Refraction of Light from Plane Surface, Critical Angle & Total Internal Reflection

Q 1 :

Critical angle of incidence for a pair of optical media is 45°. The refractive indices of first and second media are in the ratio: [2024]

$1 : \sqrt{2}$
$1 : 2$
$\sqrt{2} : 1$
$2 : 1$

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(3)

We know

$\sin c = \frac{n_{2}}{n_{1}} \Rightarrow \frac{n_{1}}{n_{2}} = \frac{1}{\sin c}$

$\Rightarrow \frac{n_{1}}{n_{2}} = \frac{1}{\sin 45} = \frac{\sqrt{2}}{1}$

$n_{1} : n_{2} = \sqrt{2} : 1$

Q 2 :

A light ray is incident on a glass slab of thickness $4 \sqrt{3}$ cm and refractive index $\sqrt{2}$ . The angle of incidence is equal to the critical angle for the glass slab with air. The lateral displacement of ray after passing through glass slab is ____ cm.

(Given sin $15^{o}$ = 0.25) [2024]

(2)

$i = θ_{c} \Rightarrow i = \sin^{- 1} (\frac{1}{μ})$

$\Rightarrow i = 45 °$

According to Snell’s law,

$1 \sin 45 ° = \sqrt{2} \sin r \Rightarrow \frac{1}{\sqrt{2}} = \sqrt{2} \sin r \Rightarrow \frac{1}{\sqrt{2}} = \sqrt{2} \sin r$

$\Rightarrow r = 30 °$

Now, $Δ x = t \sec 30 ° \sin 15 ° = 4 \sqrt{3} \times \frac{2}{\sqrt{3}} \times \frac{1}{4} = 2 cm$

Q 3 :

Two immiscible liquids of refractive indices $\frac{8}{5}$ and $\frac{3}{2}$ respectively are put in a beaker as shown in the figure. The height of each column is 6 cm. A coin is placed at the bottom of the beaker. For near normal vision, the apparent depth of the coin is $\frac{α}{4}$ cm. The value of $α$ is ____. [2024]

(31)

$h_{app} = \frac{h_{1}}{μ_{1}} + \frac{h_{2}}{μ_{2}} = \frac{6}{8 / 5} + \frac{6}{3 / 2} = 4 + \frac{15}{4} = \frac{31}{4} cm$

Q 4 :

What is the lateral shift of a ray refracted through a parallel sided glass slab of thickness 'h' in terms of the angle of incidence 'i' and angle of refraction 'r', if the glass slab is placed in air medium? [2025]

$\frac{h \tan (i - r)}{\tan r}$
$\frac{h \cos (i - r)}{\sin r}$
h
$\frac{h \sin (i - r)}{\cos r}$

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(4)

AB = h sec r

BC = h sec r sin (i – r)

BC = $\frac{h \sin (i - r)}{\cos r}$

Q 5 :

A hemispherical vessel is completely filled with a liquid of refractive index $μ$ . A small coin is kept at the lowest point (O) of the vessel as shown in figure. The minimum value of the refractive index of the liquid so that a person can see the coin from point E (at the level of the vessel) is ________. [2025]

$\sqrt{3}$
$\frac{3}{2}$
$\sqrt{2}$
$\frac{\sqrt{3}}{2}$

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(3)

For the rays from coin to reach the point E, the refracted rays must graze the surface, i.e., they must be incident at critical angle $θ_{c}$ inside the liquid.

$μ = \frac{1}{\sin θ_{c}}$

$μ$ is minimum when $θ_{c}$ is maximum.

Maximum value of $θ_{c} = 45 °$

$\Rightarrow μ$ has a minimum value of $\sqrt{2}$ .

