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Chapter 37: Problem 2899

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) \((2 \mathrm{n} / \mathrm{h})\)

Short Answer

Expert verified
The short answer to the given question is: \[\text{Apparent distance between P and its image} = \frac{2h}{n}\]

Step by step solution

01

Determine the path of light rays from P to the mirror and then to O

We have to determine the path of light rays to calculate the total apparent distance between object P and its image. When light rays travel from the object P, they reflect off the mirror at the same angle as the angle of incidence. Then the light rays travel through the liquid medium with refractive index n to reach the observer O.
02

Apparent depth in the liquid

Since the observer is looking through the liquid medium, the apparent depth of the object P due to the refractive index n will be: Apparent depth = h / n
03

Apparent distance between object P and its image in the mirror

Since we know that the depth of the object is h, and we have calculated the apparent depth of object P above the mirror, we can now determine the total apparent distance between object P and its image formed in the mirror. In plane mirrors, the object and its image have an equal distance from the mirror. Therefore, we can simply double the apparent depth of object P to get the apparent distance between object P and its image: Apparent distance between P and its image = 2 × (Apparent depth) Apparent distance = 2 × (h / n)
04

Select the correct answer

Now, let's compare our result to the options given in the problem: (A) \((2 \mathrm{~h} / \mathrm{n})\) (B) \(2 \mathrm{~h}(\mathrm{n}-1)\) (C) \(h[1+(1 / n)]\) (D) \((2 \mathrm{n} / \mathrm{h})\) Our result matches option (A): Apparent distance between P and its image = \((2 \mathrm{~h} / \mathrm{n})\) Hence, the correct answer is option (A).

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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 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) \(\mathrm{h}_{1}+\mathrm{h}_{2}\)

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
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