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Q7. Fig. Q7 depicts a beaker containing a block of glass on the bottom and wate...

Sep 25, 2024

Solution by Steps

step 1

To find the true depth of the air bubble, we will use Snell's law, which relates the angles of incidence and refraction to the refractive indices of the two media. The formula is given by:

n_1 \sin(\theta_1) = n_2 \sin(\theta_2)

, where

n_1

is the refractive index of water,

n_2

is the refractive index of glass,

\theta_1

is the angle of incidence, and

\theta_2

is the angle of refraction

step 2

Given that the refractive index of water

n_1 = 1.33

, the refractive index of glass

n_2 = 1.6

, and the angle of incidence

\theta_1 = 60^{\circ}

, we can find

\theta_2

using Snell's law:

1.33 \sin(60^{\circ}) = 1.6 \sin(\theta_2)

step 3

Calculating

\sin(60^{\circ}) = \frac{\sqrt{3}}{2}

, we substitute this value into the equation:

1.33 \cdot \frac{\sqrt{3}}{2} = 1.6 \sin(\theta_2)

. This simplifies to

\sin(\theta_2) = \frac{1.33 \cdot \frac{\sqrt{3}}{2}}{1.6}

step 4

Solving for

\sin(\theta_2)

gives us

\sin(\theta_2) \approx 0.574

. We can find

\theta_2

by taking the inverse sine:

\theta_2 \approx \arcsin(0.574) \approx 35.1^{\circ}

step 5

Now, we can find the true depth of the air bubble using the apparent depth and the relationship between the angles and depths. The apparent depth

d_a = 6.81 \text{ cm}

and the true depth

d_t

can be related by the formula:

d_t = d_a \cdot \frac{n_1}{n_2}

. Substituting the values gives us:

d_t = 6.81 \cdot \frac{1.33}{1.6}

step 6

Calculating this gives us the true depth:

d_t \approx 6.81 \cdot 0.83125 \approx 5.66 \text{ cm}

Answer

The true depth of the air bubble is approximately 5.66 cm.

Key Concept

Snell's law relates the angles of incidence and refraction to the refractive indices of different media.

Explanation

The true depth of the air bubble is calculated by considering the apparent depth and the refractive indices of water and glass, which affects how light travels through these materials.