Q 6 :

At the interface between two materials having refractive indices $n_{1}$ and $n_{2}$ , the critical angle for reflection of an em wave is $θ_{1 C}$ . The $n_{2}$ material is replaced by another material having refractive index $n_{3}$ , such that the critical angle at the interface between $n_{1}$ and $n_{3}$ material is $θ_{2 C}$ . If $n_{3} > n_{2} > n_{1}$ ; $\frac{n_{2}}{n_{3}} = \frac{2}{5}$ and $\sin θ_{2 C} - \sin θ_{1 C} = \frac{1}{2}$ , then $θ_{1 C}$ is [2025]

$\sin^{- 1} (\frac{1}{6 n_{1}})$
$\sin^{- 1} (\frac{2}{3 n_{1}})$
$\sin^{- 1} (\frac{5}{6 n_{1}})$
$\sin^{- 1} (\frac{1}{3 n_{1}})$

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(none)

$n_{2} \sin (θ_{1 C}) = n_{1} \Rightarrow s in (θ_{1 C}) = \frac{n_{1}}{n_{2}}$

$n_{3} \sin (θ_{2 C}) = n_{1} \Rightarrow s in (θ_{2 C}) = \frac{n_{1}}{n_{3}}$

Also, $n_{3} = \frac{5 n_{2}}{2} \Rightarrow s in (θ_{2 C}) = \frac{2 n_{1}}{5 n_{2}}$

$\Rightarrow \frac{n_{1}}{n_{2}} = \frac{5}{2} \cdot \sin (θ_{2 C}) \Rightarrow s in (θ_{2 C}) - \sin (θ_{1 C}) = \frac{1}{2}$

Given, $\frac{2 n_{1}}{5 n_{2}} - \frac{n_{1}}{n_{2}} = \frac{1}{2} \Rightarrow \frac{n_{1}}{n_{2}} (\frac{- 3}{5}) = \frac{1}{2}$

Coming out to be (–ve).

Q 7 :

A transparent block A having refractive index $μ$ = 1.25 is surrounded by another medium of refractive index $μ$ = 1.0 as shown in figure. A light ray is incident on the flat face of the block with incident angle $θ$ as shown in figure. What is the maximum value of $θ$ for which light suffers total internal reflection at top surface of the block? [2025]

$\tan^{- 1} (4 / 3)$
$\tan^{- 1} (3 / 4)$
$\sin^{- 1} (3 / 4)$
$\cos^{- 1} (3 / 4)$

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(3)

$r + θ_{C} = 90 ° \Rightarrow r = 90 ° - θ_{C}$

At $1^{st}$ refraction : $1 \times \sin θ = \frac{5}{4} \sin r$

$\Rightarrow \sin r = \frac{4}{5} \sin θ$ ... (i)

At point TIR:

$\frac{5}{4} \sin θ_{C} = 1 . \sin 90 ° \Rightarrow \frac{5}{4} \sin θ_{C} = 1$

$\sin (90 - r) = \frac{4}{5} \Rightarrow \cos r = \frac{4}{5}$

$\Rightarrow \sin r = \frac{3}{5}$ ... (ii)

From (i) and (ii), $θ = \sin^{- 1} (\frac{3}{4})$

Q 8 :

Two light beams fall on a transparent material block at point 1 and 2 with angle $θ_{1}$ and $θ_{2}$ , respectively, as shown in figure. After refraction, the beams intersect at point 3 which is exactly on the interface at other end of the block.

Given: the distance between 1 and 2, $d = 4 \sqrt{3}$ cm and $θ_{1} = θ_{2} = \cos^{- 1} (\frac{n_{2}}{2 n_{1}})$ , where refractive index of the block $n_{2}$ > refractive index of the outside medium $n_{1}$ , then the thickness of the block is __________ cm. [2025]

(6)

$n_{1} \sin (90 - θ_{1}) = n_{2} \sin r$

$n_{1} \times \frac{n_{2}}{2 n_{1}} = n_{2} \sin r$

$\sin r = \frac{1}{2} \Rightarrow r = 30 °$

$\tan r = (\frac{d / 2}{t})$

$t = \frac{d}{2 \tan r} = \frac{d \sqrt{3}}{2} = \frac{(4 \sqrt{3}) \sqrt{3}}{2} = 6 cm$

Q 9 :

A container contains a liquid with refractive index of 1.2 up to a height of 60 cm and another liquid having refractive index 1.6 is added to height H above first liquid. If viewed from above, the apparent shift in the position of bottom of container is 40 cm. The value of H is __________ cm. (Consider liquids are immisible) [2025]

(80)

Shift $△ t = 40 cm$

$△ t = 60 (1 - \frac{1}{1.2}) + H (1 - \frac{1}{1.6})$

$\Rightarrow 40 = (\frac{1}{6}) 60 + H (\frac{3}{8})$

$\Rightarrow 30 = H \times \frac{3}{8} \Rightarrow H = 80 cm$

Q 10 :

