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Chapter 37: Problem 2902

A bird in air looks at a fish vertically below it and inside water, \(h_{1}\) is the height of the bird above the surface of water and \(\mathrm{h}_{2}\), the depth of the fish below the surface of water. If refractive index of water with respect to air be \(\mathrm{n}\), then what is the distance of the fish observed by the bird? (A) \(\mathrm{n}_{1} \mathrm{~h}_{1}+\mathrm{nh}_{2}\) (B) \(\mathrm{nh}_{1}+\mathrm{h}_{2}\) (C) \(\mathrm{h}_{1}+\left(\mathrm{h}_{2} / \mathrm{n}\right)\) (D) \(\mathrm{h}_{1}+\mathrm{h}_{2}\)

Short Answer

Expert verified
The short answer is: \(h_1 + (h_2 / n)\).

Step by step solution

01

Visualize the problem

Draw a diagram showing the bird, the fish, and the water surface. Mark the heights (\(h_1\)) and depth (\(h_2\)), as well as the refractive index (n).
02

Use Snell's law to find the apparent depth

Snell's law states that \(n_1 \sin(\theta_1) = n_2 \sin(\theta_2)\), where \(n_1\) and \(n_2\) are the refractive indices of two media, and \(\theta_1\) and \(\theta_2\) are the angles of incidence and refraction, respectively. In our problem, \(n_1 = 1\) (air), \(n_2 = n\) (water), and \(\theta_2 = 90^\circ\) (since the bird observes the fish vertically). So we can rewrite Snell's law as: \(1 \cdot \sin(\theta_1) = n \cdot \sin(90^\circ) = n\) We can find the apparent depth of the fish below the water surface by dividing the actual depth by the refractive index: \(h_{2(apparent)} = \frac{h_2}{n}\)
03

Add the bird's height above the water to the apparent depth of the fish

To obtain the total distance the fish is observed by the bird, we add the apparent depth of the fish to the bird's height above the water: \(Distance = h_1 + h_{2(apparent)}\) Substituting the expression for \(h_{2(apparent)}\) from Step 2: \(Distance = h_1 + \frac{h_2}{n}\) So, the correct answer is (C) \(h_1 + (h_2 / n)\).

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Most popular questions from this chapter

The velocity of light in glass whose refractive index with respect to air is \(1.5\) is \(2 \times 10^{8} \mathrm{~m} / \mathrm{s}\). In a certain liquid, the velocity of light is found to be \(2.5 \times 10^{8} \mathrm{~m} / \mathrm{s}\). What is the refractive index of the liquid with respect to air? (A) \(1.44\) (B) \(0.80\) (C) \(1.20\) (D) \(0.64\)

A vessel of depth \(\mathrm{t}\) is half filled with oil of refractive index \(\mathrm{n}_{1}\) and the other half is filled with water (refractive index \(\mathrm{n}_{2}\) ). The apparent depth of the vessel when viewed from above is \(\ldots \ldots\) (A) $\left[\left\\{2 t\left(n_{1}-\mathrm{n}_{2}\right)\right\\} /\left(\mathrm{n}_{1} \mathrm{n}_{2}\right)\right]$ (B) $\left[\left\\{2 \mathrm{t}\left(\mathrm{n}_{1}+\mathrm{n}_{2}\right)\right\\} /\left(\mathrm{n}_{1} \mathrm{n}_{2}\right)\right]$ (C) $\left[\left\\{t\left(n_{1}-n_{2}\right)\right\\} /\left(2 n_{1} \mathrm{n}_{2}\right)\right]$ (D) $\left[\left\\{t\left(n_{1}+n_{2}\right)\right\\} /\left(2 n_{1} \mathrm{n}_{2}\right)\right]$

Light travels through a glass plate of thickness \(t\) and having refractive index \(\mathrm{n}\). If \(\mathrm{C}\) be the velocity of light in vacuum. What is the time taken by the light of travel this thickness of glass? (A) \(\left(\mathrm{t}_{\mathrm{C}} / \mathrm{n}\right)\) (B) \(\operatorname{tn} \mathrm{C}\) (C) (nt / C) (D) \((\mathrm{t} / \mathrm{n} \mathrm{C})\)

A beam of light is converging towards a point I on a screen. A plane parallel plate of glass whose thickness is in the direction of beam \(=\mathrm{t}\), refractive index \(=\mathrm{n}\) is introduced in the path of the beam. The convergence point is shifted by \(\ldots \ldots\) (A) \(t[1+(1 / n)]\) never (B) \(t[1-(1 / n)]\) nearer (C) \(\mathrm{t}[1+(1 / \mathrm{n})]\) away (D) \(t[1-(1 / n)]\) away

A plane mirror is placed at the bottom of a tank containing a liquid of refractive index \(\mathrm{n} . \mathrm{P}\) is a small object at a height \(\mathrm{h}\) above the mirror. An observer O, vertically above P, outside the liquid sees \(\mathrm{P}\) and its image in the mirror. The apparent distance between these two will be. (A) \((2 \mathrm{~h} / \mathrm{n})\) (B) \(2 \mathrm{~h}(\mathrm{n}-1)\) (C) \(h[1+(1 / n)]\) (D) \((2 \mathrm{n} / \mathrm{h})\)
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