Optics & Waves — CheriMathLab

⚡ Key Concepts & Laws

Snell's Law

$$n_1 \sin\theta_1 = n_2 \sin\theta_2$$ Governs refraction at the interface between two media with different refractive indices.

Wave Equation

$$v = f\lambda$$ The speed of a wave equals frequency times wavelength — fundamental to all wave phenomena.

Interference

Constructive: path difference $= m\lambda$. Destructive: path difference $= \left(m + \tfrac{1}{2}\right)\lambda$. Produces bright and dark fringes.

Diffraction

Single slit minima: $$a \sin\theta = m\lambda$$ Light bends around obstacles, revealing its wave nature.

Thin Lens Equation

$$\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$$ Relates focal length to object and image distances for converging and diverging lenses.

Doppler Effect

$$f' = f\,\frac{v \pm v_o}{v \mp v_s}$$ Frequency shifts when source or observer moves relative to the medium.

Standing Waves

Formed by superposition of two waves traveling in opposite directions. Nodes are fixed; antinodes oscillate maximally.

Polarization

Malus's Law: $$I = I_0 \cos^2\theta$$ Only transverse waves can be polarized, proving light's transverse nature.

✎ Problems

1 Easy

A light ray passes from air ($n = 1.00$) into glass ($n = 1.52$) at an angle of incidence of $35°$. Calculate the angle of refraction and determine the speed of light inside the glass.

Show Hint

Apply Snell's Law: $n_1 \sin\theta_1 = n_2 \sin\theta_2$. For the speed, use $v = \dfrac{c}{n}$ where $c = 3 \times 10^8$ m/s.

2 Easy

A guitar string of length $0.65$ m vibrates in its third harmonic. If the wave speed on the string is $320$ m/s, find the frequency and wavelength of the standing wave.

Show Hint

For the $n$th harmonic on a fixed-fixed string: $L = \dfrac{n\lambda}{2}$. So $\lambda = \dfrac{2L}{n}$, then $f = \dfrac{v}{\lambda}$.

3 Medium

In a Young's double-slit experiment, the slit separation is $0.25$ mm and the screen is $1.5$ m away. If the third bright fringe is observed $10.8$ mm from the central maximum, determine the wavelength of the light used.

Show Hint

For bright fringes: $y_m = \dfrac{m\lambda D}{d}$. Solve for $\lambda = \dfrac{y_m \, d}{m D}$. Remember to convert all units to meters.

4 Medium

An ambulance siren emits sound at $750$ Hz. The ambulance approaches you at $30$ m/s and you are walking toward it at $1.5$ m/s. What frequency do you hear? (Speed of sound $= 343$ m/s.)

Show Hint

Use $f' = f \times \dfrac{v + v_o}{v - v_s}$ where $v_o$ is observer speed toward source and $v_s$ is source speed toward observer. Both motions increase the perceived frequency.

5 Medium

A converging lens with focal length $15$ cm forms an image of an object placed $25$ cm from the lens. Find the image distance, magnification, and describe the image (real/virtual, upright/inverted, enlarged/reduced).

Show Hint

Use $\dfrac{1}{f} = \dfrac{1}{d_o} + \dfrac{1}{d_i}$, then magnification $M = -\dfrac{d_i}{d_o}$. A positive $d_i$ means real image; negative $M$ means inverted.

6 Medium

Unpolarized light of intensity $I_0$ passes through three polarizers. The first is vertical, the second is at $30°$ to the first, and the third is at $60°$ to the first ($30°$ to the second). What fraction of the original intensity passes through all three?

Show Hint

After the first polarizer: $I_1 = \dfrac{I_0}{2}$. Apply Malus's Law at each subsequent polarizer: $I = I_{\text{prev}}\cos^2\theta$ where $\theta$ is the angle between consecutive polarizers.

7 Hard

A thin film of oil ($n = 1.40$) on water ($n = 1.33$) appears green ($\lambda = 520$ nm) when viewed at near-normal incidence. What is the minimum thickness of the film for constructive interference?

Show Hint

Since $n_{\text{oil}} > n_{\text{water}}$, there's a phase change at the top surface but not the bottom. For constructive interference: $2nt = \left(m + \tfrac{1}{2}\right)\lambda$. Use $m = 0$ for minimum thickness.

8 Hard

A single slit of width $0.10$ mm is illuminated by $633$ nm laser light. A screen is placed $2.0$ m away. Calculate the width of the central maximum and the position of the second dark fringe.

Show Hint

Dark fringes: $a \sin\theta = m\lambda$. For small angles, $y = \dfrac{m\lambda D}{a}$. The central maximum width is $\dfrac{2\lambda D}{a}$ (from first minimum on each side).

9 Hard

Thought Experiment: A pipe open at both ends has a fundamental frequency of $440$ Hz (concert A). If you close one end, what is the new fundamental frequency? Explain physically why the pitch changes and describe the harmonic series for each case.

Show Hint

Open pipe: $f = \dfrac{v}{2L}$, all harmonics. Closed pipe: $f = \dfrac{v}{4L}$, only odd harmonics. Closing one end doubles the effective wavelength of the fundamental.

10 Hard

Two speakers separated by $3.0$ m emit sound in phase at $680$ Hz. A listener stands $4.0$ m in front of one speaker, perpendicular to the line connecting them. Does the listener experience constructive or destructive interference? ($v_{\text{sound}} = 340$ m/s.)

Show Hint

Calculate the distance from the listener to each speaker using the Pythagorean theorem. Find the path difference, then compare to the wavelength $\lambda = \dfrac{v}{f}$. Integer multiples of $\lambda$ give constructive interference.

11 Advanced

Derive the condition for total internal reflection and calculate the critical angle for a diamond ($n = 2.42$) in air. Then explain why diamonds are cut with specific facet angles and how this relates to their brilliance and fire.

Show Hint

Total internal reflection occurs when the refracted angle would exceed $90°$. Set $\sin\theta_c = \dfrac{n_2}{n_1}$ (with $n_1 > n_2$). Diamond's high refractive index means a small critical angle — light bounces many times internally.

12 Advanced

Estimation Problem: The Hubble Space Telescope has a mirror diameter of $2.4$ m and observes at $500$ nm. Estimate its angular resolution using the Rayleigh criterion. Could it resolve two headlights of a car $2$ m apart on the Moon ($384{,}400$ km away)?

Show Hint

Rayleigh criterion: $\theta = \dfrac{1.22\,\lambda}{D}$. Compare this angular resolution to the angle subtended by the headlights: $\theta_{\text{car}} = \dfrac{\text{separation}}{\text{distance}}$. Convert both to the same units (radians).