Interference and Wave Nature of Light — Hard Practice Quiz

A Physics cheat sheet for Interference and Wave Nature of Light — every key formula with its symbols defined — plus a hard-level practice quiz to test recall.

Practice quiz

1. A double-slit experiment with slit separation $d$ produces a first-order bright fringe at angle $\theta_{DS}$. A single-slit experiment with slit width $W$ produces a first-order dark fringe at angle $\theta_{SS}$. If $d = 2W$ and both experiments use the same wavelength of light, what is the ratio $\frac{\sin \theta_{DS}}{\sin \theta_{SS}}$?

   • $\frac{1}{2}$
   • $1$
   • $2$
   • $\frac{1}{4}$

   Answer: $\frac{1}{2}$

2. An optical instrument with an aperture of diameter $D$ just resolves two distant point sources emitting light of wavelength $\lambda$. If this light then passes through a double-slit apparatus with slit separation $d$, and the angular position of the first-order bright fringe (relative to the central maximum) is equal to the minimum resolvable angle of the instrument, what is the relationship between $D$ and $d$? Assume small angles.

   • $D = 1.22d$
   • $d = 1.22D$
   • $D = \frac{d}{1.22}$
   • $D = d$

   Answer: $D = 1.22d$

3. A diffraction grating produces a second-order principal maximum for light of wavelength $\lambda_1$ at an angle $\theta$. If the same grating is used with a different wavelength $\lambda_2$ to produce a third-order principal maximum at the *same* angle $\theta$, what is the relationship between $\lambda_1$ and $\lambda_2$?

   • $\lambda_2 = \frac{2}{3}\lambda_1$
   • $\lambda_2 = \frac{3}{2}\lambda_1$
   • $\lambda_2 = \frac{1}{2}\lambda_1$
   • $\lambda_2 = \frac{1}{3}\lambda_1$

   Answer: $\lambda_2 = \frac{2}{3}\lambda_1$

4. Consider a double-slit experiment, a single-slit diffraction experiment, and an optical instrument's resolving power. Which of the following statements is TRUE regarding the effect of changing parameters?

   • Increasing the slit separation $d$ in a double-slit experiment increases the angular separation between bright fringes.
   • Decreasing the slit width $W$ in a single-slit experiment decreases the angular width of the central maximum.
   • Increasing the wavelength $\lambda$ of light used in a diffraction grating experiment decreases the angular separation between principal maxima.
   • Increasing the aperture diameter $D$ of an optical instrument improves its resolving power, meaning it can resolve smaller angular separations.

   Answer: Increasing the aperture diameter $D$ of an optical instrument improves its resolving power, meaning it can resolve smaller angular separations.

5. In a double-slit experiment where each slit has a finite width $W$, the overall interference pattern is modulated by a single-slit diffraction envelope. If the first-order bright fringe of the double-slit interference pattern coincides with the first-order dark fringe of the single-slit diffraction pattern, what is the relationship between the slit separation $d$ and the slit width $W$?

   • $d = W$
   • $d = 2W$
   • $W = 2d$
   • $d = \frac{W}{2}$

   Answer: $d = W$

6. An optical instrument with an aperture of diameter $D$ has a minimum resolvable angle $\theta_{min}$. If a single slit of width $W$ is illuminated by light of the same wavelength $\lambda$, and the angular width of its central diffraction maximum is equal to $\theta_{min}$, what is the approximate relationship between $W$ and $D$? Assume small angles.

   • $W \approx 1.64D$
   • $W \approx 0.61D$
   • $W \approx 2.44D$
   • $W \approx 0.82D$

   Answer: $W \approx 1.64D$

7. In a double-slit experiment, the second-order bright fringe occurs at an angle $\theta_{DS}$. In a single-slit diffraction experiment, the first-order dark fringe occurs at an angle $\theta_{SS}$. If the slit separation $d$ for the double slit is three times the slit width $W$ for the single slit ($d = 3W$), and both experiments use the same wavelength of light, what is the ratio $\frac{\sin \theta_{DS}}{\sin \theta_{SS}}$?

   • $\frac{2}{3}$
   • $\frac{3}{2}$
   • $6$
   • $\frac{1}{6}$

   Answer: $\frac{2}{3}$

8. A double-slit experiment with slit separation $d_{DS}$ produces its first-order dark fringe ($m=0$) at an angle $\theta$. A diffraction grating with slit separation $d_G = 2d_{DS}$ produces its second-order principal maximum ($m=2$) at the same angle $\theta$. If the double-slit experiment uses light of wavelength $\lambda_1$ and the diffraction grating uses light of wavelength $\lambda_2$, what is the ratio $\frac{\lambda_1}{\lambda_2}$?

   • $2$
   • $\frac{1}{2}$
   • $4$
   • $\frac{1}{4}$

   Answer: $2$

9. A double-slit experiment is initially performed in air, producing a first-order bright fringe at an angle $\theta_{air}$. The entire apparatus is then immersed in a transparent liquid with a refractive index $n > 1$. What is the new angular position $\theta_{liquid}$ of the first-order bright fringe?

   • $\sin \theta_{liquid} = n \sin \theta_{air}$
   • $\sin \theta_{liquid} = \frac{1}{n} \sin \theta_{air}$
   • $\theta_{liquid} = \theta_{air}$
   • $\sin \theta_{liquid} = n^2 \sin \theta_{air}$

   Answer: $\sin \theta_{liquid} = \frac{1}{n} \sin \theta_{air}$

10. An optical instrument with an aperture of diameter $D$ can just resolve two point sources emitting light of wavelength $\lambda_1$, with a minimum angular separation $\theta_{min,1}$. If the sources instead emit light of wavelength $\lambda_2 = 2\lambda_1$, the new minimum resolvable angle is $\theta_{min,2}$. Separately, if a diffraction grating is used to observe the first-order principal maxima for $\lambda_1$ and $\lambda_2$, resulting in angles $\theta_1$ and $\theta_2$ respectively, which of the following statements is TRUE?

    • $\theta_{min,2} = 2\theta_{min,1}$ and $\sin \theta_2 = 2\sin \theta_1$
    • $\theta_{min,2} = \frac{1}{2}\theta_{min,1}$ and $\sin \theta_2 = 2\sin \theta_1$
    • $\theta_{min,2} = 2\theta_{min,1}$ and $\sin \theta_2 = \frac{1}{2}\sin \theta_1$
    • $\theta_{min,2} = \theta_{min,1}$ and $\sin \theta_2 = \sin \theta_1$

    Answer: $\theta_{min,2} = 2\theta_{min,1}$ and $\sin \theta_2 = 2\sin \theta_1$