identical coherent sources. Each alone produces at a point $P$ on a distant screen a field of amplitude $E_0$, i.e. $E_1(t)=E_0\cos\omega t$ and $E_2(t)=E_0\cos(\omega t+\phi)$, where $\phi$ is the phase difference from the path-length difference between the two slits and $P$.

   (a) By adding the two waves — either with the trigonometric sum-to-product identity or by adding phasors (vectors of length $E_0$ separated by angle $\phi$) — show that the resultant field is a single cosine of amplitude

   $$E_P = 2E_0\cos\!\left(\frac{\phi}{2}\right).$$

   (b) Intensity is proportional to the square of the field amplitude. Using the fact that a single source alone gives $I_0 \propto E_0^2$, derive

   $$I = I_0\cos^2\!\left(\frac{\phi}{2}\right),$$

   and identify the value of $I$ at the maxima in terms of $I_0$. Explain in one sentence why this is not a violation of energy conservation.

   (c) Relate $\phi$ to the path difference $\Delta r$ and, for small angles in a double-slit geometry ($\Delta r = d\sin\theta \approx d\,y/L$), show that this intensity formula reproduces bright fringes at $y_m=m\lambda L/d$.

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