Apparent DepthFoundation

Concept

Rays leaving an underwater object bend away from the normal at the surface. Your brain traces them straight back, placing the image above the object — the pool floor looks raised by a factor $n_2/n_1$. Looking from air into water: apparent = real × (1/1.33) ≈ ¾ of the true depth.

Key formula

Derivation

For near-normal viewing, Snell's law gives $n_1\sin\theta_1 = n_2\sin\theta_2 \approx n_1\theta_1 = n_2\theta_2$.

The ray from depth d crosses the surface at $x = d\tan\theta_1 \approx d\theta_1$; traced back it appears to come from $d' = x/\theta_2 = d\,n_2/n_1$. Small-angle only — oblique viewing distorts further.

Scenarios to explore

• Apparent Depth — Why pools look shallow — normal shift & slab refraction.

Real-world applications

• Spear-fishing correction: aim BELOW the fish you see.
• A glass slab shifts any object toward you by t(1 − 1/n).
• Gem cutters exploit diamond's huge n — facets look shallow & bright.

JEE exam tips

• Stacked layers: d_app = Σ (tᵢ/nᵢ) as seen from air.
• Slab normal shift = t(1 − 1/n), independent of object distance.
• Bird–fish pair: fish sees bird HIGHER by ×n; bird sees fish SHALLOWER by ÷n.

Common mistakes

• Inverting the ratio (multiplying by n instead of dividing when looking into water).
• Applying the formula for oblique viewing — it's a near-normal approximation.
• Sign confusion for the reverse case: from water, objects in air look FARTHER (×n).

Exam traps to avoid

• The image shifts toward the observer in BOTH directions of viewing — always toward the rarer-medium side? No: toward the surface when viewed from the rarer medium, away when viewed from the denser one.
• A coin under a glass slab rises by the slab shift even if there is air above the coin.