A monochromatic light wave with wavelength $λ_{1}$ and frequency $ν_{1}$ in air enters another medium. If the angle of incidence and angle of refraction at the interface are $45 °$ and $30 °$ respectively, then the wavelength $λ_{2}$ and frequency $ν_{2}$ of the refracted wave are [2023]

$λ_{2} = \frac{1}{\sqrt{2}} λ_{1}, ν_{2} = ν_{1}$
$λ_{2} = λ_{1}, ν_{2} = \frac{1}{\sqrt{2}} ν_{1}$
$λ_{2} = \sqrt{2} λ_{1}, ν_{2} = ν_{1}$
$λ_{2} = λ_{1}, ν_{2} = \sqrt{2} ν_{1}$

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(1)

Snell's law $μ_{1} \sin 45 ° = μ_{2} \sin 30 °$

$\frac{μ_{1}}{μ_{2}} = \frac{1}{\sqrt{2}}$

$\Rightarrow \frac{μ_{1}}{μ_{2}} = \frac{λ_{2}}{λ_{1}} = \frac{1}{\sqrt{2}}$

$\Rightarrow λ_{2} = \frac{λ_{1}}{\sqrt{2}}$

Frequency doesn't change on change in medium.

Q 11 :

The critical angle for a denser-rarer interface is $45 °$ . The speed of light in rarer medium is $3 \times 10^{8} ms$ . The speed of light in the denser medium is [2023]

$5 \times 10^{7} m/s$
$2.12 \times 10^{8} m/s$
$3.12 \times 10^{7} m/s$
$\sqrt{2} \times 10^{8} m/s$

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(2)

$i_{C}$ = Critical angle

$\frac{v}{C} = \frac{1}{μ} = \sin i_{C} = \sin 45 ° = \frac{1}{\sqrt{2}}$

$\Rightarrow v = \frac{C}{\sqrt{2}} = \frac{3 \times 10^{8}}{\sqrt{2}} m/s = 2.12 \times 10^{8} m/s$

Q 12 :

An ice cube has a bubble inside. When viewed from one side the apparent distance of the bubble is 12 cm. When viewed from the opposite side, the apparent distance of the bubble is observed as 4 cm. If the side of the ice cube is 24 cm, the refractive index of the ice cube is [2023]

$\frac{4}{3}$
$\frac{3}{2}$
$\frac{2}{3}$
$\frac{6}{5}$

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(2)

$d_{apparent} = \frac{d_{actual}}{μ_{rel}}$

$12 = \frac{x}{μ} \dots (i)$

$d = \frac{24 - x}{μ} \dots (i i)$

$On solving we get μ = 1.5$

Q 13 :

A vessel of depth $' d'$ is half filled with oil of refractive index $n_{1}$ and the other half is filled with water of refractive index $n_{2}$ . The apparent depth of this vessel when viewed from above will be [2023]

$\frac{d n_{1} n_{2}}{(n_{1} + n_{2})}$
$\frac{d (n_{1} + n_{2})}{2 n_{1} n_{2}}$
$\frac{d n_{1} n_{2}}{2 (n_{1} + n_{2})}$
$\frac{2 d (n_{1} + n_{2})}{n_{1} n_{2}}$

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(2)

Formula used $d_{app} = \frac{d_{1}}{n_{1}} + \frac{d_{2}}{n_{2}}$

$d_{app} = \frac{d}{2} [\frac{n_{1} + n_{2}}{n_{1} n_{2}}]$

Q 14 :

A ray of light is incident from air on a glass plate having thickness $\sqrt{3} cm$ and refractive index $\sqrt{2}$ . The angle of incidence of a ray is equal to the critical angle for glass-air interface. The lateral displacement of the ray when it passes through the plate is _______ $\times 10^{- 2} cm$ (given $\sin 15 ° = 0.26$ ) [2023]

(52)

$\sin i = \sqrt{2} \sin r$

$\sin r = \frac{1}{2} \Rightarrow r = 30 °$

$μ = \frac{1}{\sin c}$

$c = 45 °$

$Lateral displacement, x = \frac{t \sin (i - r)}{\cos r}$

$x = \frac{\sqrt{3} \sin (45 - 30)}{\cos 30 °}$

$= \frac{\sqrt{3} \times \sin 15 °}{(\frac{\sqrt{3}}{2})} = 2 \times 0.26 = 0.52 cm$

