Q.Three immiscible liquids of densities $d_1 > d_2 > d_3$ and refractive indices $\mu_1 > \mu_2 > \mu_3$ are put in a beaker. The height of each liquid column is $\dfrac{h}{3}$. A dot is made at the bottom of the beaker. For near normal vision, find the apparent depth of the dot.

Concept: Index Matching — when light passes through multiple layers, the apparent depth is the sum of each layer’s real depth divided by its refractive index.

Reasoning:

1. For near normal incidence, the apparent shift for a single layer of real depth $t$ and refractive index $\mu$ is $t / \mu$.
2. Since the layers are immiscible and stacked, the total apparent depth is the sum of the apparent depths of each layer. …

The key insight here is Index Matching — when light travels from a denser to a rarer medium (or vice versa), the apparent shift depends on the refractive index of the medium the light is leaving. For near-normal viewing from air above, each liquid layer acts like a "compressor" of the apparent depth beneath it.

Think of it this way: if you have a single liquid of refractive index $\mu$ and actual depth $h$, the apparent depth when viewed from air is $h/\mu$. Why? Because light bends away from the normal when exiting into air, making the bottom look shallower. Now, when you stack multiple liquids, the light passes through each interface sequentially. The total apparent depth is just the sum of the apparent thicknesses of each layer — because each layer's effect is independent when the viewing angle is near normal.

Let's work it out step by step.

1. Set up the problem.

   The beaker has three immiscible liquids. Densities decrease upward ($d_1 > d_2 > d_3$), so the densest liquid ($d_1$) sits at the bottom, then $d_2$, then $d_3$ on top. The refractive indices also decrease upward ($\mu_1 > \mu_2 > \mu_3$). Each liquid column has actual height $h/3$. A dot is at the very bottom of the beaker. We view from above, near normal incidence.

2. Understand the apparent depth for a single layer.

   For a single medium of refractive index $\mu$ and actual depth $H$, the apparent depth when viewed from air (refractive index 1) is $H/\mu$. This comes from Snell's law for small angles: $n_1 \sin\theta_1 = n_2 \sin\theta_2$ reduces to $n_1 \theta_1 \approx n_2 \theta_2$, and geometry gives $\text{apparent depth} = \frac{\text{actual depth}}{\mu}$.

3. Extend to multiple layers — the principle of additivity.

   When light from the dot passes through the bottom liquid (index $\mu_1$), it first encounters the interface with the middle liquid ($\mu_2$). For an observer in the middle liquid, the bottom of the beaker appears at depth $(h/3)/\mu_1$ as measured from that interface. But we're not stopping there — the light then enters the middle liquid, then the top liquid, then air.

   The neat trick: treat each layer as if it were the only one, and add the apparent thicknesses. Why does this work? Because at each interface, the apparent depth of everything below gets "compressed" by the ratio of the refractive indices. For near-normal rays, the total apparent depth from the top is:

where $h_1, h_2, h_3$ are the actual heights of the layers from bottom to top. …

Method: Apparent Depth Through a Stack of Immiscible Media

Use this method whenever an object is viewed (near-normally) through two or more layers of different transparent media stacked on top of it, and you need the total apparent depth or shift.

Steps

Step 1: Recall the single-layer rule

For one medium of refractive index $\mu$ and real thickness $t$, viewed from air along a near-normal direction, the apparent depth contributed by that layer alone is

This comes from applying Snell's law at small angles ($n_1\theta_1 \approx n_2\theta_2$) and the resulting shift-toward-the-observer geometry.

Step 2: Recognise that the layers add independently

For near-normal viewing, each interface only compresses the apparent distance to everything below it by the ratio of the indices on either side. Because this compression is local to each layer, the total apparent depth as seen from above is simply the sum of each layer's own apparent-depth contribution — you never need to explicitly track the ray angle changing layer to layer.

Step 3: Sum the contributions, bottom to top …