A science museum shows light in a tall glass tank that holds two clear liquids which do not mix…

A science museum shows light in a tall glass tank that holds two clear liquids which do not mix. The bottom layer is sugar syrup, of refractive index $1.50$ and depth $15.0\ \text{cm}$ . Above it is a layer of water, of refractive index $1.33$ and depth $12.0\ \text{cm}$ . The figure shows the set-up, which is not drawn to scale. A tiny lamp L sits on the floor of the tank and sends light in all directions. A visitor looks straight down at the lamp from above the water. The glass walls and the thin floor can be ignored. Take the refractive index of air as $1.00$ .

Side view of a tall rectangular glass tank, not drawn to scale. The bottom layer is sugar syrup, shaded orange, with a depth of 15.0 cm and refractive index 1.50. Above it is a layer of water, shaded blue, with a depth of 12.0 cm and refractive index 1.33, and air above the water surface. A small lamp L sits at the middle of the floor. Three rays leave it, one straight up and two slanting, bending slightly at the join of the layers. An eye above the tank looks straight down.

How far below the water surface does the visitor think the lamp is?
Light leaves the water surface only if its angle of incidence there is smaller than a limit. What is this limit, the critical angle of the water and air boundary?
What angle does the limiting ray make with the vertical while it travels in the syrup?
The light that escapes forms a circular patch on the water surface, centred above the lamp. What is the radius of the patch?
The water is now poured out, leaving only the syrup. What is the radius of the circular patch on the syrup surface?

The worked solution

Have you tried it?

Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.

Plan. Treat each layer on its own, from the lamp up. The product $n \sin\theta$ is the same in every layer, and it equals $1.00$ at the top for a ray that just grazes the surface.

(a) Apparent depth. For a view straight down, the apparent depth is $\dfrac{h}{n}$ for each layer: $\dfrac{15.0\ \text{cm}}{1.50} + \dfrac{12.0\ \text{cm}}{1.33} = 10.00\ \text{cm} + 9.02\ \text{cm} = 19.02\ \text{cm}$ .

(b) Critical angle. At the water and air boundary, $\sin C = \dfrac{1.00}{1.33}$ , so $C = 48.8^\circ$ .

(c) Angle in the syrup. The limiting ray has $n \sin\theta = 1.00$ in the syrup too. So $\sin\theta_1 = \dfrac{1.00}{1.50}$ and $\theta_1 = 41.8^\circ$ . In the water it then makes $48.8^\circ$ , because $1.50 \sin 41.8^\circ = 1.33 \sin 48.8^\circ = 1.00$ .

(d) Radius of the patch. The ray moves sideways $h\tan\theta$ in each layer: $15.0\ \text{cm} \times \tan 41.8^\circ + 12.0\ \text{cm} \times \tan 48.8^\circ = 13.42\ \text{cm} + 13.69\ \text{cm} = 27.1\ \text{cm}$ .

(e) Syrup only. The first term alone: $15.0\ \text{cm} \times \tan 41.8^\circ = 13.4\ \text{cm}$ .

Answer: (a) $19.02\ \text{cm}$ . (b) $48.8^\circ$ . (c) $41.8^\circ$ . (d) $27.1\ \text{cm}$ . (e) $13.4\ \text{cm}$ .

Check. Send rays up from the lamp and increase their angle in the syrup. The widest one that still leaves the water is at $41.8^\circ$ , the angle found in (c), and it meets the surface $27.1\ \text{cm}$ from the axis.

Watch out. Using the water's critical angle for the whole depth gives $(15.0 + 12.0)\ \text{cm} \times \tan 48.8^\circ = 30.8\ \text{cm}$ , which is too large. The syrup bends the ray less. Adding the depths without dividing by $n$ gives $27.0\ \text{cm}$ in (a). The syrup and water join does not limit the beam, because $1.50 \sin 41.8^\circ = 1.00$ is smaller than the water's index $1.33$ .

← All Problems

← A glass bulb holds hydrogen gas. The atoms are excited to the level n = 4 and fall back to lower…A student examines a slide of pond algae with a compound microscope… →

More onOptics › Refraction of Light

Discussion (0)

Loading…

A science museum shows light in a tall glass tank that holds two clear liquids which do not mix…

The worked solution

Discussion (0)

Want to go further in Optics › Refraction of Light?