$= 52 \times 10^{- 2} cm$

Q 15 :

A pole is vertically submerged in swimming pool, such that it gives a length of shadow 2.15 m within water when sunlight is incident at an angle of $30 °$ with the surface of water. If swimming pool is filled to a height of 1.5 m then the height of the pole above the water surface in centimeters is ( $n_{w} = 4 / 3$ ) _______. [2023]

(50)

$By Snell's law$

$1 \sin 60 ° = \frac{4}{3} \sin r \Rightarrow \sin r = \frac{3 \sqrt{3}}{8} \Rightarrow \tan r = \frac{3 \sqrt{3}}{\sqrt{37}}$

$By the diagram$

$x \sqrt{3} + 1.5 \tan r = 2.15$

$\approx 0.50 meter \Rightarrow x = 50 cm$

Q 16 :

A fish rising vertically upward with a uniform velocity of $8 {ms}^{- 1}$ observes that a bird is diving vertically downward towards the fish with the velocity of $12 {ms}^{- 1}$ . If the refractive index of water is $\frac{4}{3}$ , then the actual velocity of the diving bird to pick the fish, will be _____ ${ms}^{- 1}$ . [2023]

(3)

$\frac{V_{b / f}}{(\frac{4}{3})} = \frac{- 8}{(\frac{4}{3})} + \frac{(- v)}{1} \Rightarrow \frac{- 12}{(\frac{4}{3})} = \frac{- 8}{(\frac{4}{3})} + \frac{(- v)}{1}$

$\Rightarrow v = 3 m/s$

Q 17 :

Consider light travelling from a medium A to medium B separated by a plane interface. If the light undergoes total internal reflection during its travel from medium A to B and the speed of light in medium A and B are $2.4 \times 10^{8} m/s$ and $2.7 \times 10^{8} m/s$ respectively, then the value of critical angle is: [2026]

$\tan^{- 1} (\frac{8}{\sqrt{17}})$
$\cot^{- 1} (\frac{3}{\sqrt{13}})$
$\sin^{- 1} (\frac{9}{8})$
$\cos^{- 1} (\frac{8}{9})$

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(1)

$μ_{A} \sin c = μ_{B} \sin 90$

$\Rightarrow \sin c = \frac{μ_{B}}{μ_{A}} = \frac{v_{A}}{v_{B}}$

$∴$ $\sin c = \frac{2.4 \times 10^{8}}{2.7 \times 10^{8}} = \frac{8}{9}$

$\Rightarrow \tan c = \frac{8}{\sqrt{81 - 64}} = \frac{8}{\sqrt{17}}$

$c = \tan^{- 1} (\frac{8}{\sqrt{17}})$

Q 18 :

The wavelength of light, while it is passing through water is 540 nm. The refractive index of water is $\frac{4}{3}$ . The wavelength of the same light when it is passing through a transparent medium having refractive index of $\frac{3}{2}$ is ______ nm. [2026]

840
380
480
540

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(3)

$\begin{matrix} \frac{μ_{1}}{μ_{2}} - \frac{v_{2}}{v_{1}} = \frac{λ_{2}}{λ_{1}} | \end{matrix}$ $, v = f λ$

$\frac{μ_{1}}{μ_{2}} = \frac{λ_{2}}{λ_{1}}$

$\frac{μ_{B}}{3 / 2} = \frac{λ}{540}$

$λ = (\frac{4 \times 2 \times 540}{3 \times 3})$

$λ = 480 nm$

Topic Question Set

Q 1 :

Critical angle of incidence for a pair of optical media is 45°. The refractive indices of first and second media are in the ratio: [2024]

Q 4 :

What is the lateral shift of a ray refracted through a parallel sided glass slab of thickness 'h' in terms of the angle of incidence 'i' and angle of refraction 'r', if the glass slab is placed in air medium? [2025]

Q 11 :

The critical angle for a denser-rarer interface is $45 °$ . The speed of light in rarer medium is $3 \times 10^{8} ms$ . The speed of light in the denser medium is [2023]

Q 12 :

