Apparent Depth and Shift in Layered Media

 

❓ Concept Question

How do we calculate the apparent shift of bottom when light passes through multiple liquid layers?

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🖼 Concept Image

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✍️ Short Concept

In multi-layer systems, each layer affects light separately.

Final apparent depth is obtained by adding contributions of each layer.

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🔷 Step 1 — Apparent Depth Rule 💯

When viewed from air:

$$\text{Apparent depth} = \frac{\text{Real depth}}{\mu}$$

👉 Higher refractive index ⇒ object appears closer

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🔷 Step 2 — Multiple Liquids = Layer-wise Effect

For multiple layers:

$$\text{Total apparent depth} = \frac{h_1}{\mu_1} + \frac{h_2}{\mu_2} + \cdots$$

⚠️ Never combine into a single refractive index

👉 Each layer acts independently

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🔷 Step 3 — Real Depth vs Apparent Depth

Total real depth:

$$h_{total} = h_1 + h_2$$

Shift:

$$\text{Shift} = \text{Real depth} - \text{Apparent depth}$$

👉 JEE usually gives net shift directly

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🔷 Step 4 — Ray Path Understanding

Light travels:

Bottom → lower liquid → upper liquid → air

👉 Every medium contributes to refraction

Effects add up, not cancel

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🔷 Step 5 — Unknown Height Trick

Common JEE setup:

• One layer height given
• Other layer unknown

Use:

$$\text{Shift equation}$$

👉 Solve directly for unknown height

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✅ Final Takeaway

For layered liquids:

$$\boxed{\text{Apparent depth} = \sum \frac{h}{\mu}}$$
$$\boxed{\text{Shift} = \text{Real} - \text{Apparent}}$$

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⭐ Golden JEE Insight

Biggest mistake:

❌ Using single μ for entire system

Correct approach:

👉 Treat each layer separately