The Timpson Lyceum · University Physics II (2026) – Electricity & Magnetism

Chapter 33 — The Nature & Propagation of Light

Problem sheet · 13 points · 3 extra credit

Chapter 33 — The Nature & Propagation of Light

Core: 13 points · Extra credit: 3 points. Show your reasoning; partial credit is given.

1. 1. (1 pt •) A beam of light travels through a transparent liquid whose index of refraction is $n=1.36$.

   (a) Find the speed of the light in the liquid.

   (b) If the light has vacuum wavelength $589\,\text{nm}$, find its wavelength in the liquid. Does the frequency change?

2. 2. (2 pts ••) A ray of light in air strikes the flat surface of a block of crown glass ($n=1.52$) at an angle of incidence of $40^\circ$ measured from the normal.

   (a) Find the angle of refraction inside the glass.

   (b) The ray then travels through the block and hits the far face, which is parallel to the first. Find the angle at which it emerges back into the air, and state its direction relative to the original incoming ray.

3. 3. (2 pts ••) Light inside a block of glass ($n_1=1.52$) reaches a boundary.

   (a) Find the critical angle for total internal reflection if the other side of the boundary is air.

   (b) Find the critical angle if the other side is instead water ($n_2=1.33$). Explain physically why this critical angle is larger.

4. 4. (2 pts ••) An optical fiber has a glass core of index $n_\text{core}=1.48$ surrounded by cladding of index $n_\text{clad}=1.46$.

   (a) Find the critical angle at the core–cladding boundary.

   (b) A ray travels down the fiber and strikes the core–cladding wall at $82^\circ$ from the normal. Is it totally internally reflected? Justify with a comparison to part (a).

5. 5. (2 pts ••) A narrow beam of white light in air strikes the surface of a piece of glass at $60^\circ$ from the normal. The glass has index $n=1.514$ for red light and $n=1.528$ for violet light.

   (a) Find the angle of refraction for each color inside the glass.

   (b) Find the angular spread between the red and violet rays inside the glass. Which color bends more, and why?

6. 6. (2 pts ••) Unpolarized light of intensity $I_0$ passes through two ideal polarizing filters. The transmission axis of the second is rotated $30^\circ$ relative to the first.

   (a) Find the intensity transmitted through the first filter alone, in terms of $I_0$.

   (b) Find the intensity transmitted through both filters.

   (c) At what angle between the two axes would the final intensity be exactly $10\%$ of $I_0$?

7. 7. (2 pts ••) A beam of light in air reflects off the flat top surface of a pane of glass ($n=1.52$).

   (a) Find Brewster's angle — the angle of incidence at which the reflected light is completely polarized.

   (b) When light is incident at Brewster's angle, find the angle of refraction of the transmitted ray, and verify that the reflected and refracted rays are perpendicular to each other.

8. Extra Credit (3 pts •••)

   Three ideal polarizing filters are stacked. Unpolarized light of intensity $I_0$ enters the first, whose transmission axis defines the reference direction $0^\circ$. The third filter's axis is fixed at $90^\circ$ (crossed with the first). The middle filter's axis is at an adjustable angle $\theta$ (with $0^\circ<\theta<90^\circ$).

   (a) Using Malus's law at each stage, derive an expression for the final transmitted intensity $I(\theta)$ in terms of $I_0$ and $\theta$.

   (b) Note that with the middle filter removed, the crossed first-and-third pair transmits nothing. Explain in one or two sentences how inserting the middle filter lets light through.

   (c) Use calculus to find the angle $\theta$ that maximizes $I(\theta)$, and compute the maximum transmitted intensity as a fraction of $I_0$.