An ice cube has a bubble inside. When viewed from one side the apparent distance of the bubble is 12 cm. When viewed from the opposite side, the apparent distance of the bubble is observed as 4 cm. If the side of the ice cube is 24 cm, the refractive index of the ice cube is [2023]

Q 13 :

A vessel of depth $' d'$ is half filled with oil of refractive index $n_{1}$ and the other half is filled with water of refractive index $n_{2}$ . The apparent depth of this vessel when viewed from above will be [2023]

Q 18 :

The wavelength of light, while it is passing through water is 540 nm. The refractive index of water is $\frac{4}{3}$ . The wavelength of the same light when it is passing through a transparent medium having refractive index of $\frac{3}{2}$ is ______ nm. [2026]

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Topic Question Set

Q 1 : Critical angle of incidence for a pair of optical media is 45°. The refractive indices of first and second media are in the ratio: [2024]

Q 2 : A light ray is incident on a glass slab of thickness 43 cm and refractive index 2. The angle of incidence is equal to the critical angle for the glass slab with air. The lateral displacement of ray after passing through glass slab is ____ cm. (Given sin15o = 0.25) [2024]

Q 3 : Two immiscible liquids of refractive indices 85 and 32 respectively are put in a beaker as shown in the figure. The height of each column is 6 cm. A coin is placed at the bottom of the beaker. For near normal vision, the apparent depth of the coin is α4cm. The value of α is ____. [2024]

Q 4 : What is the lateral shift of a ray refracted through a parallel sided glass slab of thickness 'h' in terms of the angle of incidence 'i' and angle of refraction 'r', if the glass slab is placed in air medium? [2025]

Q 10 : A monochromatic light wave with wavelength λ1 and frequency ν1 in air enters another medium. If the angle of incidence and angle of refraction at the interface are 45° and 30° respectively, then the wavelength λ2 and frequency ν2 of the refracted wave are [2023]

Q 11 : The critical angle for a denser-rarer interface is 45°. The speed of light in rarer medium is 3×108 ms. The speed of light in the denser medium is [2023]

Q 12 : An ice cube has a bubble inside. When viewed from one side the apparent distance of the bubble is 12 cm. When viewed from the opposite side, the apparent distance of the bubble is observed as 4 cm. If the side of the ice cube is 24 cm, the refractive index of the ice cube is [2023]

Q 13 : A vessel of depth 'd' is half filled with oil of refractive index n1 and the other half is filled with water of refractive index n2. The apparent depth of this vessel when viewed from above will be [2023]

Q 16 : A fish rising vertically upward with a uniform velocity of 8 ms-1 observes that a bird is diving vertically downward towards the fish with the velocity of 12 ms-1. If the refractive index of water is 43, then the actual velocity of the diving bird to pick the fish, will be _____ ms-1. [2023]

Q 18 : The wavelength of light, while it is passing through water is 540 nm. The refractive index of water is 43. The wavelength of the same light when it is passing through a transparent medium having refractive index of 32 is ______ nm. [2026]

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Q 1 :

Critical angle of incidence for a pair of optical media is 45°. The refractive indices of first and second media are in the ratio: [2024]

Q 4 :

What is the lateral shift of a ray refracted through a parallel sided glass slab of thickness 'h' in terms of the angle of incidence 'i' and angle of refraction 'r', if the glass slab is placed in air medium? [2025]

Q 11 :

The critical angle for a denser-rarer interface is $45 °$ . The speed of light in rarer medium is $3 \times 10^{8} ms$ . The speed of light in the denser medium is [2023]

Q 12 :

An ice cube has a bubble inside. When viewed from one side the apparent distance of the bubble is 12 cm. When viewed from the opposite side, the apparent distance of the bubble is observed as 4 cm. If the side of the ice cube is 24 cm, the refractive index of the ice cube is [2023]

Q 13 :

A vessel of depth $' d'$ is half filled with oil of refractive index $n_{1}$ and the other half is filled with water of refractive index $n_{2}$ . The apparent depth of this vessel when viewed from above will be [2023]

Q 18 :

The wavelength of light, while it is passing through water is 540 nm. The refractive index of water is $\frac{4}{3}$ . The wavelength of the same light when it is passing through a transparent medium having refractive index of $\frac{3}{2}$ is ______ nm. [2026]